Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters to the

editor should be sent to either of the

editors-in-chief, Chandler Davis or

Marjorie Senechal.

Euler Diagrams and Venn Diagrams In his review of my book Cogwheels of the Mind, Hamburger [1] incorrectly says, 'The famous three-circle Venn di­agram, which is known to most peo­ple, had already been used by Euler.' It does not occur in either Euler's orig­inal Lettres a une Princesse d'Allemagne of 1768 or Hunter's English translation of 1795, the two works cited by Ham­burger. It should have done, but it didn't.

Euler simply made a mistake in drawing his diagrams for three sets. When he added a third set C to a two­set diagram with overlapping sets A and B so as to overlap A partially, he drew the cases C wholly in B and C wholly outside B, but muddled the case C par­tially in B, which would have given him a Venn diagram for three sets.

An 1823 reprint of the erroneous di­agram can be seen at math.dartmouth. edu/� euler/, E343 Plate 3 figure 27, which is the same as figure 26. Venn owned a copy of this book; I have ex­amined it and he did not mark the er-

ror. Recent editors of Euler's work have corrected it.

The relationships between Euler di­agrams and Venn diagrams were well understood by Venn himself and are, I believe, correctly described in my book.

Euler did draw a Venn diagram for two sets, but then, so did writers in the eleventh century [2] .

In other respects it is best to draw a veil over Hamburger's tirade against my "dabblings."

REFFERENCES

[ 1 ] P. Hamburger, "Cogwheels ofthe Mind. The

Story of Venn Diagrams by A.W.F. Ed­

wards" (review), The Mathematical lntelli­

gencer 27 (2005), 36-38.

[2] C. Nolan, Music theory and mathematics, in

T. Christensen (ed.) The Cambridge History

of Western Music Theory, Cambridge Uni­

veristy Press (2002), 272-304.

A.W.F. Edwards

Gonville and Caius College

Cambridge CB2 1 TA, U.K.

e-mail: awfe@cam.ac.uk

ERRATUM

Thanks to Gregory Kriegsmann of New Jersey Institute of Technology for spotting an error in the article "e: The Master of All" by Brian J. McCartin (Math Intelligencer 28 (2006), no. 1, 10-21). In equation (5), the denominator reads (2k)!; it should be (2k+1)!.

The author acknowledges the typographical error, and reassures us that the erroneous formula was not used in arriving at Figure 2. Just so! If it had been, the figure would have made the claims of great accuracy for the formula (5) looked odd indeed.

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 3

An Elementary Proof of the Gregory­Mengoli-Mercator Formula

In this note we present a novel, elementary proof of "the remarkable formula

1 1 1 1 - - + - - - + - = log 2 2 3 4

. . . '

(1)

one of the relations whose discovery made a deep impression on the earliest pi­oneers of differential and integral calculus" (Richard Courant [C)). The formula goes back at least to Pietro Mengoli (1626-1686) [Men] , James Gregory (1638-1675) [G], and Nicolaus Mercator (1620-1687) [Mer] ; it provided an unexpected link between the antique world view, with its well-ordered Pythagorean natural numbers, and the emerging seventeenth-century culture of mathematics as a tool for exploring the material world, with its transcendental concepts of logarithms, infinitesimals, and the number e.

1

In modem times, the left-hand side of (1) is likely to have been first encountered by readers of this note in an undergraduate course on calculus or analysis, in the early chapters on sequences and series: it provides a classic example of a series which is convergent but not absolutely convergent. We are taught why the limit ex­ists, usually in connection with Leibniz's more general criterion that an alternating series whose terms decrease monotonically to zero in absolute value is convergent.

But what is the limit? The textbooks inform us that it will later in the course be shown to equal log 2 , using differential/integral calculus. This is somewhat discouraging: neither left-hand side nor right-hand side appears to require cal­culus. So why can't one establish equality directly?

The answer is that one can. Our proof below requires only the definition of the natural logarithm as the logarithm to base e, i .e . , the inverse of the expo­nential to base e,

elog X= X, (2)

and Euler's famous formula well known from compound-interest calculations that

e = lim (1 + _!_ )n n�(l; n

(one of several standard definitions of e).

(3)

We do not require tools from calculus, unlike the usual derivations. One of the latter starts from the power series representation of the logarithm, obtained for instance by termwise integration of a geometric series,

Lx dt x2 x3 x4 log(l + x) = -- = x - - + - - -+ -

o 1+t 2 3 4 (4)

An additional argument such as Abel's theorem [A] is then employed to justify equality of left-hand side and right-hand side at the point x = 1 , at which the underlying geometric series fails to converge. Alternatively one uses Taylor's the­orem (see, e .g . , [R]) to obtain the series, and a careful remainder analysis to jus­tify its validity at x = 1 .

We thank R . Burckel for pointing out another calculus-based treatment:

One shows that the quantity DN = I 1/ n - JN(l! t)dt, which compares a Rie-n=I 1

mann sum to an integral, converges (the limit being known as Euler's con-stant) . Using this, the first equation in (4), and the identity (6) below, we 2N have I (-l)n+ljn- log(2N) + log(N) = DzN- DN � 0, which together

n=l with the addition rule log(2N) - log N = log 2 proves ( 1 ) . See [K].

1Which was already known empirically at the time from the so-called quadrature problem for the hyperbola,

as the base which makes the first equation in (4) below valid.

4 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Business Media. Inc.

We proceed to give our own derivation of the following

THEOREM 1

1 1 The alternating harmonic series 1 - - + --2 3 -+-4

converges to log 2.

PROOF 1. Preliminaries. We need to show that the partial sums SN := I;Y=1 (-1)"+1/n converge to log 2. Existence of a limit follows, e.g., from Leibniz's criterion, as discussed above. We denote the limit by S. By the definition of the natural log­arithm, ( 2), and by continuity of the exponential function x� ex, 2 it suffices to show that

lim eS,v = 2. (5) N-->oo

2. Simplifying the partial sums. We note that the alternat­ing harmonic series from 1 to 2N is equal to the non­alternating harmonic series from N + 1 to 2N, I c -l)n+l

= I l_. n=l n n=N+l n (6)

This is because the left-hand side is a sum of positive odd fractions and negative even fractions, and hence equal to the sum of all fractions minus twice the sum of the even fractions, I�0\ lin - 2I;Y=1 l/2n. 3. Using the law of exponents. It suffices to consider, in (5),

partial sums from 1 to even integers 2N, because SzN+l -1 S2N = -- converges to zero. By (6) and the law of ex-2N+l ponents ea+h = eaeh, the left-hand side of (5) becomes

2N e52lv = IT e� .

n=N+l (7)

4. Using a different approximation for each factor. Now the key idea is to use a different approximation for each of the above factors, in such a way that the exponent 2_ always cancels. If N is large, e = ( 1 + �)n for each n E {N + 1, . . . , 2Nl, and hence, heuristically, 2N 2N

( 1 )nlu

nll+lelln =

nll+l 1 +

--;; N+ 2 N+ 3 2N+ 1 =---·---· N+l N+2 2N

But the numerator of each factor cancels the denominator 2N+1 of the next, so the right-hand side equals--, which ap-N+1

proaches 2 as Nbecomes large. To make this rigorous, we use the well-known fact ( e.g., [H, Chapter 1.10)) that the sequences en : = ( 1 + �)n and En : = ( 1 + n� Jn, which both converge to e by ( 3), are increasing and decreasing, respectively. ( Proof that the first sequence is increasing:

1 + I en+l = (1 + l_) ( �)n+ l en n 1 +-n

= (1 + l_) (1 - 1 )n+l 2: (1 + 1_)(1 - -1 ) = 1, n (n + 1)2 n n + 1

by Bernoulli's inequality ( 1 + x)11 2: 1 + nx for x > -1 and natural numbers n. To show that the second sequence is

decreasing, one considers � and argues similarly. ) Con­En+! sequently we can estimate the right-hand side of (7) from above and below by

2N ( 1 )n 2N

1 2N ( 1 )!1 IT 1 + - n � IT en � IT 1 + n - 1 ".

n=N+l n n=N+l n=N+l The left product was already evaluated above, and the right . N+l N+2 2N . product gives--·--· . . . · -- = 2. This reduces the N N+l 2N-l inequality to

1 2----� N+1 2N IT 1 en� 2.

n=N+l As N approaches infinity, the term on the left converges to 2, and therefore so does the term in the middle. This es­

tablishes the theorem.

As the reader may have guessed, our motivation for devis­ing this proof arose in the context described in our intro­ductory comments. We were teaching the contemporary canon of convergence theory for sequences and series, to second-month mathematics undergraduates, and wanted to share relation ( 1) with our students.

But since our main arguments are explicit manipulations of finite sums and products, our proof should be elementary enough to be appreciated without a precise theory of limits.

In fact, it is the latter situation which resonates more closely with the context of discovery of formulae such as ( 1) by the early pioneers: underlying finite calculations, such as evalu­ation of what we nowadays call Riemann sums in quadrature problems, were based on firm mathematical concepts, and "passage to the limit" was performed intuitively.

REFERENCES

[A] Niels Henrik Abel, Recherches sur Ia serie 1 + '!'_ x + m(m+1l x2 + 1 1·2 m(m�.�(;+2l x3 + . . . , Crelles Journal 1, 3 1 1 -339, 1 827. Reprinted

in L. Sylow & S. Lie (eds), Oeuvres completes deN. H. Abel, Tome

1 , 2 1 9 -250, Grondell & Son, 1 881 .

[C] Richard Courant, Vorlesungen uber Differential- und lntegralrech­

nung. Erster Band. Springer 1 955

[G] James Gregory, Vera Circuli et Hyperbolae Ouadratura, in Propria

Sua Proportion is Specie, lnventa & Demonstrata, Padua, 1 667. [H] Stefan Hildebrandt, Analysis 1 , Springer, 2002

[K] Max Koecher, Klassiche elernentare Analysis , Birkhauser, 1 987.

[Men] Pietro Mengoli, Novae quadraturae arithmeticae, seu de addi-

tione fractionum. Bologna 1 650.

[Mer] Nicolaus Mercator, Logarithmotechnia, 1 668.

[R] Walter Rudin, Principles of Mathematical Analysis, 2nd edition,

McGraw-Hill, 1 964

Gero Friesecke

Zentrum Mathematik

Technische Universitat Munchen

D-85747 Garching b. Munchen Germany

Mathematics Institute University of Warwick

Coventry CV4 7AL, U.K.

e-mail gf@ma.tum.de

Jan Christoph Wehrstedt Zentrum Mathematik

Technische Universitat Munchen

D-85747 Garching b. Munchen Germany e-mail: jwehr@ma.tum.de

2or using the more elementary argument that the exponential function is increasing, and that SN < S < SN+J for N even, whence e5N:,; e5:,; e5N+1

© 2005 Spnnger Science+ Business Media, Inc., Volume 28, Number 3, 2005 5

A Heuristic for the Prime Number Theorem HUGH L. MONTGOMERY AND STAN WAGON

l \ lhy does e play such a central role in the distri­\ / ·. \ · · bution of prime numbers? Simply citing the Prime Number Theorem ( PNT), which asserts that

1T(x) � x/ln x, is not very illuminating. Here"�" means "is asymptotic to" and 1T(x) is the number of primes less than or equal to x. So why do natural logs appear, as opposed to another flavor of logarithm?

The problem with an attempt at a heuristic explanation is that the sieve of Eratosthenes does not behave as one might guess it would from pure probabilistic considerations. One might think that sieving out the composites under x using primes up to Vx would lead to x IIp<Yx(l- �) as an asymptotic estimate of the count of numbers rlmaining ( the primes up to x; p always represents a prime). But this quantity turns out to be not asymptotic to xlln x, for F. Mertens proved in 1874 that the product is actually asymp­totic to 2 e-'Y /In x, or about 1.1 2/ln x. Thus the sieve is 1 1 o/o ( from 1/1. 1 2) more efficient at eliminating composites than one might expect. Commenting on this phenomenon, which one might call the Mertens Paradox, Hardy and Wright [5, p. 37 2] said: "Considerations of this kind explain why the usual 'probability' arguments lead to the wrong asymptotic value for 1T(x)." For more on this theorem of Mertens and related results in prime counting, see [ 3; 5; 6, exercise 8.27; 8].

Yet there ought to be a way to explain, using only ele­mentary methods, why natural logarithms play a central role in the distribution of primes. A good starting place is two old theorems of Chebyshev ( 1849).

CHEBYSHEV'S FIRST THEOREM For any x 2': 2, 0.9 2 x/ln x < 1T(x) < 1.7 x/ln x.

CHEBYSHEV'S SECOND THEOREM .if 1T(x) -x/Iogc x, then c = e.

A complete proof of Chebyshev's First Theorem ( with slightly weaker constants) is not difficult, and the reader is encouraged to read the beautiful article by Don Zagier [9], the very first article published in this magazine ( see also [ 1, §4.1]). The first theorem tells us that x/logc x is a rea­sonable rough approximation to the growth of 1T(x), but it does not distinguish e from other bases.

The second theorem can be given a complete proof us­ing only elementary calculus [8]. The result is certainly a partial heuristic for the centrality of e because it shows that, if any logarithm works, then the base must be e. Further, one can see the exact place in the proof where e arises Cf 1/x dx =In x). But the hypothesis for the second the­orem is a strong one; here we will show, by a relatively simple proof, that the same conclusion follows from a much

weaker hypothesis. Of course, the PNT eliminates the need for any hypothesis at all, but its proof requires either an understanding of complex analysis or the willingness to read the sophisticated "elementary proof". The first such was found by Erdos and Selberg; a modern approach ap­pears in [7].

Our presentation here was inspired by a discussion in Courant and Robbins [ 2]. We show how their heuristic ap­proach can be transformed into a proof of a strong result. So, even though our original goal was just to motivate the PNT, we end up with a proved theorem that has a simple statement and quite a simple proof.

THEOREM .[fxhr(x) is asymptotic to an increasing/unction, then 1T(x) � x/ln x.

Figure 1 shows that x/1T( X) is assuredly not increasing. Yet it does appear to be asymptotic to the piecewise linear func­tion that is the upper part of the convex hull of the graph.

This work was supported in part by National Science Foundation grant DM8-0244660 to H. L. M.

6 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Scrence+Business Media, Inc.

5

4

3

2

300 400 500 Figure I. A graph of xhr(x) shows that the function is not purely increasing. The upper convex hull of the graph is an increasing piecewise linear function that is a good approxi­mation to x!'lT(X).

Indeed, if we take the convex hull of the full infinite graph, then the piecewise linear function L(x) corresponding to the part of the hull above the graph is increasing (see Con­clusion section) . If one could prove that x/1r(x) � L(x) then, by the theorem, the PNT would follow. In fact, using PNT it is not too hard to prove that L(x) is indeed asymptotic to x/1r(x) (such a proof is given at the end of this article) . In any case, the hypothesis of the theorem is certainly be­lievable, if not so easy to prove, and so the theorem serves as a heuristic explanation of the PNT.

Nowadays we can look quite far into the prime realm. Zagier's article of 29 years ago was called The First Fifty Million Prime Numbers. Now we can look at the first 700 quintillion prime numbers. Not one at a time, perhaps, but the exact value of 1r(lOi) is now known for i up to 22; the most recent value is due to Gourdon and Sabeh [4] and is 1T(4 · 1022) = 783 964 1 59 847 056 303 858. Fig­ure 2 is a log-log plot that shows the error when these stratospheric 1T values are compared to x/ln x and also the much better logarithmic integral estimate li(x) (which is fa 1/ln t dt) .

Two Lemmas The proof requires two lemmas. The first is a consequence of Chebyshev's First Theorem but can be given a short and elementary proof; it states that almost all numbers are com­posite.

LEMMA 1 limx--->oc 1T(X)/x = 0.

10-10

• . . , ..... • • • • • • • • • • • • • •• • • •

• •

• • • •

• • • •

• • •

• ••

10 105 1010 1015 10201022 Figure 2. The large dots are the absolute value of the error when x/ln x is used to approximate 'lT(x), for x = lQi The smaller dots use li(x) as the approximant.

PROOF. First use an idea of Chebyshev to get 1T(2n) -1r(n) < 2 nlin n for integers n. Take log-base-n of both sides of the following to get the needed inequality.

4 n = (1 + 1)2 n > (2n) 2: II p > II n = n1T(2n)-1T(n)

n n<po52n n<po52n

This means that 1T(2 n) - 1r(n):::; (In 4) n/ln n. Suppose n is a power of 2 , say 2k; then summing over 2 :::; k:::; K, where K is chosen so that 2K:::; n < 2K+l , gives

K 2k In 4 1r(n) :::; 2 + I -- .

k�2 k Here each term in the sum is at most 3/4 of the next term, so the entire sum is at most 4 times the last term. That is , 1r(n):::; c n/ln n, which implies 1T (n)/n� 0. For any pos­itive x > 0, we take n to be the first power of 2 past x, and then 1r(x)/x:::; 2 1T(n)/n, concluding the proof. D

By keeping careful track of the constants, the preceding proof can be used to show that 1r(x):::; 8 . 2 x!ln x, yield­ing one half of Chebyshev's first theorem, albeit with a weaker constant.

The second lemma is a type of Tauberian result, and the proof goes just slightly beyond elementary calculus. This lemma is where natural logs come up, well, naturally. For con­sider the hypothesis witii loge in place of ln. Then the con­stant In c will cancel, and so the conclusion will be unchanged!

LEMMA 2 If W(x) is decreasing and f3 W(t) in(t)/t dt �

In x, then W(x) � 1 /ln x.

······························································································································································································································································

HUGH L. MONTGOMERY studied at the Uni­versity of Illinois and the Universrl:y of Cambridge. He worl<s in analytic number theory, particularly

on distribution of prime numbers.

Department of Mathematics University of Michigan Ann Arbor, Ml 481 09 USA e-mail: hlm@umich.edu

STAN WAGON studied at McGill in Montreal and at Dartmouth. Much of his work currently is on using Mathematica to illustrate various concepts of mathematics, from the Banach-T arski Paradox to dynamical systems; see, for instance, his recent lnte/ligencer cover (vol. 27, no. 4). Another enthu­siasm of his which has been reported in this mag­azine is snow sculpture (see vol. 22, no. 4).

Department of Mathematics Macalester College, St. Paul, MN 55 I 04 USA e-mail: wagon@macalester.edu

© 2006 Spnnger Science+ Business Media, Inc., Volume 28, Number 3, 2006 1

PROOF. Let E be small and positive; let j (t) = ln(t)/t. The hypothesis implies g' +• W(t) j (t) dt - E In x ( to see this, split the integral into two: from 2 to x and x to x1 +€). Since W(x) is decreasing,

1+€ E In x- r W(t) j (t) dt

: W(x) rl+€ In t dt = E (1 + �) W(x) (In x)2. X f 2

Thus lim infx_,oo W(x) In x � 1/( 1 + �). A similar argument 2 starting with g�-, W(t) j(t) dt - E In x shows that lim supx_,oo W(x)ln x :S 1/(1 - 1) . Since E can be arbitrar­ily small, we have W(x)ln x-1. D

Proof of the Theorem

THEOREM If x/7T(X) is asymptotic to an increasing function, then 7T(x) -x/ln x.

PROOF Let L(x) be the hypothesized increasing function and let W(x) = 1/ L(x), a decreasing function. It suffices to show that the hypothesis of Lemma 2 holds, for then L(x) - In x. Letj(t) = ln(t)lt. Note that if ln x- g(x) + h(x) where h(x)/ln

x � 0, then In x - g(x); we will use this several times in the following sequence, which reaches the desired conclusion by a chain of 11 relations. The notation pk II n in the third ex­pression means that k is the largest power of p that divides n; the equality that follows the II sum comes from consider­ing each pm for 1 ::s m ::s k.

In x - _!_ I In n X n-:::;x (can be done by machine; note 1)

= _!_ I I k In p = _!_ I I In p X n�x pklln X n�x pmln,m""i -1 '\' X 1 '\' X

L In p [----;;;-]- - L In p-X pm:::;;x, J=::;;m P X pm-:::;x, l:Sm pm (error is small; note 2)

-I f(p) + I In!- I f(p)

p:5x pm:sx, 2:Sm p p:sx (geometric series estimation; note 3)

= 7TclxJ) j(x) - r 7T(t) j'(t) dt 2

_ _ r 7T(t) j'(t) dt 2

- - {x t W(t) ( __!_ - In t ) dt )2 t2 t2

- {x

W(t) In t dt )2 t

NOTES

(partial summation; note 4)

because 7T(lxJ) j(x)/ln x :5 ?T(x)/x-> 0 by Lemma 1)

(because 7T(t) - t W(t))

(l'H6pital, note 5) D

1. It is easy to verify this relation using standard integral test ideas: start with the fact that the sum lies between If In t dt and x In x. But it is intriguing to see that Mathe­matica can resolve this, using symbolic algebra. The sum is just lnClxJ!) and Mathematica quickly returns 1 when asked for the limit of x In x/ln(x!) as x � oo.

8 THE MATHEMATICAL INTELLIGENCER

error 1 1 2. -1 - :S -1- kpmsx In p :S -1- k�x logp X In p n x x n x x n x 1 = -1-- kp�x In x = 7T(x)!x� 0 by Lemma 1. x n x

3 . The second sum divided by In x approaches 0 because:

'\' In p '\'oo In n L�x p(p-1 ) :::; Ln�2 n(n-1 )

Ioo (n -1 )1/2 :::; <oo

n�z (n - 1 )2

4. We use partial summation, a technique common in an­alytic number theory. Write the integral from 2 to x as a sum of integrals over [n, n + 1] together with one from lxJ to x, and use the fact that 7T(t) is constant on such inter­vals and jumps by 1 exactly at the primes. More precisely:

r 7T(t) j'(t) dt 2

lx l J 1 fn+l = 7T(t) J' (t) dt + I x_- 7T(t) j'(t) dt

lxJ n-2 n

= 7T(lxJ) (j(x) - JClxJ) ) + I�x��1 7T(n)(j(n + 1) -j(n))

= 7TclxJ) j(x) - Ip�x j(p)

5. L'Hopital's rule on -1- Lx t W(t) � dt yields W(x)/ x

In x 2 t 1/x W(x) = 1/L(x), which approaches 0 by Lemma 1.

No line in the proof uses anything beyond elementary cal­culus except the call to Lemma 2 . The result shows that if there is any nice function that characterizes the growth of 7T(x), then that function must be asymptotic to x/ln x. Of course, the PNT shows that this function does indeed do the job.

This proof works with no change if base-c logarithms are used throughout. But as noted, Lemma 2 will force the natural log to appear! The reason for this lies in the indef­inite integration that takes places in the lemma's proof.

Conclusion Might there be a chance of proving in a simple way that

x/7T(x) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. Let L(x) be the upper convex hull of the full graph of x/7T(x) ( precise definition to follow ). The piecewise linear function L(x) is increasing because x/7T(X) � oo as

x � oo. Moreover, using PNT, we can give a proof that L(x) is indeed asymptotic to x/7T(x). But the point of our work in this article is that for someone who wishes to under­stand why the growth of primes is governed by natural log­arithms, a reasonable approach is to convince oneself via computation that the convex hull just mentioned satisfies the hypothesis of our theorem, and then use the relatively simple proof to show that this hypothesis rigorously im­plies the prime number theorem.

We conclude with the convex hull definition and proof. Let B be the graph of xhr(x) : the set of all points (x, xhr(x)) where x 2: 2. Let C be the convex hull of B: the intersec­tion of all convex sets containing B. The line segment from (2, 2) to any (x , xhr(x)) lies in C. As x � oo, the slope of this line segment tends to 0 (because 1T(X) � oo) . Hence for any positive a and E, the vertical line x = a contains points in C of the form (a, 2 + E) . Thus the intersection of the line x = a with C is a set of points (a, y) where 2 < y ::5 L(x). The function L(x) is piecewise linear and xhr(x) ::5 L(x) for all x. This function is what we call the upper convex hull of xhr(x).

THEOREM The upper convex hull of x/1T(x) is asymptotic to 1T(x) .

PRooF. For given positive E and ,X{), define a convex set A(.x{)) whose boundary consists of the positive x-axis, the line segment from (0, 0) to (0, ( 1 + E)(ln .x{))), the line seg­ment from that point to (e ,X{), (1 + E)(l + In .x{))), and fi­nally the curve (x, (1 + E) In x) for e x0 ::5 x. The slopes match at e ,X{), so this is indeed convex . The PNT implies that for any E > 0 there is an x1 such that, beyond x1, (1 - E) In x < x/1T(x) < (1 + E) In x. Now chose .x{) > x1 so that 1T(x1) < (1 + E) In x0 . It follows that A(.x{)) contains B: be­yond .x{) this is because x0 > x1; below x1 the straight part is high enough at x = 0 and only increases; and between x1 and .x{) this is because the curved part is convex down,

and so the straight part is above where the curved part would be, and that dominates 7T(x) by the choice of x1 . This means that the convex hull of the graph of x/1T(x) is contained in A(.x{)), because A(.x{)) is convex. That is, L(x) ::5 (1 + E) In x for x 2: x0. Indeed, (1 - E) In x ::5 x/7T(x) ::5 L(x) ::5 (1 + E) In x for all sufficiently large x . Hence x/1r(x) � L(x) . D

REFERENCES

1. D. Bressoud and S. Wagon, A Course in Computational Number

Theory, Key College, San Francisco, 2000.

2. R. Courant and H. Robbins, What is Mathematics?, Oxford Univer­

sity Press, London, 1941.

3. J. Friedlander, A. Granville, A. Hildebrand, and H. Maier, Oscillation

theorems for primes in arithmetic progressions and for sifting func­

tions, J. Amer. Math. Soc. 4 (1 991) 25-86.

4. X. Gourdon and P. Sebah. The 7T(X) project, http://numbers.

computation. free. fr /Constants/constants. html.

5 . G. H. Hardy and E. M. Wright, An Introduction to the Theory of Num­

bers, 4th ed. , Oxford University Press, London, 1 965.

6. I . Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction

to the Theory of Numbers, 2nd ed. , Wiley, New York, 1991.

7. G. Tenenbaum and M. Mendes France, The Prime Numbers and

their Distribution, Amer. Math. Soc. , Providence, R . I . , 2000.

8. S. Wagon, It's only natural, Math Horizons 13:1 (2005) 26-28.

9. D. Zagier, The first 50,000,000 prime numbers, The Mathematical

lntelligencer 0 (1977) 7-19.

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© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 9

it!Jfjjrf§ .. 6hl¥11§1§4£11h!•i§•id Michael Kleber and Ravi Vakil, Editors

TipOver Is N P-complete ROBERT A. HEARN

This column is a place for those bits

of contagious mathematics that travel

from person to person in the

community, because they are so

elegant, suprising, or appealing that

one has an urge to pass them on.

Contributions are most welcome.

Please send all submissions to the

Mathematical Entertainments Editor,

Ravl Vakil, Stanford University,

Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA

e-mail: vakil@math.stanford.edu

T ipOver™ is a popular puzzle in which the goal is to navigate a layout of vertical crates, tipping

some over to reach others, so as to eventually reach a target crate. Crates can only tip into empty space, and you can't jump over empty space to reach other crates. The challenge is to tip the crates in the right directions and the right order.

TipOver began life as an online puz­zle created by james Stephens, called the "The Kung Fu Packing Crate Maze" [7]. Now it also exists in physical form, produced by ThinkFun, the makers of Rush Hour and other multilevel chal­lenge puzzles. Like Rush Hour, TipOver comes with a board, a set of pieces, and 40 challenge cards, each with a differ­ent puzzle layout. TipOver was the Games Magazine puzzle of the year for 2005, and after playing it, it is easy to appreciate its simplicity, ingenuity, and challenge.

Just how hard a puzzle is TipOver? Is there, perhaps, a clever algorithm for coming up with a solution to a given puzzle instance? Probably not. Like

1 Q THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+Business Media. Inc.

Minesweeper [5] and some other puz­zles, TipOver is NP-complete. Nobody knows how to solve an NP-complete problem efficiently; it is the most fa­mous unsolved problem of computer science. If we are given a TipOver puz­zle on an n X n grid, we don't know how to do better than a brute-force search of all possible move sequences in order to solve the puzzle, and that takes time exponential in n.

I will show that TipOver is NP-com­plete by reducing the Boolean Formula Satisfiability problem (SAD to TipOver. For any given instance of SAT, there is a corresponding TipOver puzzle that can be solved just when the SAT prob­lem can be solved. Therefore, TipOver must be at least as hard as SAT, and SAT is the prototypical NP-complete problem. It's easy to show that TipOver is also no harder than SAT, so TipOver must be NP-complete as well.

The Puzzle In its starting configuration, a TipOver puzzle has several vertical crates of var­ious heights (1 X 1 X b) arranged on a

Figure 2. A sample TipOver puzzle and its solution.

grid, with a "tipper"-representing a person navigating the layout-standing on a particular starting crate. There is a unique red crate, 1 X 1 X 1 , elsewhere on the grid; the challenge is to move the tipper to this target red crate.

The tipper can tip over any vertical crate that it is standing on, in any of the four compass directions, provided that there is enough empty space for that crate to fall unobstructed and lie flat. The tipper is nimble enough to land safely on a newly fallen crate. The tip­per can also walk, or climb, along the tops of any crates that are adjacent, even when they have different heights. However, the tipper may not jump over empty space to reach another crate or jump to a diagonally neighboring crate; sides of the crates must be touching.

Surprisingly, it does not take many crates to make an interesting puzzle. The number of tips required can never be more than the number of crates­once a crate has been tipped over, it stays fallen-yet finding the correct se­quence can be quite a challenge. A sample puzzle and its solution are shown in Figure 2. The first layout is the initial configuration, with the tip­per's location marked with a red square outline, and the height of each vertical crate indicated. In each successive step, one more crate has been tipped over.

The Gadgets Solving puzzles is a great pleasure for many people. Trying to prove that a puzzle is hard is another way of deriv­ing enjoyment from puzzles-a meta­puzzle, if you will, akin to puzzle de­sign, but also a part of theoretical computer science. Typically, it involves the construction of "gadgets" which transform the puzzle into some kind of abstract computer [4). Sometimes the gadgets can be extremely complex, and great ingenuity is needed to come up with them. In the case of TipOver, the gadgets we will need are not so com­plicated.

What we need are gadgets that let us build a Boolean-formula-satisfiability tester out of a TipOver configuration. So, let's begin by defining the SAT prob­lem. A Boolean formula is a logical ex­pression composed of ANDS (/\), 0Rs (V), and literals. A literal is either a Boolean variable (x, y, z, w, . . . ) or its negation (:X, y, z, w, . . . ). Here is a sample Boolean formula:

((x V x V y) 1\ (y V z V w) 1\ w). Boolean variables must be assigned

to either true or false. If a variable x is true, then its negation :X is false, and vice-versa. Similarly, (P 1\ Q) is true if both P and Q are true, where P and Q are arbitrary Boolean formulas, and

(P V Q) is true if either P or Q is true. If we pick an assignment for all the variables in a Boolean formula, then the entire formula is true or false, depend­ing on the values we choose.

The SAT problem is this: given a Boolean formula, is there any assign­ment of truth values to its variables that will make the formula true? This prob­lem is NP-complete [2). If there is such an assignment, we say that that formula is satisfiable. The above formula is sat­isfied by (for example) setting x true, y false, z true, and w false.

Because Boolean formulas are made out of variables, ANos, and 0Rs, we'll need to make variable, AND, and OR gadgets. More fundamentally, we'll need wires to propagate signals and connect the gadgets. We'll also need a way to split signals, so that one vari­able can be used multiple times. Finally,

t Q

Figure 3. TipOver OR gadget. If the tip­per can reach either P or Q, then it can reach P V Q. All crates are height 2.

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 1 1

· · · I I w ITIJ···

ITIJ···

· · ·rra:.o I I I I -- ----

Figure 4. A wire that must be initially traversed left to right.

we must find a way to cross wires over each other, so we can wire variables to logic gates arbitrarily.

Once we have all the gadgets, we can use them to make a TipOver puz­zle corresponding to any given Boolean formula, so that the puzzle can be solved just when the formula is satisfi­able. The signals flowing from one gad­get to the next are represented in the puzzle by reachability: the idea is that the tipper will be able to reach certain regions only when corresponding subexpressions of the formula are true. To solve the puzzle, the tipper must use a satisfying assignment to set the vari­able gadgets appropriately, enabling it ultimately to reach the target crate.

Wiring is the first concern; fortu­nately this is easy. A wire will just be a line of vertical crates, placed adjacent to each other, so that the tipper can walk along the tops of them. Crates of height 2 will be sufficient here. In fact, we won't need any crates taller than 2

for the entire construction. Next, let's build an OR. What we

want is a configuration that can be ex­ited if it can be entered from either of

p . �L.� �w-+- -1-.1

two directions. All we need to do to arrange this is 'T' one wire into another (Fig. 3). If the tipper enters from the left or from the bottom, it can exit via the right. Now, there is one slight problem with this arrangement-the tipper can also enter along one of the inputs, and exit on the other input. This is a prop­erty we don't want our gadgets to have. Once signals start flowing down wires in the wrong direction, anything can happen.

We'll solve this problem by making a one-way gadget. This is a wire that can only be traversed in one direction, the first time it is traversed. After that, it can be traversed back and forth. (We'll need the tipper to retrace its path on occasion.) The sequence of steps in a left-right traversal is shown in Figure 4. Once it's been so traversed, a one­way gadget can be used as an ordinary wire. But if it is first approached from the right, there's no way to bridge the gap and reach the left side. We'll pro­tect the entrances to our multiple-input gadgets with one-way devices, so that the tipper can never exit along an en­trance it couldn't otherwise reach. An OR gadget protected like this is shown in Figure 5 .

Before moving on to the AND gad­get, let's take another look at Figure 3 . Suppose we reinterpret the left input as a second output. Then this same con­figuration serves to split a signal com­ing from the bottom input: if the tipper enters at the bottom, it can exit to the left or the right. The only difference from an OR is that we don't attach one­way gadgets (because a split gadget has only one input).

Building an AND is a bit more chal­lenging. My solution is shown in Figure 6. (The one-way gadgets on the inputs

Q Figure S. OR gadget with one-way inputs.

12 THE MATHEMATICAL INTELLIGENCER

p ... 1 - -

� ; t Q

Figure 6. TipOver AND gadget. If the tip­per can reach both P and Q, then it can reach P 1\ Q.

are not shown.) This time the tipper must be able to exit to the right only if it can independently enter from the left and from the bottom. This means that, at a minimum, it will have to enter from one side, tip some crates, retrace its path, and enter from the other side. Ac­tually, the needed sequence will be a bit longer than that.

To begin our proof that the AND gad­get works, note that the crate labeled C in Figure 6 is the only one that can possibly be tipped so as to reach the right side; no other crate will do. If the tipper is only able to enter from the left, and not from the bottom, it can never reach the right side. The only thing that can be accomplished is to tip crate C down, so as to reach the bottom input from the wrong direction. But this doesn't accomplish anything, because once C has been tipped down it can never be tipped right, and the exit can never be reached. Suppose, instead, the tipper can enter from the bottom, but not from the left. Then again, it can reach the left input from the wrong di­rection, by tipping crate A right and crate Z up. But again, nothing is ac­complished by this, because now crate B can't be gotten out of the way with­out stranding the tipper.

The correct way to use the AND is shown in Figure 7. First the tipper en­ters from the bottom, and tips crate A right. Then it retraces its steps along the bottom input, and enters this time from the left. Now it tips crate B down, con­necting back to the bottom input. From there, it can again exit via the bottom, return to the left input, and finally tip crate C right, reaching the right side. The right side winds up connected to the bottom input, so that the tipper can still return to its starting point as needed later in the puzzle.

···I I lclsl UJJ···

Figure 7. How to use the AND gadget.

Next, let's build a variable gadget. Boolean variables can only be true or false, so this is just a two-way choice device. The split gadget is not good enough, because the tipper can exit to the left, perform some actions, then re­turn and exit to the right. We need the tipper to be able to reach either side, but not both sides. That is easy enough to do, but we also have to arrange for the tipper to be able to get back to the variable input from the selected output; otherwise it will not be able to reach both inputs of the AND gadgets. We solve this problem by adding a return pathway back to the input from each output, protected by a one-way gadget, as shown in Figure 8. If the tipper en-

x true

ters from the bottom, it can reach the left output by tipping crate A left, or the right output by tipping it right. Either way, it can connect back to the input, but it can never reach the other side.

Finally, we have to address how to cross wires over each other; often this is one of the trickiest parts of hardness proofs. An important early result [6] shows that SAT remains NP-complete even when the graph representing the formula is planar. However, that con­struction did not represent the ANDs ex­plicitly, and the result is not applicable to this problem. Fortunately, the gad­gets we have still suffice. It is possible to build a crossover gadget using only AND, OR, split, and two-way choice (i.e . ,

x false

choose x

Figure 8. TipOver variable gadget.

(a) Half crossover. If either input activates, then either output can; if both inputs activate, then both outputs can.

(b) Full crossover. Plus symbols represent half-crossover gadgets. If the left input activates, then the right output can; if the bottom input activates, then the top output can.

Figure 9. Crossover gadgets. AND and OR are represented by conventional digital logic

symbols; split is represented by a forking symbol; choice is represented by a question mark.

variable) gadgets; the construction is shown schematically in Figure 9. The reader is invited to verify that the gad­gets have the necessary properties. The proof that they work may be found in [3].

Putting it All Together Now we have all the pieces; do they fit together? A SAT formula is satisfiable if and only if there is a true/false assign­ment to its variables that makes it true. Given a formula, we construct a corre­sponding puzzle such that variable choices correspond to the directions chosen in the variable gadgets. Using split gadgets, we connect the puzzle's starting point to all the variables. Then we wire each variable output to all the ANDs and 0Rs in which it is used in the formula, again using split gadgets. We also wire the AND and OR outputs to further inputs as needed, to match the structure of the formula. Whenever we need to cross wires, we use the con­struction given in [3]. Finally, there will be one AND or OR that corresponds to the entire formula's truth value. We put the target red crate at the end of its out­put wire. A TipOver representation of our sample Boolean formula above is shown in Figure 10.

Now, if there is a satisfying assign­ment, the tipper can go to each vari­able gadget in turn, activate the correct direction, and return to the start to reach the next variable. Then, for every AND or OR corresponding to a true subex­pression of the formula, the tipper can reach that gadget's output, because, re­cursively, it can reach the needed true inputs. In particular, it can reach the tar­get crate.

But if there is no satisfying assign­ment, then no choice in the variable gad­gets will allow the puzzle to be solved. Therefore, if we had a TipOver algorithm we could use it to solve SAT problems, so TipOver is at least as hard as SAT.

The one thing remaining to show that TipOver is NP-complete is show-

© 2006 Spnnger Science+ Business Media, Inc., Volume 28, Number 3, 2006 13

§ § m

start Figure I 0. TipOver puzzle for ((x V x V y) 1\ (y V z V w) 1\ w).

ing that it is also no harder than SAT, i.e. , that TipOver is itself an NP prob­lem. A problem is in NP if it can be solved nondeterministically (allowing lucky guesses) in polynomial time, so a simple litmus test for an NP problem is this: if you are given a proposed solu­tion, can you verify it in polynomial time? In TipOver, you can certainly ver­ify it quickly enough: each crate is tipped at most once, and it's easy to check that each successive move of a purported solution is legal, and that the target crate is finally reached. Therefore, TipOver is in the class NP, and thus is NP-complete.

Beyond NP-Complete Earlier I mentioned Rush Hour, another puzzle made by ThinkFun. In Rush Hour, cars and trucks slide backwards and forwards on a grid; a particular tar­get car has to escape the grid. Perhaps Rush Hour is also NP-complete? Actu­ally, it is (we think) even harder! The problem is that our method for verify­ing a proposed solution in polynomial time doesn't work for Rush Hour. Cars can move backwards and forwards many times, unlike crates, which can

14 THE MATHEMATICAL INTELLIGENCER

only tip over once. This has the effect that there can be solution sequences which require exponentially many moves, so we can't check such se­quences in polynomial time. It turns out that Rush Hour is PSPACE-complete [1] .

(We can solve it using polynomial space.) Just as we're not positive that NP-complete problems are harder than ones solvable in polynomial time, we're not positive that PSP ACE-complete problems are harder than NP-complete problems. But in both cases, the smart money says yes.

In any case, it is a curious fact that many, if not most, interesting games and puzzles seem to be NP-complete or harder. It seems as though the very fea­tures that make puzzles challenging also tend to give them a kind of computa­tional power, which is reflected in their formal complexity. What, if anything, this says about the nature of intelli­gence, and the appeal of puzzles, is an interesting question.

REFERENCES

[1 ] Gary William Flake and Eric B. Baum. Rush

Hour is PSPACE-complete, or "Why you

should generously tip parking lot atten-

finish

dants." Theoretical Computer Science,

270(1-2):895-91 1 , January 2002.

[2] Michael R. Garey and David S. Johnson.

Computers and Intractability: A Guide to the

Theory of NP-Completeness. W. H. Free­

man & Co. , New York, NY, 1 979.

[3] Robert A. Hearn. Amazons, Konane, and

Cross Purposes are PSPACE-complete. In

R. J. Nowakowski, editor, Games of No

Chance 3, 2006. To appear.

[4] Robert A. Hearn. Games, Puzzles, and

Computation. PhD dissertation, Massachu­

setts Institute of Technology, 2006. To ap­

pear.

[5] Richard Kaye. Minesweeper is NP-corn­

plete. Mathematical lntelligencer, 22(2):9--1 5,

2000.

[6] David Lichtenstein . Planar formulae and

their uses. SIAM J. Comput. , 1 1 (2):329-

343, 1 982.

[7] James W. Stephens. The kung fu packing

crate maze. http://www.puzzlebeast.com/

crate/ index.html.

MIT Computer Science and Artificial Intelligence Laboratory

Cambridge, MA 021 39

USA e-mail: rah@csail .mit.edu

lj¥1(9ii·M David E. Rowe , Ed itor l

Augustus De M organ1s I naugu ra l Lecture of 1828 RONALD ANDERSON

Send submissions to David E. Rowe,

Fachbereich 17 -Mathematik,

Johannes Gutenberg University,

055099 Mainz, Germany.

The lecture reproduced here in print for the first time was given by Augustus De Morgan in No­

vember of 1828 as one of the intro­ductory lectures delivered at the open­ing of the University of London. The University, later to be named University College London (UCL), had been founded two years earlier with the in­tent of providing a more accessible uni­versity education in England than Ox­ford and Cambridge and one more suited to the professional worlds of Lon­don. De Morgan had been appointed professor of mathematics at the univer­sity in February 1828 at the remarkably young age of 2 1 , having just graduated from Cambridge the previous year. De­tails about De Morgan's appointment and the foundation of the University may be found in Rice [1997bl, and on his life and work in general in Rice [1996).

The steady theme of the lecture is mathematics in an educational setting. The theme is a fitting one for the start of the new type of university in a context of a concern for general education and as well the currents of unrest and reform in England at the time that were to cul­minate in the political and other reforms of the 1830s. The Society for the Diffu­sion of Useful Knowledge (SDUK) in par­ticular, which De Morgan had contact with, had been established in 1826, and one of its purposes was to spread "use­ful information" to all parts of society through suitable self-educational texts.

De Morgan's stress on the language of mathematics that starts the lecture echoes a theme of the early 19th-cen­tury reformers of mathematics at Cam­bridge, a group who formed the "Ana­lytical Society" dedicated to this purpose, some of whom were his tu­tors. Herschel, Babbage, Peacock, Whewell, and others sought to import the more efficient Continental notation for calculus to Cambridge, and the man­ifesto style of the "Preface" of the Mem­oirs of the Analytical Society, written by Babbage and Herschel, celebrates the power of a symbolic language for math-

16 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Business Media, Inc.

ematical reasoning [Babbage, 1813) . Then late in the lecture De Morgan wilt enlist John Locke to contrast the clarity of definitions in mathematics with the subtle diffuse shades of meaning that occur in regular language.

In an argument with strikingly mod­em resonances, De Morgan argues that an understanding of how the mind works and a theory of education suit­able for mathematics should weave to­gether. In the light of Locke's arguments against innate ideas, and the discredit­ing of notions of the mind as a "blank sheet" of paper, the early cultivation of reasoning powers of the mind were now to be seen as important in new ways. Here lies De Morgan's central point: that mathematics stands unique in cultivating such powers by its demonstrations of how conclusions fol­low from self-evident starting points. The "links of the chain" in an argument are particularly visible in mathematics. Moreover, the confidence in abstract reasoning that beginners get from math­ematics is valuable for other disciplines where the conclusions need not be so clear and obvious, such as natural phi­losophy (essentially our discipline of physics). Irrespective of the composi­tion of his classes, to De Morgan his job will be to "form the mind to habits of reasoning. "

I n the earlier part of the lecture De Morgan remarks that the "human mind cannot bring all its power to bear on two objects at the same time, it can­not surmount two distinct obstacles at once" [ 1 2) . One cannot learn a lan­guage while attending at the same time to its style and beauty, and one can­not get knowledge from authority while at the same time attending to the steps of reasoning. The two-fold division reflects a similar division at the end of the "Preface" of the Mem­oirs of the Analytical Society between observing the mind directly and at the same time observing the "links" that connect ideas, and moreover: "power­ful indeed must be the mind, which can simultaneously carry on two

processes, each of which requires the most concentrated attention . " [Bab­bage, 1813, p. xxi]. For the authors of the Preface such a dual attention is needed to discover a "philosophical theory of invention. " The issue for De Morgan is attending to the operations of the mind when doing mathematics.

One can locate here the charter that was to drive De Morgan's life-long concern with education. His energies were spent in teaching at UCL through to 1 866 (apart from 1831 to 1836 when he had resigned) and in writing for general audience periodicals and En­cyclopedia articles on the history and nature of mathematics. His more spe­cialized and abiding concern, the foundations and meaning of algebra, resides here as well [Richards, 2002] . As the mathematician ]. ]. Sylvester at one stage remarked about De Morgan, "He did not write Mathematics, he wrote about Mathematics. " [quoted in Halsed, 1 900] The lecture illustrates the beginning of this meta-discourse, one which was to emerge generally in mid-Victorian England in other disci­plines.

From an annotation on the title page, the text before us appears to have been deposited at the UCL li­brary in 1920 by Sir Walter Wilson Gregg (1875-1959). The occasional awkward construction and punctua­tion no doubt reflects a manuscript in­tended to be read from in a lecture setting rather than one prepared for publication. While other talks given for the opening session were either published or commented on in papers at the time, De Morgan's was not (on others see Bellot [ 1929] , p. 75f, and for other talks, Conolly et al. [ 1829]) . Adrian Rice [ 1997] notes that the day before its delivery, De Morgan had been informed by Leonard Horner,

the University Warden, that if he had any intention of publication "it is the wish of the Council that you should previously obtain their sanction." That may have discouraged De Morgan fol­lowing up on publication. One can speculate that the dense argumenta­tion in places may have made the lecture hard to follow and served to discourage comment or publication. And offence may have been taken at the slight implication that others (in­cluding others who had given lectures at the opening) had injudiciously promoted their own subjects; or his mention that mental laziness is the ori­gin of the universal distaste for math­ematical studies. In these features, though, one can see a youthful energy, one captured in his wife's comment on the lecture in her memoir:

It is not only a discourse upon men­tal education, but upon mind itself. It was the work of a young man of twenty two years and four months old, and the earnestness and san­guineness of youth may be seen in the strong determination with which his work was begun, and the high hope which he felt of the work the University had to do. [De Morgan, 1882, p. 31]

ACKNOWLEDGMENTS

I am grateful for the assistance of the staff of UCL Library Special Collections and Susan Stead in particular on mat­ters to do with the manuscript. Ex­changes with Joan Richards and Adrian Rice on De Morgan and this lecture have been most valuable and their en­couragement to explore publication es­pecially acknowledged. The assistance of Noah Moss Brender in transcribing the manuscript is also gratefully ac­knowledged.

REFERENCES

1 ] Babbage, Charles and John Herschel

(1 81 3). "Preface." Memoirs of the Analyti­

cal Society: i-xxii.

2] Be/lot, Hugh Hale (1 929). University Col­

lege, London, 1 826-1926. London, Uni­

versity of London Press, Ltd.

3] Conolly, John, et a/. (1 829). Ten Introduc­

tory Lectures Delivered at the Opening of

the University of London, session, 1 828-9. London, Taylor.

4] De Morgan, Sophia Elizabeth (1 882). Mem­

oir of Augustus De Morgan. London, Long­

mans Green and Co.

5] Halsed, George Bruce (1 900). "De Morgan

to Sylvester." The Monist 1 0: 1 88.

6] Lardner, Dionysius (1 829). "A discourse on

the advantages of natural philosophy and

astronomy as part of a general and pro­

fessional education being an introductory

lecture delivered in the University of Lon­

don, on the 28th October, 1 828." Pub­

lished in Conolly, 1 829.

7] Rice, Adrian (1 996) . "Augustus De Morgan

(1 806-1 871 ) ." The Mathematical lntelli­

gencer 1 8: 40-43.

8) Rice, Adrian (1 997a). "Augustus De Morgan

and the development of university mathe­

matics in nineteenth-century London, Ph.D.

dissertation." Middlesex, University, London.

9] Rice, Adrian (1 997b). "Inspiration or Des­

peration?: Augustus De Morgan's appoint­

ment to the Chair of Mathematics at Lon­

don University in 1 828." British Journal for

the History of Science 30: 257-274.

1 0] Richards, Joan L (2002). "In A Rational

World All Radicals Would Be Exterminated:

Mathematics, Logic and Secular Thinking

in Augustus De Morgan's England." Sci­

ence in Context 1 5: 1 37-1 64.

Department of Philosophy Boston College Chestnut Hill, MA 02467 USA e-mail: ronald.anderson@bc.edu

Tbe lecture follows overleaf

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 1 7

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Title page of De Morgan's manuscript.

18 THE MATHEMATICAL INTELLIGENCER

An Introductory Lecture delivered to the opening of the Mathematical Classes in the University of London

Nov. 5th, 1828 By Augustus De Morgan

Professor of Mathematics in that University1

[1 ] An introductory Lecture must always be a matter of difficulty, whatever may be the source from whence the materi­als are to be drawn. It is not easy to bear in mind, that though this may not be the case, in every instance a low state of knowledge must be supposed in those who are addressed, and that the subject must not be entered to a depth which the beginner cannot be ex­pected to fathom. The duty which de­volves upon me this day is rendered more than commonly difficult by the peculiar nature of the sciences which I am appointed to teach. Had the math­ematics ever possessed that degree of general interest which is attached to the other branches of education, I should still have felt, that to select the most forcible arguments in favour of their cul­tivation and to support those arguments in the manner which the subject de­serves, would have required a judgment and power of expression far superior to my own; but when I consider how few, even among highly educated persons, have thought it necessary to make themselves acquainted with more than the merest elements of these branches of learning, I feel that I cannot hope to attach an interest to the subject which

[2] circumstances seem to have denied, or to make an impression favorable to these sciences on the minds of any who have not studied them for themselves.

The mathematics labour under pe­culiar disadvantages from the manner in which they are usually brought to the notice of those who are ignorant of them. If a man of education take up a book on general literature, law, politi­cal economy, or experimental philoso­phy, he can seldom fail to derive some instruction, even from a few minutes' perusal of the part to which he may

chance to open, although the subject of the book may never have formed a part of his studies; at least he will meet with something to excite his curiosity, and fix his attention. He will therefore close the volume with some idea, although an imperfect one, of the nature and ob­ject of its contents. He will become per­fectly prepared to believe in the utility of the studies; which it presents, and cannot help feeling prepossessed in fa­vor of a science, from which by a few minutes' desultory reading, he has ob­tained both pleasure and information. [3] Let us suppose that one who is ig­norant of the first principles of algebra and geometry, has chanced to light upon a book which treats of any math­ematical science. What conception can he form of the nature of the informa­tion which lays open to his sight? what opinion as to its general utility? what conclusion upon the question whether it would be desirable for him to com­mence the study? none whatever. The page which lies before him is filled with characters repulsive in their appear­ance, and conveying no meaning to one not previously acquainted with them. They are in fact to him, a foreign lan­guage, a system of hieroglyphics, which

there is no more probability of his be­ing able to decipher, than of his read­ing at first sight, the language of an Arab or Chinese. Under these circumstances many have left the subject with an ig­norant contempt for what was to them useless and unintelligible, many with an equally ignorant admiration for those who could fathom what appeared so profound and incomprehensible. [4] I might proceed to show that this novel system of writing, this compendious language (for such in fact it is) contains, in its very formation, the germ of the

most valuable improvements which the mathematics have ever received, and has been from its peculiar structure, a never failing guide to new discoveries. I might lay before you the opinions of men the most distinguished for acute­ness of reasoning as well as fertility of imagination, in support of the assertion that the study of this language, without reference to any of its applications, is instrumental in furnishing the mind with new ideas, and calling into exercise some of the powers which most pecu­liarly distinguish man from the brute creation. But this is not the point to which I wish to direct your attention to clay. Neither is it my object to attempt to exalt the sciences, which I am ap­pointed to teach, as high as some have injudiciously wished to raise them, by an uninstructive and useless panegyric. I proceed to lay before you the argu­ments which in this age, I consider as most forcible [5] demonstrating the util­ity of the mathematics, as well in gen­eral, as professional education, to an­swer the most formidable of the objections which are usually urged against their general reception, to bring to your notice those which cannot be fairly answered, and to point out how

to remedy the inconveniences arising from them. I also intend to give you some information upon the method by which I propose to teach them.

It is a maxim inculcated on every one in his infancy, and constantly repeated in his maturer years, that God had disc tinguished man from the brutes by what is denominated the gift of reason. This was formerly understood to mean, that man, in whatever situation he may be born, and independently of all external circumstances whatever, possesses at his birth, most of the ideas as well as

1 University College London, Special Collections: MS ADD 3. Numbers in [ ] indicate pages in the original manuscript and text in { } are annotations made on the back

of pages intended for insertion on the following page.

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Augustus De Morgan

the powers, which distinguish him from the inferior animals. Such was, until the time of Locke, the opinion of those who had [6] made the human mind their pe­culiar study; and although his masterly refutation of the doctrine of innate ideas, as it was called, soon caused the contrary opinion to gain ground among philosophers and men of learning, it was some time before his enlightened doctrine had spread sufficiently to pro­duce any remarkable result. Neverthe­less, men were at last convinced of this truth, till then unknown or unconsid­ered, that we are not indebted to na­ture, either for our most simple ideas, or for any sure method of combining them, so as to arrive at conclusions which may be relied on as the results

of accurate reasoning. When that part of the community whose knowledge enables it to direct the sentiments of the rest, was once practically convinced, that not only opinions themselves, but the methods of forming them and dis­tinguishing between the right and the wrong, depend upon previous educa­tion, and are not natural endowments, a powerful [7] and settled conviction of the necessity for the early cultivation of the reasoning powers could not fail to be the consequence. The world began to perceive, that its notions as to the proper end of education, were perni­cious in practice, and absurd in theory; that the human mind was not as had been supposed, like a blank sheet of

paper fit and ready to receive any im­pression and retain it for its own use and that of others, but that the recepta­cle itself was to be prepared, its pow­ers of arranging and combining what was presented to it were to be devel­oped, and that it was only the difficulty of the subject, combined with their ig­norance of moral phenomena, which prevented an earlier exposure of the fact, that their plan was no less absurd than the attempt to nourish a newly born infant, with the food of a grown man.

This alteration of public opinion as to the objects of education, produced a corresponding [8] change in the num­ber of those to whom it was considered necessary. The reason is extremely ob­vious. If to acquire the bare knowledge of facts be the sole intent with which a youth is sent to a school or university, it may be plausibly argued that unless that knowledge be immediately con­nected with his future profession, it can be of no utility whatsoever to him in life: that Greek is therefore only desir­able for a clergyman, and mathematics for an engineer or landsurveyor. {If rea­son be like a state carriage, an article only fit to be used upon solemn occa­sions, they were right who inferred that its culture was expedient only for those who are destined to overawe a multi­tude by the pomp of their acquirements, or to maintain a particular station in So­ciety. l But when the gradual alteration of opinion which I have described in­duced men to turn the light with which extended knowledge had furnished

them, on the neglected theory of edu­cation, it appeared most clearly that the success of every individual in the world must depend on his power of reason­ing on the events which immediately concern him, and the circumstances in which he is placed, {a power whose na­ture was concealed, and which was rep­resented as independent of education, under the specious title of judgment, and common sensel.2 They saw that the reflecting powers [9] by the aid of which all will be required to distinguish be­tween right and wrong, are not ade­quate to this purpose, unless a habit of using them rightly have been previously framed; and that this habit can never be acquired but in early life . When these

opinions came to be entertained, not in the rapid manner which often marks the ephemeral reception of brilliant and plausible fallacies, but in the course of time, in spite of opposition directed through many and powerful channels, with that gradual and silent accession of converts from different quarters which has characterized the progress of all the great truths that have now no longer anything to fear from opposition, can it be doubted that the result would be a conviction, that no rank or condi­tion, whether high or low, ought to be exempt from the benefits of mental cul­tivation, and that [10] he who effectu­ally promotes the cause of education is the benefactor of his species, and con­fers a lasting and substantial benefit on the country which can boast the hon­our of having given him birth. The pres­ent age is reaping the salutary fruits of this conviction; they are gathered in the cottage of the husbandman no less than in the Senate of the Nation, every day, every hour augments the number of those who feel its influence, and profit by its effects; who is willing to be the last in the general advance, in the march of intellect, the object of scorn to the few who still remain to offer it, an ob­ject of scorn! to which nevertheless they owe the power of truly declaring their sentiments when they are at variance with public opinion. [ 1 1] I have tres­passed much upon your time in dwelling upon the progress of opinion with regard to education, on account of the partial and detached view which is

usually taken of the mathematics as a branch of it.

An opinion highly unfavourable to their progress prevails still among many who are unacquainted with these sci­ences, which supposes them to be valu­able only in a professional point of view, and not advantageous, or at least not at all necessary, to the scholar and man of general information. By consid­ering the subject closely in connexion with education in general, I hope to contribute towards removing this im­pression, if it exist in the mind of any one here present. I now proceed to the consideration of those peculiarities of the mathematics which make them so valuable to the general student, but be-

2Between these two additions on the back of page 7 of the original manuscript is a further phrase whose place of insertion is unclear: "that every faculty of the human

being is intended for the use of all."

20 THE MATHEMATICAL INTELLIGENCER

fore I begin, I would wish to observe, that in the comparisons which I shall have occasion to institute between this and other branches of knowledge, noth­ing is further from my intention than to depreciate any of them. An attempt of this description would indicate deprav­ity both of taste and feeling, surrounded as I am by those, who, in the preced­ing introductory lecture [1 1 a] have so ably vindicated the utility of their re­spective studies. And here let me espe­cially disclaim the intention of invali­dating the arguments which were used the other day by my distinguished col­league, the Professor of Natural Philos­ophy, when he described the advan­tages which the study of Natural Philosophy possesses over that of Math­ematics.3 All who heard these admirable lectures will recollect, that it was the demonstrative study of Natural Philoso­phy, the application of mathematical reasoning to the sensible qualities of matter, which he recommended to the attention of his hearers. His arguments therefore form the proper continuation of mine, and this would be the place to urge considerations of a similar na­ture were it not that time would fail [if in addition to what I have proposed to myself in this Lecture, I were in to re­count] the numerous and daily increas­ing applications which may be made of mathematical knowledge.

The fact is, that every description of information has advantages peculiar to itself, and I cannot more forcibly point out those of the mathematics, than by comparison of them with various sci­ences, whose superiority I admit in other points but whose defects I shall make use of, wherever they will illus­trate my argument.

It is obvious that whatever be the subject which is selected as the instru­ment by which the act of reasoning should be taught to those unaccustomed to demonstration, it should if possible be free from extraneous difficulties, that is from difficulties which are indepen­dent of the reasoning itself. The human mind cannot bring all its powers to bear on two objects at the same time, it can­not surmount two distinct obstacles at once. For instance, no one can be ex­pected to see the beauties of a foreign language, while he is toiling to master

the difficulties of its construction. It is not until he is so far accustomed to these as to read with fluency that he can at­tempt to distinguish one style from an­other. On the same principle, the youth who has hitherto acquired all his knowl­edge on the word and the authority of his instructors [ 13] and who has never before lent himself to the task of trac­ing, one by one, the links of a chain of ideas by the means of which some truth is deduced from an obvious first princi­ple, has ample employment for all his intellects in merely following the steps of a demonstration. But there are cir­cumstances connected with most sci­ences, which prevent the attention from fixing itself on this one object. In some the terms which are used are frequently imperfectly defined, sometimes so ab­stract as to be totally incapable of defi­nition, and so complex as to require it in no ordinary degree. In others the first principles, the axioms which must be admitted, and declared satisfactory with­out demonstration, before the student can proceed, are not self-evident, or even very definite in the ideas which they convey. Frequently, the conclu­sions themselves, far from carrying with them such an air of truth as to form the least verification of the process by which they were obtained, are so contradictory to previously acquired notions, that the mind revolts from their admission, and would rather suppose some concealed flaw in the reasoning or falsehood in the original positions, than assent to a re­sult so monstrous in its appearance. In this case it is evident, that nothing but the most perfect [14] assurance of the truth of the method employed in the demonstration, an assurance only to be obtained by previous experience of its efficacy, can produce the required con­viction, or give any confidence in the result. Such a study is therefore unfit to be the medium of instruction in the first principle of reasoning.

The mathematical sciences are en­tirely free from these defects. This as­sertion I shall proceed to establish as far as it can be done without entering into technical details, or sacrificing, for the amusement of those who understand the subject, the benefit of those who do not.

The object of the mathematics is the consideration of the properties of space,

3The reference here is no doubt to the opening lecture by Dionysius Lardner [Lardner, 1 829.

number, and magnitude in general, with a view to the application to natural phi­losophy and the arts of life. It is noto­rious that the first ideas which any hu­man being secures are derived either from the figure or number of the ob­jects which surround him. From the ap­pearances of the material world, certain distinct notions are gathered, which though their prototypes have no [ 1 5] real existence in nature, are the clear­est and most definite which our minds contain. Thus, a straight line, needs no definition, nor will the mention of it leave the least doubt as to what is meant in the mind of any person present. The word triangle contains a definite idea, and though the number and species of triangles is unbounded, yet that which constitutes a triangle is so perfectly clear, that there is not the least danger of the term being misapplied so as to create any confusion.

But this is not the greatest advantage which mathematical definitions possess. They have one peculiar to themselves, of which hardly any traces are to be found in those of other sciences. I mean the certainty with which terms that are in any degree synonymous, can be dis­tinguished from one another, however slight the difference may be. When I ar­rived at this part of my subject, I took up Locke on the Human Understand­ing, with a view to draw from some part or other an illustration of my meaning. This profound work, as you are [ 16] aware, is remarkable for correct appli­cation of terms, and the disposition which prevails throughout never to sac­rifice clearness of expression to beauty

In mathematics there are no terms exactly synonymous, and those between which there is resemblance can

be as clearly distinguished as others.

of composition. All difficulties of words may therefore be fairly attributed to the subject, and not to the author. In the first page to which I turned, I found the four following terms, all used to signify the same thing, solidity-resistance­hardness-and impenetrability. Here are four expressions, so much alike in their

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Page one of De Morgan's manuscript.

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meaning, that in ordinary conversation, they might be used indiscriminately one for the other. Yet there are no two of them which convey precisely the same idea, and the difference between them cannot be expressed in (simple) words, so as to leave in the mind of the stu­dent no doubt as to the limits of the meaning of each. In mathematics, on the contrary, there are no terms exactly synonymous, and those between which there is some resemblance, can be as clearly distinguished from one another as others which have the most contrary significations. Locke himself [17) takes notice of this, as far as numbers are con­cerned in the following words;

"The simple modes of numbers are of all others the most distinct; every the least variation, which is an unit, making each combination as clearly different from that which approacheth nearest to it, as the most remote; and the idea of two as distinct from the idea of three as the magnitude of the whole earth, is from that of a mite. This is not so in other simple modes, in which it is not so easy, nor perhaps possible for us, to distinguish between two approaching ideas, which yet are really different. For who will undertake, to find a difference between the white of this paper, and that of the next degree to it." He might perhaps have said with more meaning who will undertake to express this dif­ference, so as to convey to another an exact notion of its quantity.

I come now to the first principles which must be admitted before any at­tempt to reason upon the nature of the objects to which names have been

There can he no science which has not some

undeniable truths, too simple to he either estab­

lished or destroyed by demonstration .

given can be attended with success. An edifice [18) may indeed be erected by the skill of the builder but the soil un­der the foundation, must be rendered secure by the hand of nature, and adapted to the purpose by that of man, or no stability can be expected. If rea­son be the art of deducing complex re­sults from those which are simple by satisfactory demonstration, and if a sci-

ence be, as it has been defined to be, a systematic body of knowledge whose parts are connected by sound reason­ing, there can be no science which has not some undeniable truths as its groundwork, some propositions which are too simple to be either established or destroyed by demonstration. I have already observed that it is of the great­est importance that these elementary truths should be really self-evident, for if they are not so, no subsequent result, however sound may be the demonstra­tion which connects it with its first prin­ciple can be depended on as true with any degree of certainty. When the in­termediate reasoning is correct, the ev­idence of the result is the same as that of the first [19) principle, if the former be dubious, a corresponding degree of uncertainty attaches itself to the latter. Of all the errors whose combination has retarded the progress as well of morals as of real philosophy. no one has been so effectual as the admission of first principles which have no claim to be self-evident. The greatest minds, when trained to this pernicious habit have been affected by it in a manner which is hardly credible. {Who has not heard of the fame of Galileo, of his splendid discoveries, his strenuous efforts to es­tablish truth, and overturn error, and of the persecutions which he endured on this account. There is not a wreath in this crown which has not been most justly awarded, and if this great mind should afford the most striking exam­ple of the habit arising from careless ad­mission of first principles, which other instance will be necessary?! When Galileo was in the zenith of his repu­tation, some Florentine mechanics dis­covered that a pump would not work if its piston were more than 32 feet in height, and they applied to Galileo to explain this phenomenon. Now the philosopher had been accustomed, in common with those of his time, to con­sider as perfectly obvious, the position, that nature abhors a vacuum. He there­fore found no difficulty in modifying this principle by an alteration which was not less evident than the original itself, and he accordingly answered the application by saying that [20) nature abhors a vacuum, and therefore the wa­ter rises in the piston when the air is exhausted, and that she continues to ab­hor it, until {a column of water 32 feet

in height has been raised I , after which she is content that it should exist.

The propositions which are assumed in Geometry are of the simplest possi­ble character, and really come up to the notions which I have given of first prin­ciples. Some of them such as the fol­lowing one, the whole is greater than its part, are mere consequences of the meaning of the terms which convey them. Some, such as this one, that two figures which entirely coincide with one another are equal, have rather the char­acter of definitions than of principles. Other such as this, Two straight lines cannot enclose a space, constitute our most obvious notions of figure. I have, I am sure, instanced enough to shew those who are unacquainted with Geom­etry that whatever obstacles they may have to encounter in pursuing this study, their difficulties will at least not [21) arise from any ambiguity in the terms to which they are introduced, or any doubt as to the truth of the preliminary propo­sitions which they are required to admit.

I must now direct your attention to the nature of the conclusions which are derived by means of mathematical demonstration. I have already observed, of what importance it is that beginners should have some test constantly at hand, by which they may try the accu­racy of their first conclusions, and thereby acquire that confidence in the usual processes, which will induce them to trust the results of abstract reasoning, when all other methods of demonstra­tion fail . This confidence will be par­ticularly necessary when the mathe­matical student begins to apply his knowledge to the study of Natural Phi­losophy. There are many truths in me­chanics which appear highly paradoxi­cal when they are heard for the first time. Such are the following proposi­tions, that no force, how great soever, will stretch [22) a thread, that has any weight, into a horizontal straight line, however nearly it may appear such to the eye. That it requires as much man­ual labor to raise a weight by means of the most powerful machine, as it would do if the machine were not in existence. That the earth communicates every in­stant as great an impetus to the sun as the sun does to the earth, and many others which strange as they may ap­pear, are absolutely demonstrable.

There are two verifications which

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may be easily obtained of the truth of every mathematical proposition. The first arises from the capability which ex­ists in almost every theorem, of being demonstrated in more than one way, that is, the first principle and the con­clusion may generally speaking be con­nected by several trains of reasoning. Each of these is a confirmation of the others and there is no exercise which I know of, that in my opinion tends so much to strengthen the mathematical powers of the student, as the attempt to discover different demonstrations of the truths which are presented in his books. Every demonstration which he thus discovers makes the proposition with which it is connected peculiarly his own and furnishes the proof of his progress in the study. [23]

The second test of the accuracy of the reasoning is ocular demonstration. This may be applied, if necessary, to all the propositions of Geometry and Algebra. Indeed, many of them derive from mere inspection of correctly drawn figures, a high degree of evidence, almost amount­ing to certainty. Who would doubt for an instant that the straight line is the shortest distance between two points? Or that two parallel straight lines make the same angle with a third straight line which crosses them both? This last proposition is as obvious as the former to every one who understands the terms, and inspects the figure.

I have now offered to your notice the principal advantages which the mathematics appear to me to possess, considered as the medium of instruc­tion in the art of reasoning. In addition, it may be mentioned, that there is no species of sophism, of which the annals of mathematics do not furnish instances. This is no reflection either on the sci­ences themselves, or on those who have cultivated them; for of all the fallacies which have been promulgated by men whose imagination has exceeded their knowledge, ninety nine out of a hun­dred have been fully exposed almost in the instant of their appearance. The rec­ollection of these will furnish [24] abun­dant opportunities for a very important exercise, the detection of incorrect rea­soning, an exercise which will be the more instructive, as by the occurrence of great names, in connection with fal­lacies and misconceptions, the student will perceive that brilliant intellect, un-

24 THE MATHEMATICAL INTELLIGENCER

accompanied by habits of correct think­ing has often led its possessor to the di­rect path of error, and that if he neglect the constant improvement of the men­tal faculties, he may perhaps acquire profound knowledge, but will never reason with accuracy.

The catalogue of those who object to mathematical studies comprises a very large portion of the human race viz: all who dislike the labor of serious thought. I do not mean by this to as­sert that no person of a thinking turn has ever arrived at a conclusion unfa­vorable to scientific pursuits but I am convinced, could the causes of this op­position be registered, and the com­parative influence of each be fairly as­certained, it would be seen that though prepossession in favor of other studies, a desire to check the progress of knowledge, ignorance of the [25] nature and objects of mathematics frequently exert their power, yet that the great agent in producing the almost univer-

The great agent in producing the almost universal distaste for

mathematics is mental indolence.

sal distaste for mathematical studies, is mental indolence.

Against this disposition it is needless to argue; I will therefore proceed to the several objections which have been urged against the study of mathematics with a wish that it were as easy to erad­icate the cause as to refute the argu­ments to which it has given birth.

There are two objections which I have generally heard from beginners, and which shew in the strongest light the disposition to which I have alluded. It is frequently said that the difficulty of the mathematical studies is so great as to make them require more time and study than their importance warrants the instructors of youth in bestowing upon them. This objection proceeds upon a false notion both as to the im­portance of the knowledge, and the dif­ficulty of its acquisition. Of the former, I shall say nothing more; with respect to the latter, I shall remark, that the dif­ficulty of mathematical studies is gen­erally greatly overrated. You have al-

ready seen what childish simplicity reigns throughout its first principles. The links of the chain of argument which is fastened on these principles are connected by others as simple and possessing equally, in every respect the character of being self-evident. What difficulty therefore can [26] such a study offer to the beginner, except that which must arise from the novelty of follow­ing connected reasoning. And is this dif­ficulty peculiar to the Mathematics alone? Will the mind that has revolted from severe thought, when connected with the properties of space and num­ber, seek the page of history, or the phenomena of nature, with any object except mere amusement unqualified by the least desire of improvement.

This objection may be dismissed with confidence as arising from a prin­ciple whose tendency is to substitute the entertaining in place of the useful, and to render the acquisition of knowl­edge nothing more than a sensual grat­ification. In addition to this, I may re­mark, that even the plausibility of the objection is daily diminishing. Within the last century, much has been placed within the reach of the elementary stu­dent, which in the days of Newton, could only be learnt and appreciated by a mathematician of the highest order. A better confirmation of this fact could not be desired than you will receive when I say that Newton's Great Work the Prin­cipia, which in that century, was barely understood by the most profound men of the day, is now read with the com­parative facility by many who have not attained the age of nineteen. [27]

Of a similar character to the last is the reflection which is often conveyed in the statement of the fact that some have made discoveries of practical ben­efit and have done that which the great­est mathematicians have left undone, with out any, or at most with only a very limited knowledge of pure math­ematics. Even supposing this to have been invariably the case and that the professed mathematicians had added no one single discovery to our stock of practical knowledge, this would in no degree affect what I have said upon the utility of mathematics, as a branch of general education. That must stand or fall with the truth of what I have as­serted upon the nature of the study it­self, and not with its effect on com-

merce manufacturers and the fine arts, numerous and beneficial as they are. But let us see how the case really stands. Every one who has traced the history of the application of human powers to the arts of life, cannot but have observed, that for one discovery which has sprung solely from natural intelligence, a hundred have resulted from the application of acquired knowl­edge. To the beginner who urges the objection and especially to him whose future profession is in the least degree connected with [28] any phenomenon of nature I would say do you feel within yourself the power which has enabled many illustrious men to triumph over the defects of education? Even in that case I could not recommend the neglect of these sciences, for your utility will be more extensive, and your fame more certain, if you extend the boundaries of knowledge than if you are merely con­tent with shewing how much can be done without it. But if analogy be any argument you cannot possess the pow­ers which distinguished those of whom we are speaking. For none of the emi­nent men, under whose involuntary de­fects you are endeavoring to shield yourself from the appalling task of thought, ever felt or reasoned in this way. There is no tribute of which the abstract sciences can boast so fairly or which is so conclusive in their favor, as the regret of these great men who for the want of them, have been obliged to forego the attainment of a height which they saw, but could not reach. If there­fore such are the sentiments of men who lie under great temptations to de­preciate these sciences, all who are not conscious of extraordinary talent, should feel that if knowledge be power, if reputation be valuable, they cannot afford to neglect any probable means of acquiring either. Let not any one wait for a lucky accident to bring him in the way of discovery. When the cases in [29] which this is reported to have hap­pened come to be closely examined, it will appear that the share of fortune has generally been very small . Thousands had bathed before the time of Archimedes, but it required his habits of mind to discover from a common cir­cumstance the method of detecting the proportions of gold and silver in a mix­ture of the two. The lovers of nature have admired the setting sun from their

windows from time immemorial, but the polarization of light remained undis­covered, until fortune, if you will, led the profound research of Malus to the observation of this phenomenon. Whichever way we turn, the success of knowledge is apparent[:] all therefore, who have this to direct their efforts, may confidently hope; if here and there an isolated instance of success without knowledge should occur, all it proves is that those who have not this reliance need not absolutely despair.

A third and very common objection to mathematical studies, is the assertion that they have a tendency to deaden the imagination, and to destroy the taste for literature. Those who raise this objec­tion are not aware that the most pro­found mathematical discoveries have been due, not less to the [30] reasoning powers than to the fertile imaginations of the gifted men who first promulgated them. Never was the talent of invention so brilliantly displayed as in the various successful attempts by which, from the time of Newton to that of Laplace, all the phenomena of the solar system were mathematically demonstrated to result from the operation of the Newtonian Law of Gravity. This part of the objec­tion therefore falls to the ground. With regard to the rest of it, that mathemati­cal demonstration destroys the taste for poetry and the belles lettres, I take it for granted that the supporters of this as­sertion refer to the habits of mind which such a course of study is likely to form, and not to the actual information which it communicates. For I cannot suppose them to mean that the mere knowledge of the facts which are established in Geometry can have the effect above stated. In this case the reverent retort that could be made would be, to allow the truth of the objection in its fullest extent. This would amount to an ad­mission that habits of thinking are in­compatible with all lighter pursuits, and that Shakespeare and Milton can never be tolerated by anyone who is accus­tomed to arrange his ideas [31] system­atically. Even supposing this to be the case, I will ask which ought to hold the highest place, reason or fancy? Which is the most peculiar attribute of man, thought or feeling? In which part of ed­ucation should the most strenuous ef­forts of instruction be directed, to the establishment of a standard of beauty,

or of truth? But the question need never be raised, the alternative which it im­plies, need not be considered, for the objection will prove, on examination, to be entirely without foundation. It is ren­dered extremely improbable by the na­ture of the case, if the opinion be cor­rect, that imagination, unrestrained by habits of observation and thought, is apt to indulge in excesses which good taste must condemn. But it is entirely refuted by the number and eminence of the mathematicians whose fame depends no less on their literary merit, than on their scientific skill. Need I instance Plato, whose name would have been immortalized as the first promoter of the method of investigation to which mod­ern science owes its proudest triumphs, had it not been that the splendor of his writings, which are in the hands of all, has rendered almost imperceptible the merit of his geometrical discoveries, which are known [32] only to mathe­maticians. The writings of Fontenelle, Pascal, Descartes, Leibnitz, and D'Alem­bert, are as distinguished for spirit, taste, and beauty, as their authors are for sci­entific talent. Add to these considera­tions that in all ages science and litera­ture have flourished together. It is well remarked by Montucla, a man himself distinguished in both characters, as a writer and a philosopher, that the re­vival of literature gave to Italy her most distinguished mathematicians, in the same age which produced Tasso and Aristo, that in France, the day of Pascal, Descartes, Fermat, and De L'Hopital, was also that of Corneille, Moliere, and Racine; and that in England, Newton, Wallis, and Halley lived in the same cen­tury with Milton, Addison, and Pope. It is believed by some that mathematical demonstration contributes to the forma­tion of a self sufficient and arrogant spirit, and that these studies combined with those of Natural Philosophy, tend to give extravagant ideas of human power, and exalted notions of human importance, And as these objections have been specially directed against the mathematics and all [33] their applica­tions, so there exist in these very sciences peculiar circumstances from which a most conclusive answer may be drawn. I refer to their immeasurable ex­tent, and to the facility with which unan­swerable questions may be proposed. This is not the case in all branches of

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 3, 2006 25

learning. Let us take for an example the study of the literature of any well known country. Suppose that an individual has acquired, from an attentive perusal of remaining authors, all the knowledge which is practicable, with respect to the ancient constitution of this country, and the manners of its people. Suppose also that in addition he has carefully studied its language, has conversed with its in­habitants, has diligently inspected its monuments of antiquity, and has traced with his own eyes the scenes of all the memorable events which have distin­guished its history. Are there not many to whom this description will apply, whether we suppose that country to have been Greece, Italy, England, or any other civilized region on the face of the globe. If such a person should boast of his information and vaunt the powers which have placed him at the head of the department of knowledge to which his attention has been directed, however absurd the feeling might be, he would clearly be right in this point, that unless some most unprecedented occurrence should happen, such as was [34) the dis­covery of Herculaneum and Pompeii in Italian history, he might be said to have attained a degree of knowledge in his favorite pursuit, which no man could reasonably be expected to surpass. It would be difficult, for any other person not equally accomplished to propose a question which the first could not either answer, or give a satisfactory reason, from the failure of authorities or some other cause, why it must always remain unanswerable. But if the greatest math­ematician that ever flourished, if Laplace, if Newton himself, were to hint one word on the greatest of his own dis-

In the mathematics, the higher one advances, the

more widely does the horizon open around him.

coveries, and the superiority of his own powers, it would require very little knowledge either of Geometry or Physics, to stop all such boasting by the most conclusive argument that could be urged against it, the proposi­tion of one of the numerous problems whose solution has not yet been dis­covered. Here would be no possibility

26 THE MATHEMATICAL INTELLIGENCER

of evasion, no opportunity of throwing the failure upon any external source; the only alternative would be, a solu­tion of the question proposed, or a con­fession of inability. Nor are such ques­tions difficult to be conceived.

Every one who [35) is acquainted with the mathematics is aware that the higher he advances, the more widely does the horizon open around him, that for one difficulty which he is able to conquer, a hundred remain to baffle his utmost efforts, that should he even suc­ceed in discovering the clue to one of these, that very success would create matter for new enquiries and furnish fresh ground for the confession, that af­ter all the accumulated labors of three thousand years, in spite of all the bril­liant discoveries which have kept alive the admiration of the world for the last two centuries, human knowledge is still as limited in its extent, as small in its proportions as are the atoms in a sun­beam in comparison with the sun itself. I do not mean to assert that a different conclusion would be derived, from lit­erary or other pursuits, because I am well aware the tendency of all knowl­edge is to bring the mind to a sense of the feebleness of its own powers, when measured by the magnitude of the ob­jects on which they should be exerted, but I will maintain, that a science which has no perceptible limits, except those which bound the human intellect, is better adapted [36) to shew its follow­ers how circumscribed the range of that intellect really is, and less likely to cre­

ate extravagant notions of human at­tainments than others, in which the im­possibility of any further progress, may be thrown on the nature of the case, and the unconquerable difficulties of the subject by that desire of hiding from himself his own defects, which is in­herent in the constitution of men. The probability, nay, the absolute certainty, that a future age will far surpass this one in scientific discovery can never fail to strike every one who is competent to look at the present state of the sci­ences. This is the conclusion at which the most eminent philosophers have sooner or later arrived. Throughout their writings may be observed, (and never more distinctly than in those of the latest era) a spirit of anticipation which, in the midst of the most suc­cessful exertions, looks forward to the

time, when the results of their labors, instead of shining with their [37) pres­ent unrivalled brilliancy, shall be con­spicuous among many others of equal magnitude, which will then adorn the firmament of science. If their anticipa­tions should be realized, if Newton should be the Archimedes of a future generation, whose discoveries as much exceed his own, as they in their tum surpass those of the Grecian philoso­pher, the enlightened of that age will still have a task before them, which will check all feelings of arrogance or self­sufficiency, and which will convince them, much more effectually than if they had remained in ignorance, that the universe of knowledge is as bound­less as the universe of matter.

This it may be allowed, is perfectly undeniable, and yet the supporters of the original objection may contend that they do not mean to apply their obser­vations to the mind which has fathomed the utmost depths that can be sounded by the plummet of discovery, but that the great mass of those who are in­structed, and whose attainments cannot be expected to rise above mediocrity, are exposed to the fatal errors above mentioned without the corrective influ­ence which the view from the summit cannot fail to exert. They argue [38) that the reason when daily treated as an ab­solute monarch, and made the arbiter of all the questions which education brings before the mind, will act the tyrant, and endeavor to reduce territo­ries which were never meant to come

under its sway. Weak as is the founda­tion of this argument, it is derived en­tirely from an erroneous notion of the manner in which the elements of math­ematical knowledge are conveyed to the student. This error consists in the supposition that the student himself, in every stage of his progress, is the judge of truth or falsehood, in all the ques­tions to which his attention is directed; that his reasoning faculty, informed as it is, becomes the instructor and not the pupil. This is very far from being the case. I do not deny that it is highly de­sirable that the student should if possi­ble, see the foundation and appreciate the accuracy, of all the reasoning whose steps he is required to follow. To en­able him to accomplish this, should be the main object of every instructor; but should cases arise (as they will some-

times) in which the pupil, after atten­tive examination, cannot feel convinced of the truth of any proposition or fol­low the reasoning [39) which has led to it, should he on that account allow his own judgement to turn the scale against his instructor, and form a decision on the point in question? His duty in such a case has been well prescribed by one of the most illustrious members of the fraternity of science, D'Alembert, a man equally distinguished as a mathemati­cian, a metaphysician and a natural philosopher. His authority on this point is peculiarly valuable, because, to him we owe it, that the difficulties of whose nature I am now speaking, are much less numerous than they were before his time. Had he possessed the self con­ceit which has distinguished some of his inferiors, he would have said, I have cleared the paths of mathematics from the obstacle which formerly presented themselves, whenever therefore the stu­dent meets with any unconquerable dif­ficulty, let him doubt the proposition which has caused it. But what really was the opinion of this illustrious philoso­pher upon the course to be pursued in such circumstances. It was delivered in the following emphatic words "Go for­ward and faith will follow." He recom­mends the beginner in all his embar­rassments, [40) to rely on the word of his instructor, and to proceed with con­fidence, and he assures him from ex­perience, that the time will come, when, on a renewal of his exertions, he will find that the difficulty has vanished. There is a period at which the student becomes a proficient, and may fairly constitute himself a judge of the truth or falsehood of the results of his previ­ous education, but during the first years of initiation, the Pythagorean maxim of silence is the only one whose adoption will ensure efficient instruction, or ren­der success certain.

The data of ordinary life are of a doubtful cast, no ways resembling the

certain axioms of Geometry.

The last objection which I shall no­tice, is one which though it has been brought against the mathematics, ought to have been directed against the ex-

elusive study of them. It bears in no de­gree upon these sciences themselves, nor should I have mentioned it, but for the opportunity it will afford of re­minding the student, that the mere study of mathematics alone, is inadequate to make a finished, or even a competent reasoner. It is urged with great truth that the data of ordinary life are of a doubt­ful and disputable cast, no ways resem­bling the certain and simple [41) axioms of Geometry: that mathematical reason­ing is not of that nature which will be required in the business of the world: that the distinguished mathematician has sometimes shewn a childish degree of incompetence, when endeavoring to reason on the common occurrences of life, or on the political event respecting which every man of education is ex­pected to be able to deliver his opin­ions in a sensible and luminous man­ner. This is all in a great degree correct, and it is no invalidation of the arguments which I have urged in favor of mathe­matics as a part of education, to assert, that did they constitute the whole, the march of education and that of intellect would be no longer, as they are at pre­sent, absolutely identical. Those who have noticed the course of Instruction marched out by the Council of this Uni­versity, will perceive that there is no cause of apprehension of its cramping the powers of the mind, by too exclu­sive an attention to any particular branch of knowledge. In placing mathematical studies at the beginning of the Course, a direct and indirect application of them is contemplated, [42) the latter of which, though not equally obvious, is no less important than the former.

The direct application of mathemat­ics is plainly perceivable, in the Studies of Nat. Philosophy, Astronomy, and Chemistry. In the latter its utility is great and daily increasing, it is the sole key to any effectual attainment of the two for­mer. Between the investigations of Nat­ural Philosophy and those of Mathemat­ics there appears a marked difference, arising from causes which free the for­mer science from the objections which I have asserted to exist against the exclu­sive study of the latter. The axioms of mathematics are as I have stated evident of themselves, those of Natural Philoso­phy are deduced from experiment and induction. A new species of exercise for the mind is thus introduced, namely the

balance of evidence for and against con­tested propositions. Absolute certainty is no longer held to be necessary to the admission of a principle, a high degree of probability is sufficient, but the axiom is subject to rejection, if the [43) results to which it leads are found to be untrue. All these circumstances constitute the in­vestigations of Natural Philosophy nearly akin to those of less exact sciences and under them peculiarly fit to become the step by which the mind may safely de­scend from the certain demonstration of mathematics, to the extensive induction, and balance of probabilities whence arises the evidence of truths of meta­physics, of jurisprudence, or of political economy.

The demonstration of the indirect utility of mathematical habits would lead me into too wide a discussion, to do justice to which would require the most varied and extensive knowledge. I will not therefore take up your time with endeavoring to prove, what is pretty generally admitted, that wherever sophism is to be exposed or the strength of strong reasoning duly ap­preciated, wherever previous formed habits of abstraction [44) and general­ization are valuable, the preparation of mathematical studies is useful in the highest degree. This opinion has gained ground daily, and what better proof of its increasing power can be given, than the fact that in a University the first in­stance of one founded in this immense metropolis for the promotion of useful, practical, education, the mathematical sciences have been selected to form a part of the foundation of the truly mag­nificent superstructure of knowledge which it is proposed to erect.

[45] I now proceed to some account of the manner in which I intend to dis­charge the duties of the office which has been entrusted to me, and if I do not express at length my sense of its re­sponsibility, it is because I consult your time, and not my own feelings. The prospectus of the Mathematical Course which has already been published con­tains a detail of the subjects which will occupy the two years allotted to these studies. Of this prospectus I need only say, that I shall not consider myself bound to carry the class through the whole of what is contained in it if it shall appear that their interests will be more effectually consulted by my con-

© 2006 Springer Science+ Business Media, Inc. , Volume 28, Number 3, 2006 27

fining myself to the more prominent parts of it. The course [46] there marked out is of great extent and variety; it re­mains to see whether the time allotted to it can be made sufficient; but I shall certainly not proceed with any new branch of the subject, until I am thor­oughly satisfied that the class has ac­quired a competent knowledge of the preceding ones.

It is my intention to devote at least half of the [class}

time to examinations written & oral.

Some misconceptions have prevailed with regard to the nature of the in­struction to be given in this University, on account of the use of the term Lec­ture, which considering the meaning generally attached to this word, is in some respects inappropriate. Oral in­struction will not form the sole duty of any Professor in this place, least of all of them whose Departments belong to general education. [47] It is my inten­tion to devote at least one half of the time allotted to me, to examinations both written & oral, and in the Junior Class the proportion will probably be greater. The attendance on these ex­aminations will be imperative on all those who desire a Certificate at the end of the Session, as they furnish the prin­cipal means by which I can judge of the proficiency of any pupil. On the re­

sults of the weekly examinations in writ­ing which I propose to institute, will depend in a great measure, the value of this certificate, as much perhaps, as on the final examinations at the end of the Session. The advantage of this arrangement is evident, for were the lat­ter the only criterion, the student might be [48] disposed to neglect some Lec­tures, with the expectation of recover­ing lost time before the end of the Ses­sion; this the present arrangement will render unavailing as every week will have its value: at the same time were there no final trial of the Student's progress, a powerful stimulus would be wanted to induce him to keep up and retain the information which he has ac­quired from week to week.

28 THE MATHEMATICAL INTELLIGENCER

The latter part of every Lecture will be devoted to oral examination in the subject which has preceded. The stu­dents will here be invited to state any difficulty which they may feel and I rec­ommend it to all of them, particularly the younger ones, to over-[49] come any reluctance which they may feel to thus publicly expressing what may have ap­peared obscure since, independently of other advantages, the (onerous) attempt to put into precise language the sub­stance of their difficulty, will frequently shew of itself from what cause the mis­conception has arisen.

At the close of each lecture I intend to deliver to the class, some exercises for their leisure hours, which will con­sist mostly of simple deductions from the Theorems which they will find in their Books. From the answers to them, which will be delivered to me in writ­ing at the next meeting of the class, I shall obtain the best materials for future examinations. [50] Considerable obsta­cles generally present themselves to the beginner, in studying the elements of Solid Geometry, from the practice which has hitherto uniformly prevailed in this country, of never submitting to the eye of the student, the figures on whose properties he is reasoning, but of drawing perspective representations of them upon a plane. The council of the University with that desire of procuring every thing necessary to the most efficient system of instruction which has distinguished them through­out, has placed at my command, the

means of constructing such apparatus, as will be [51] sufficient to remove this difficulty, and I hope that I shall never be obliged to have recourse to a per­spective drawing of any figure where parts are not in the same plane.

The description of persons who de­sign to study Mathematics in this uni­versity may be divided into two classes, the first composed of those who are commencing the course of education recommended by the Council, and for whom it is necessary that the hours de­voted to this pursuit should not inter­fere with those appropriated to Classi­cal Literature, the second consisting of such as are desirous to add to their pre­viously acquired [52] knowledge, the el­ements of a scientific education, to

whom it is also indispensable that they should be able to attend the classes of Natural Philosophy and Chemistry. To accomplish this double object, it has been proposed to open two Mathemat­ical Classes, the first from twelve to half past one, intended for the latter class of students, the second from half past two to four, for the former. I should there­fore wish all who are [in which] not studying the Greek & Roman Languages. As the class which meets first will prob­ably be composed of older students whose faculties are more developed [53] and whose attention is more particularly devoted to the Subject they will proba­bly proceed somewhat more rapidly and I may be able to direct their attention to some points which I could hardly ex­pect the younger students to under­stand. It is I hope unnecessary to say, that in all Lectures addressed to the Reg­ular Students of the University, my ob­ject must be to form the mind to habits of reasoning, and without reference to the qualifications for any profession, or the wishes of those to whom a partial Mathematical education may have made a repetition of the first principles un­necessary.

[54] I have thus gentlemen laid be­fore you the reasons which induce me to uphold the Mathematics as a branch of Education, and my views as to the best method of rendering them efficient in this respect. That which is incom­plete in these views will be shewn by time & experience, & I hope that these reasons will be reinforced by argu­ments drawn from the success of the system of education to be adopted in this university, of which the sciences form so distinguished a part. And why should we not hope for this success? Why may we not anticipate the day when it will be the boast of many [55] who are the objects of admiration in public, and of respect in private life, that they were educated in the Univer­sity of London? Why should we not look forward to the time when some portion of that ray shall illuminate these walls, which is reflected on the exist­ing academical institutions of Great Britain by the perspicuity & elegance of Playfair or Paley, the profound re­search of Locke, or the unrivaled dis­coveries of Newton.

M a thern a tic a l l y B e n t

The proof is in the pudding.

Opening a copy of The Mathematical Intelligencer you may ask yourself

uneasily, "What is this anyway-a

mathematical journal, or what?" Or

you may ask, "Where am I?" Or even

"Who am I?" This sense of disorienta­

tion is at its most acute when you

open to Colin Adams's column.

Relax. Breathe regularly. It's

mathematical, it's a humor column,

and it may even be harmless.

Column editor's address: Colin Adams,

Department of Mathematics, Bronfman

Science Center, Williams College,

Williamstown, MA 01267 USA

e-mai l : Colin.C.Adams@williams.edu

Colin Adam s , Editor

The Cohomo logy of Proofs BY COLIN ADAMS AND

STEVEN G. KRANTZ

o you wake up in the middle of the night, sweat soaking your pa­jamas, sheets in a bunch, wony­

ing about whether the proof of your lat­est Lemma 4.3.7 might contain an error that could bring your entire oeuvre to its knees? Do you fear that the proof-check­ing theories of Babai are too complex for you to have applied them correctly in your latest paper? Do you have a nightmare wherein some National Acad­emy Mandarin stands up in the middle of your lecture at Harvard, thumbs his nose, and stalks out of the room?

Well, return to your slumbers. Yes, you can finally shelve the Ambien. There is now a solution for your trou­bles, a panacea to soothe your wrinkled brow. And it comes from an unlikely source-none other than the Eilenberg­Steenrod Axioms!

You may already be familiar with how these humble generators of ho­

mology and cohomology theories can spice up a dull dinner party. Many is the time you may have used E-S to en­able idle chitchat during a long subway ride. And what about their incisive ap­plication during a boring calculus lec­ture?

But today we are talking about their use in the new theory of the cohomol­ogy of proofs. Yes, the cohomology of proofs is an amazing device for detect­ing holes in your proofs. And it will de­tect a hole of any dimension! What is more, the Steenrod cohomology oper­ations allow you to combine holes, patch up holes, and perform surgery on holes. What you can do with Poincare duality almost defies description.

Of course a one-dimensional hole in your proof is an all-too-familiar hap­penstance-this is just the classic in­stance of sticking your entire foot in your mouth, and it is easy to imagine a co-boundary encircling that ankle. But now we can discern higher-dimensional gaps in your reasoning, indeed ones that are impossible to imagine philo­sophically, or to describe using accept­able anatomical imagery.

How could such a paradigm-shift possibly work? The answer is simplic­ity itself. Let us define the first Godel­Tarski (G-T) cohomology group. To start, we identify all the lines of rea­soning in your paper that have neither beginning nor end. They dribble (as if by osmosis) into existence, set up a life­support system, and then ooze hack out of focus into some fractal netherworld. These are the cochains. An important submodule of the cochains is the set of coboundaries: the instances of circular reasoning. Coboundaries are often eas­ily identified, as they usually enclose nuggets of nonsense. Now we just mod out the cochains by the coboundaries and-voila!-the result is the first G-T cohomology group of your paper. The higher-dimensional G-T cohomology groups involve more convoluted in­stances of self-referential reasoning, and we cannot treat them here. But now we have the wherewithal for creating the entire cohomological algebra of your

latest paper. In the case that the groups-and therefore the algebra­are all trivial , you have a paper with no holes, the logical equivalent of one spicy meatball . But even if your paper has holes the size of Skewes number and wouldn't be published by Popu­lar Mechanics, you can still write an­other paper about calculating its co­homology groups, and publish that in the Annals.

Of course applying topological con­siderations to a mathematical treatise is not a new idea. Consider if you will the Pythagoreans' passion for the number 8. This is in fact based on a calculation of the first Godel-Tarski Betti number of Pythagoras's initial incorrect proof of

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 29

the Pythagorean theorem. It is a little­known fact that the Pythagoreans dis­covered the concept of Godel-Tarski cohomology and then accidentally mis­placed it for the next two millennia (c.f. [Cangaloop] . ) Their discovery is partic­ularly impressive when one considers that modules were not invented for an­other 1900 years.

In the latter part of the nineteenth century, significant effort was expended on elucidating the topology of mathe­matical papers, (c.f. [Gauss], [Mobius]). Definitions of connected open subsets of a mathematical paper, the possible metrics that the topological structure could sustain, and their rigidity were all well understood. Quite a few careers were destroyed upon the realization that a given paper had infinitely many path components. But it wasn't until Weier­strauss proved by construction that a se­quence of subsequent drafts of a given paper might converge to a final version that the entire mathematical community stood up in unison and took notice.

All of this work culminated in 1897 with the proof by Poincare [Poincare]) that the set of all theorems is open in the Publish-or-Perish topology. This amazing result heralded the beginning of a golden age in mathematics, allow-

ing as it does the publication of infi­nitely many slight variations on any given idea. However, it represented the death knell for the field of topological proof theory, convincing many practi­tioners that there were no worthy prob­lems left to pursue. This area of research fell into disfavor for over 100 years.

Since that time, the available topo­logical tools have expanded dramatically. Now one can pursue the properties of the resulting cohomological algebra it­self, with no concern over the implica­tions for the original paper whatsoever. Opportunities are virtually boundless.

However, we close with a warning. There is a temptation to use the theory of G-T cohomology groups to generate an infinite stream of papers from a sin­gle pseudo-trivial idea. This would be a mortifying mistake. For the Hilbert­Einstein stability theorem tells us that the cohomology algebra eventually sta­bilizes: If paper k + 1 is iteratively written to calculate the cohomological structure of the arguments in paper k, then, eventually, after some k = k,;, the results will be equal in all subsequent papers. Considering the backlog at most mathematical journals these days, this is certainly a state of affairs for which we can all be grateful.

MOVING? We need your new address so that you

do not miss any issues of

REFERENCES

1 . Adams, C. C. , and Krantz, S. G . , The co­

homological dimension of "The Cohomology

of Proofs, " Journal of Topological Proof The­

ory and its Ramifications, to appear.

2. Adams, C. C. , and Krantz, S. G . , The co­

homological dimension of "The cohomolog­

ical dimension of 'the Cohomology of

Proofs, ' " Topology, to appear.

3. Cangaloop, C. F . , Despair at the Pythagorean

School: Who Lost the Cohomology? Bulletin

of the Topological Proof Theory Historical

Society Vol. 227 (2003), 1 24-236.

4. Gauss, Karl Friedrich, Disquisitiones De

Operum Mathematicorum Geometria Situs,

Journal fOr Reine und Angewandte Mathe­

matik 70 (1 865) 1 1 6-1 48.

5. Mobius, A.F., Ober die Nicht-orientierbarkeit

eines Artikels von Gauss, Abhandlungen

Sachsischer Gesel/schaft der Wissenschaften,

1 866.

6. Poincare, J . Henri, Methodes nouvelles de

Ia topologie des papiers mathematiques,

Bulletin Societe Mathematique de France,

Vol. 29(1 897), 251 -255.

Steven G. Krantz Department of Mathematics Washington University in St. Louis St. Louis, MO 631 30 e-mail: sk@math.wustl.edu

THE MATHEMATICAL INTELLIGENCER

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30 THE MATHEMATICAL INTELLIGENCER

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llft@U.t§ .. fhi£11Q.i .. i,iii.ilh¥J Marjorie Senechal, Editor

M ath and Art i n the M ou ntai ns

DORIS SCHA TTSCHNEIDER

This column is a forum for discussion

of mathematical communities

throughout the world, and through all

time. Our definition of "mathematical

community" is the broadest. We

include "schools" of mathematics,

circles of correspondence,

mathematical societies, student

organizations, and informal

communities of cardinality greater

than one. W'bat we say about the

communities is just as unrestricted.

We welcome contributions from

mathematicians of all kinds and in

all places, and also from scientists,

historians, anthropologists, and others.

Please send all submissions to the

Mathematical Communities Editor,

Marjorie Senechal, Department

of Mathematics, Smith College,

Northampton, MA 01063 USA

e-mail: senechal@minkowski.smith.edu

The name 'Banff' floods memory with jagged mountain peaks, clear air scented with pine, startling

blue sky, rustic lodging, a leading cen­ter for the arts and conference center, and last summer, an unforgettable gath­ering dedicated to connections between mathematics and the arts. The confer­ence "Renaissance Banff: Mathematics, Music, Art, Culture" July 30-Aug. 3 , 2005, was the eighth annual Bridges conference, and was also the first in­terdisciplinary conference on mathe­matics and the arts to be held at the famed Banff Centre. Sponsored by the Pacific Institute for the Mathematical Sciences (PIMS) and the Canadian Mathematical Society and organized by Reza Sarhangi (the leading organizer of all Bridges conferences) and Robert V. Moody (the first Scientific Director of the Banff International Research Station [BIRS], a North American version of Oberwolfach located at the Banff Centre and a component of PIMS), Renaissance Banff brought together a diverse group of about 200 participants from the US, Canada, and abroad.

When Moody accepted the director­ship of BIRS in 2002 (postponing early retirement from the University of Al­berta), it was his dream to have not only high-level mathematics workshops, but also to integrate BIRS with the Banff Centre. The first move was to hold cre­ative writing workshops in math and science, and the Bridges Conference was the second, "Renaissance" in the conference title captured its objectives: "to foster a new era for drawing the worlds of the arts and the sciences closer together." At this event, mathematicians, computer scientists, artists, poets, writ­ers, educators, musicians, cultural histo­rians, dramatists-and many fit more than one of these categories-gathered to celebrate and explore connections be­tween mathematics and the atts. The Bridges conferences are normally three days, but this special event had an added "Coxeter Day" to celebrate the legacy of Canada's legendary geometer.

With "art" interpreted in its most broad sense (the arts) , you might ask if this conference was about seeing math in art, or utilizing math in the creation

The instigators Robert V. Moody and Reza Sarhangi, shown in front of part of the

exhibit of Coxeter memorabilia.

© 2006 Springer Science+ Business Media. Inc., Volume 28. Number 3, 2006 31

Assembling the Zome-Tool models.

of art, or using math to analyze art, or displaying mathematical objects as art, or illuminating mathematics through art (or vice-versa), or teaching mathemati­cal ideas using art (or vice-versa), or other possible connections between math and art, and the answer would be "all of the above." Each day there was a smorgasbord of talks and workshops, punctuated with breaks for viewing the art exhibition or visiting the display room filled with models, puzzles, math­ematical toys, and works in progress; each evening offered a special event. In the lobby outside the lecture hall, there

were always a few children, partiCI­pants, and spouses seated or kneeling on the floor, busy joining 3,720 lome­Tool connector balls to 10,680 struts in modules that each hour were added to two growing monster models, shadows of the 4D "cantellated" 600-cell. David Richter and Daniel Duddy orchestrated the project. And the first-class smorgas­bord of food at every meal in the airy dining room, whose walls of windows framed the mountain view, threatened to send us all home 10 pounds heavier.

Invited talks on the opening day, given by mathematician Bart de Smit

DORIS SCHATISCHNEIDER is Professor Emerita of Mathematics at Moravian College where she taught for many years. Most of her nu­merous publications and lectures have concerned links between math­

and art, and between mathematics and artists. Thus her book M. C. Escher (Abrams, 2004) and the book of essays on Escher

she co-edited (Springer, 2003). She was a member of the pro­ject which created the popular software The Geometer's Sketchpad. She loves travel, and walking in the mountains.

2038 Sycamore St. Bethlehem, PA 1 80 1 7 e-mail: schattdc@moravian.edu

32 THE MATHEMAnCAL INTELLIGENCER

("The Droste-Effect and the Exponen­tial Transform"), philosopher-poet Jan Zwicky [3), and writer Siobhan Roberts [4), aptly illustrated some of the diverse ways in which connections between art and mathematics are manifest. De Smits's talk filled in details of his Lei­den University teams' mathematical ad­venture of "completing" M.C. Escher's Print Gallery, Zwicky compared a writer's use of metaphor in poetry and prose to a mathematician's elucidating a proof through a visual model, and in her evening public lecture, Roberts, bi­ographer of H.S.M. (Donald) Coxeter [4), gave a lively account filled with per­sonal anecdotes and highlights of the geometer's life, including his interac­tions with artists. In the lobby, partici­pants could browse through an array of photos and other memorabilia of Cox­eter's life contributed by his daughter Susan Coxeter Thomas and the Coxeter archives at Robarts Library, University of Toronto. The exhibit was arranged by Coxeter's son Edgar, who manned the exhibit during the entire conference

and fielded any questions (Susan had been present for the Coxeter Legacy symposium in Toronto a year earlier (2] , but she was unable to attend this event).

One highlight of opening day was a 'barn-raising' by about 25 volun­teers, who were initially faced with sep­arate piles of three different patterns of laser-cut birch pieces resembling boomerangs--60 identical pieces of each type, 180 in all. In less than an hour, under the guidance of George Hart, they had assembled the pieces without glue or screws to form his large spherical sculpture "Spaghetti Code. '' Later, Hart also gave a workshop on making paper 'polylinks' , one of the seven participatory workshops de­signed to actively engage teachers (and others), providing ideas for the class­room. Other speakers addressed the topics of playing mathematics and do­ing music, the Platonic solids, aspects of tiling and patterns, a thousand cranes and statistics, and connecting gross-mo­tor movement, dance, and mathematics.

The conference program was packed-each day there were two or three hour-long plenary lectures, and except for the leisurely 2-hour break for lunch, the rest of the day was filled with three parallel sessions of 20-minute talks-60 short talks over four days, many by artists whose work was on ex­hibit (and these in competition with the workshops) . Always there were pic­tures, models. and music-the techni­cians of the Banff Centre were con­stantly (competently and cheerfully) busy sorting out technical snafus for presenters who brought every possible device; somehow they always got them working with the Centre's equipment in time for presentations.

Although many of the themes of the talks were familiar-symmetry (and symmetry-breaking), proportion, and scale in art; harmony, scales, meter, and syncopation in music; mathematical transformations in art (anamorphosis, perspective); tilings and patterns; as­pects of geometric and topological

form; recursion in mathematics and an; the use of art in the teaching of math­ematics-there were often surprising new connections and insights offered. Most talks exemplified the different kinds of math-arts connections men­tioned earlier. A few titles serve to il­lustrate.

Mathematics creates art. Talks on this theme included the familiar use of fractals to create artistic images ("Re­cursion in Nature, Mathematics and Art", Anne Burns) as well as new techniques such as altering traditional polygonal tilings to create intricate artistic patterns ("Spiral Tilings with C-Curves: Using Combinatorics to Augment Tradition," Chris K. Palmer) . The cover of the con­ference program featured a Celtic-like labyrinth of repeating whorls produced by Palmer using this technique. That mathematics sometimes creates art un­expectedly was vividly demonstrated in "Aliasing Artifacts and Accidental Algo­rithmic Art" by Craig Kaplan, whose computer presented him with an inex-

The display room. Hart's assembled Spaghetti Code; also Termesphercs (back of room) and various models on tables.

© 2006 Springer Science+Susiness Media. Inc . . Volume 28, Number 3. 2006 33

haustible series of intricate abstract works of art (rather than an expected blank page). This was the result of an algorithm that computed distances in order to generate Voronoi diagrams; the computer's limited precision in repre­senting numbers led to aliasing, which created patterns like fine Persian car­pets .

Mathematics is art. In his talk, "Split­ting Tori, Knots, and Moebius Bands," Carlo H. Sequin showed a seemingly endless variety of beautiful sculptural forms, all realizations of topological variations found by splitting pure geo­metric forms. His forms first see life dig­itally, and then are often produced in "real" media, from computer-produced models (made on rapid prototyping ma­chines) to enormous snow carvings (in the annual snow-sculpting competition in Breckenridge, Colorado). In "Mosaic Art: From Pebbles to Pixels" Irene Rousseau explained how a purely math­ematical entity-a hyperbolic tiling-is interpreted (rather than accurately rep­resented) by a mosaic artist.

Mathematics renders artistic images. In reproducing black-and-white images, usually a screen of dots ("halftone") produces the levels of light and shadow. In their talk "TSP [Traveling Salesman Program to render] Art," Robert Bosch and Craig Kaplan explained how a con­nected single curve, following a pre­scribed path, could accomplish the same effect while additionally provid­ing interesting texture to the image. In "Symmetry, Proportion and Scale: Tools for the Jacquard Designer and Weaver," Barbara Setsu Pickett reminded us of the importance of the most fundamen­tal geometric tools used in design, es­pecially essential in weaving on a loom.

Art inspires mathematical investiga­tion. Planar graphs formed by edges and vertices of certain polygonal tilings were the origin of Rinus Roelof's fruit­ful investigation of "Three-dimensional and Dynamic Constructions Based on Leonardo Grids." Named for the Re­naissance master whose notebooks contained the only sketches of similar grids, these assemblages of overlapping sticks led to dome systems, layered 3D structures, and curved sculptures of interwoven components. Conversely, Mathematical investigation inspires art: John Sharp's talk, "D-forms and Devel­opable Surfaces," featured many in-

34 THE MATHEMATICAL INTELLIGENCER

triguing sculptures that resulted from exploring a single mathematical ques­tion: What happens when two simple closed planar forms that have the same perimeter are joined at their boundaries? A simple case is joining two ellipses so that the minor axis of one meets the ma­jor axis of the other-a baseball-like join-in which the seam is a bit like a saddle.

Hidden mathematics can be discov­ered in art. In each of their talks, Robert Moody ("Alice Boner and the Geome­try of Temple Cave Art of India") and Marcia Ascher ("Malekula Sand Trac­ings: A Case in Ethnomathematics") made clear the difficulty, and pitfalls, of reading mathematics into art. Yet each revealed the hidden (and documented) mathematical structure of works of art. Moody outlined how Alice Boner (1889-1981) had conjectured that a hid­den scaffold of three strategically-placed circles was the main element of com­position in Indian Temple cave art (a theory first rejected by critics, but later confirmed by early records). Intrigued by this, he made his own study of her important painted triptych and showed that she had also been guided by this geometric device. Ascher carefully doc­umented how intricate and highly sym­metric Malekula sand tracings of a sin­gle continuous line were guided by algorithms which directed how an ini­tial design was to be continued through repetition of transformed copies. The repetition could be a horizontally or vertically reflected copy, a rotated copy (by a multiple of 90°), or a "backwards" tracing of the initial design-in the end, each algorithm produced a flowing symmetric closed path. However, As­cher cautioned, although we know how they did it, we do not know what they were thinking as they did it.

Mathematics analyzes art. In her talk "The Complexity of the Musical Vo­cabulary of the Nzakara Harpists, " Bar­bara Gregory applied several different mathematical measures of complexity to a piece of harp music, each an attempt to evaluate the amount of movement of a harpist's hands required in playing the piece. In a similar vein, F. Gomez, A. Melvi, D. Rappaport, and G.T. Toussaint ("Mathematical Measures of Syncopa­tion") proposed a new measure that at­tempts to capture the essence of syn­copation, and compared it with other

mathematical measures that use combi­natorics or group theory. How a sym­metry analysis of designs can reveal cul­tural preferences was discussed by Donald Crowe ("Geometrical, Percep­tual, and Cultural Perspectives on Fig­ure/Ground Differences in Bakuba Pat­tern") and Duncan J. Melville ("Aspects of Symmetry in Arpachiyah Pottery").

Mathematical ideas can be taught through art. In his talk "Looking at Math: Using Art to Teach Mathematics, " Pau Atela gave examples of how works of art (by both Renaissance and con­temporary artists) are a starting point to learning mathematical concepts in his courses at Smith College. Using these, students develop computer simulations, make complex computations and geo­metric analyses, and carry out hands-on model construction projects. Con­versely, Art can be understood through mathematics. Michael Frantz, known for his courses and teacher workshops that demystify the mathematics of lin­ear perspective in drawing and paint­ing, chose to demystify the curious dis­torted art known as anamorphosis. In "A Perspective on Infinity: Anamor­phism and Stereographic Projection," Frantz addressed different types of anamorphism, and offered the sugges­tion that stereographic projection (be­tween plane and sphere) might offer an­other interesting type of anamorphism for art.

Many of the talks showed models and forms that begged to be held, ex­amined close up, and manipulated. Speakers placed informal displays on ta­bles around the workshop room for ex­actly that purpose and invited hands-on play. Greg Frederickson's piano-hinged wooden puzzles were flipped and folded into different forms. Rinus Roelofs's ny­lon computer-produced models of in­terlocked polyhedra, mysterious poly­hedral dissections (one a notably "screwy" tetrahedron that twisted along a spiral path to disassemble into two identical parts), and notched sticks to build 'Leonardo grid' domes had to be handled to be understood. Michael Longuet-Higgins's "Rhombo" magne­tized blocks (golden rhombohedra) were arranged and rearranged countless times (often by children). George Hart's models of "polylinks" (some produced by his own computer software) and im­possibly intricate puzzles based on

Greg Frederickson's heart-ellipse transformer.

them asked to be turned again and again to figure out how they hung to­gether. Origami fractals, 'spidron' mod­els, and amazing folded tessellations in­vited close-up examination. And you could stand beneath Dick Termes's painted spheres and be mesmerized as they slowly turned, revealing their 360-degree scenes.

The art exhibition, organized by Robert Fathauer and held in a gallery adjacent to the main lecture hall, fea­tured 25 artists and 90 art works dis­playing a plethora of media and styles. Three-dimensional works were fash­ioned from paper (hand-cut and glued, or folded and joined), plastic (woven in strips), wood, clay, metal, and cloth. Most artists interpreted familiar geo­metric forms (polyhedra, skew polyhe­dra Mobius band, and knot) in new ways, and yet there were some sur­prising forms that defied classification. Bradford Hansen-Smith's sculptures, produced by folding and interleaving common paper plates, suggested grace­ful backbones and spiraling shells. Two-dimensional works included drawings, paintings, and prints on flat and on spherical surfaces (done by hand, or by photographic techniques,

or computer-rendered), fabric quilts, a Jacquard-loom handwoven pattern, and glass and marble tile mosaics. The location of the conference attracted

some western Canadian artists partici­pating in a Bridges conference for the first time. The colorful paper geomet­ric models of Robert Stowell , a sculp­tor from Calgary, Alberta, were cut and assembled with such precision that it was hard to believe that these were real forms, not digital holograms. The geo­metric quilts fashioned by Gerda de Vries, an applied mathematician at the University of Alberta, were guided by her approach to structural design. All works in the exhibition can be viewed in a virtual gallery mounted by Fathauer at www.mathartfun.com/shopsite_sc/ s tore/h tml/BridgesO 5 /Bridge sEx­hibit2005.html.

Drama and music provided two spe­cial evening events. On Monday, three participants gamely joined four profes­sional actors in a reading of the musi­cal-theater work "Delicious Rivers" by noted playwright Ellen Maddow, pro­duced by her husband Paul Zimet (they are cofounders of The Talking Band in New York City). This presented the hi­larious predicament of several harried folks trapped in line in a post office run by a dysfunctional staff. Jarring music by a trio of trombone, bass fiddle, and bass voice connected the scenes; sub­

tle repeated lines ("it's the same, yet it's not the same") and a strange postal clerk were oblique references to Robert Ammann and his work in aperiodic

Rinus Roelofs's "screwy tetrahedron" puzzler.

tiling. Actors' lines and gestures also subtly echoed one another through re­flection, rotation, and translation. This staging was a first test of readiness for an off-Broadway run at La MaMa Ex­perimental Theater in January and Feb­ruary, 2006. And a most suitable venue for the play's first steps, since it was conceived by Ellen at the first BIRS cre­ative writing workshop in 2003 and nur­tured through a collaborative effort with mathematician Marjorie Senechal. A mu­sical evening wrapped up the confer­ence at the end of Coxeter Day (more below).

To ensure that everyone could en­joy (guilt-free) the beauty of the sur­rounding Canadian Rockies, the after­noon of the third day was reserved for an excursion to two spectacular moun­tain lakes, Moraine Lake and Lake Louise. Although in the gallery, the frac­tal-generated mountain scene by Anne Burns could easily fool the eye into be­lieving it was real, there was no mis­taking that, on this outing, we were en­veloped by the real thing. As our bus traveled to Moraine, we had the excel­lent luck to see, grazing close to the road, a prize elk buck sporting a large rack of antlers. On this afternoon, na­

ture provided all the art-turquoise lakes framed by snow-capped moun­tains and bounded by shaded mean­dering trails. An evening barbecue back

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 35

Bradford Hansen-Smith's sculpture Tendril (Helix Spiral) .

at the Centre capped the day, and con­versations continued late into the evening.

Coxeter Day "Coxeter Day" began with Coxeter­that is, Edgar. Donald Coxeter's son had been an active participant through the previous days, and by now was a fa­miliar face. Edgar opened with some personal observations about his father and family (emphasizing he had not in-

herited his father's prodigious mathe­matical mind), and set the stage for a day in which many sides of Donald Coxeter were revealed.

Christiane Rousseau, past president of the Canadian Mathematical Society and chair of the Scientific Committee for the conference, presided at the four plenary sessions devoted to top­ics that had occupied Coxeter's life, and that had an artistic component. In "Coxetering Crystals" (Marjorie Senechal) we learned how Donald in

A precision paper geometric model by Robert Stowell.

36 THE MATHEMATICAL INTELLIGENCER

his later years had been fascinated with the golden rhombohedra and the possibility of aperiodic tiling of space. His life-long love for symmetry and polytopes in three and four dimen­sions was recognized in "Symmetrical Hamiltonian Manifolds on Regular 3D and 4D Polytopes" (Carlo Sequin). Tiling and spheres came together in "Some New Tilings of the Sphere with Congruent Triangles" (Robert ] . MacG. Dawson) . And the not-sa-well-known lifelong interactions of Coxeter with many artists, both mathematical and mathphobic, was documented in "Coxeter and the Artists: Two-Way In­spiration" (Doris Schattschneider). In

three parallel sessions, short talks cov­ered several other topics that reflected Coxeter's interests: hyperbolic tessel­lation (interpreted in mosaic by artist Irene Rousseau, depicted in Tony Bamford's hooked rugs, explained by Doug Dunham), intertwined hollow polygons (George Hart), transforma­tions (of motifs-Gary Greenfield; of Escher patterns-Joshua Jacobs; of polygons into sculptures-Douglas Burkholder), explosion-implosion of polyhedra (Robert McDermott), the 600-cell (David Richter), and polygo­nal dissections into fractal tilings (Robert Fathauer) .

The last evening of the conference was one of relaxed celebration. It be­gan with a memorable reception (fea­turing a chocolate fountain!) by BIRS and PIMS in honor of Robert Moody,

Two quilts by Gerda de Vries: challenges to perception of regularity, in the tradition of Vasarely.

continued with retirement tributes to him, and ended with a musical melange organized by David Fulmer, billed as "The Last Gathering. " There were pi­ano solos by Robert Craig and cham­ber music by a string quartet formed during the conference (with David Ful­mer, violin, Marian Moody, viola, Jan Zwicky, second violin, and Diana Nut­tall, cello). Trudy Morse read Lewis Car­roll's poem "Jabberwocky" to David Fulmer's violin accompaniment, John Belcher drummed a conga solo, and Rachael Hall played a small concertina, joined by Belcher for an impromptu duet. There would have been more-

there were many talented partiCI­pants-but time literally ran out.

The next day, we all scattered to our destinations, and the day after that, I re­ceived an e-mail from sculptor Rinus Roelofs, who had traveled more than 22 hours to his home in the Netherlands. This said it all : "''m tired, but full of in­spiration."

PHOTO CREDITS

Doris Schattschneider: Figs. 1 -4 and Fig. 8; Ri­

nus Roelofs: Fig. 5; Bradford Hansen-Smith:

Fig. 6; Carlo Sequin: Fig. 7. Christmas City Stu­

dio: Author's photo.

REFERENCES

[ 1 ] Bridges: Mathematical Connections in Art,

Music, and Science, Proceedings 2005. Eds. Reza Sarhangi and Robert V. Moody,

Winfield, Kansas, 2005.

[2] The Coxeter Legacy-Reflections and Pro­

jections. Eds. Chandler Davis and E. W.

Ellers, Fields lnst. Comm. ser. no. 46, Amer.

Math. Soc. , 2006.

[3] Zwicky, Jan, Mathematical Analogy and Metaphorical Insight. Mathematical lntelli­

gencer 28 (2006), no. 2 , 4-9. [4] Roberts, Siobhan, King of Infinite Space:

Donald Coxeter, the Man Who Saved

Geometry, New York, Walker & Company, 2006.

The relevant websites for more information on BIAS and on the Bridges conferences (including the 2006 Bridges London conference) are www.banffcentre.ca/

partnerslbirsl and www.sckans.edu/-bridges/. The Proceedings of Renaissance Bridges (550 pages, with CD Rom) [ 1 ] can be ordered through Robert Fathauer's

website www.mathartfun.com.

© 2006 Springer Science+Business Media. Inc . . Volume 28, Number 3. 2006 37

An E lementary Remark on the Accuracy of Approximations by Regu lar Conti nued Fractions

F. l. BAUER

"Continued fractions are the 'lost sons' of mathematical classroom teaching. They are re­

garded as too advanced for secondary schools and too elementary for universities, as a

result of which they generally fall between the cracks of both syllabuses. "

Petr Beckmann, 19821

Continued fractions have been used since the times of Wallis and Euler for the approximation of ir­rational real numbers, the Euler constant e and quadratic surds like v2 and CVs + 1)/2 being prominent examples. Common lore says that truncated continued fractions give 'best approxima­

tions'. Even scholarly excellent books like the one by Oskar Perron2 do not give practical advice as to what this means, although (for 1 :::::; x < 10) a rather simple quantitative statement like

the number of correct (decimal) places in the approximating quotient for x is (roughly) equal or even larger than the sum of the numerator's and the denominator's number of digits

would be a lore easy to understand and to remember. In fact, the situation . . . even larger than . . . is shown by the following example of the well-known

ordinary (regular) continued fraction for 7T (in Wallis's Tractatus de algebra of 1685)3:

[3; 7' 15 , 1 ,292 , 1 , 1 , 1 ,2, 1 ,3, 1 '14,2, 1 , 1 ,2,2,2,2, 1 ,84,2, 1 , 1 , 15,3, 13, 1 ,4,2,6,6,99, 1 ,2 , . . . ] ,

more conventionally written

1 3 + ---------

1 7 +--------

1 1 5 + ------

1 1 + -----

292 + -1-

1 + . . .

1 Petr Beckmann, A History o f 71'. 5th ed., Boulder, Colorado: The Golem Press, 1 982

20skar Perron, Die Lehre von den Kettenbruchen. Leipzig und Berlin: Teubner 1 9 1 3

3Jorg Arndt and Christoph Haenel, Pi Unleashed. Berlin: Springer 2001 , p. 232

38 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+Business Media, Inc.

For its third approximation

to be compared with

333

106

7T = 3..14150943 . . .

= 3 . 14159265 . . . ,

there are five decimal places in agreement: the normal case, only one less than the sum 3 + 3 of the number of digits in the denominator and 3 in the numerator.

For its fourth approximation

to be compared with

355

1 13

7T = 3.. 1415229203 . .

= 3. 1415926535 . .

there are seven decimal places in agreement, even one more than the sum 3 + 3 of the number of digits in the denominator and 3 in the numerator. And

for the tenth approximation

to be compared with

1146408

364913 7T

= � 141592653591403 . .

= 3 . 141592653589793 . .

there are 1 1 decimal places in agreement, two less than the sum 13 of the number of digits in the de­nominator and the number in the numerator. A closer look shows that actually the disagreement starts with 91 . . . and 89 . . . , rounded to 12 places there is the identical result 3 . 14159265359, which means that one more place, i .e . , 1 2 places should be counted as agreeing! The conclusion is: the plain number of agreeing places is not a trustworthy indicator for the relative error; its use should better be avoided.

Anyhow, the number of agreeing places is not a very precise measure and we should prefer to use the relative error instead. Thus, we take for the fourth approximation

I (��� - 7T)/7T I = 0.84914 . . . X w-7 = w-7·07102 . . . and compare it with

-1- = 0.24928 . . . x w-4 = w-46om . ·

355· 1 13

showing how . . . even larger than . . . was justified here. For the third approximation, however, we have

I (��� -7T)/7T I = 0.26490 . . . X w-4 = w-457692

. . . , and the comparison with

--1- = 0.28330 . . . X 10-4 = 10-4· 54775 · comes much closer. 333·106

Thus, we should better say in these cases the negative decimal logarithm of the relative error in the ap­proximating quotient is close to the sum of the decimal logarithms of the numerator and the denomina­tor. Is this generally true?

By the way, the lore also says that a continued fraction approximation is particularly good if the break is made immediately before a large partial denominator, like 292 in the case of 355

, or 84 in the case of 21053343141 113

6701487259 .

lilii�IJIFJi� FRIEDRICH L. BAUER wrote his doctoral the­sis on group representations (Munich, 1 952), but since then has been known especially for his many contributions to computing. He was one of the developers of the language ALGOL60; with Klaus Samelson, he developed the stack

principle. More recently, he set up the Com-puter Science collection of the Deutsches Mu­

seum in Munich. He is now Professor Emeritus of the lnstitut fUr lnformatik der T echnischen

Universitit Munchen.

Ni:irdliche Villenstrasse 1 9 D-82288 Kottgeisering Germany

© 2006 Springer Science+ Business Media. Inc. , Volume 28, Number 3, 2006 39

Regular Continued Fractions and Error Bounds We use the notation K = [�; � , hz, . . . , bn, bn+1 , . . . ) for converging infinite continued fractions of the form

� + 1

1 b1 + hz + 1

1 b:, + 1 b4 + bs + . . .

and restrict our attention here to 'regular' o r 'simple' continued fractions (called regelmafliger Ketten­bntch by Perron) where all the bi following the semicolon are positive integers, like the one above for 7T.

A Let --....!!:.. = [�;�,hz, . . . , b,J denote the n-th approximation. From elementary theory2 with A-1 = 1 , E_1 = 0,

En Ao = �' Eo = 1 ,

A classical result b y Lagrange 0798) for the absolute error says

and can be derived as follows. We decompose K into a section and a remainder,

(*) bn+1 + 1 > f3n+1 2:: bn+ 1 and f3n+1 · An + An-1 K = '

(1)

f3n+1 · En + En-1 An · En-1 - En · An-1 . Cf3n+1 · En + En-1) -En

' thus

i.e. ,

Since En-1 > 0, there is also the slightly weaker upper bound

(2)

Furthermore, since bn+1 ::::: 1 ,

(3) I An - K l < � - quite a lot weaker the larger bn+1 is.

En En

Bounds for the Relative Error Using again (*), we now obtain

where IAn · En- 1 - En · An-11 = 1 . Thus

I (�: - K )/K I = _Cf3_n_+_l_·_A_n_+_1

_A_n __ -1)-· -E-n ::::; Cbn+1 · An � An-1) · En ' i .e. ,

40 THE MATHEMATICAL INTELLIGENCER

(1 ') (= 1 I A n+ l

En+l ' En En+ l , cf. (1)) ·

Since A n-l > 0, there is also the slightly weaker upper bound

(2') (= cf. (2)) · Furthermore, since bn+ l 2:: 1 , there is the simple, but still weaker bound

(3 ') ( 1 / A n = E� En

,

Moreover, there is a lower bound for the relative error:

1 I ( A n ) I 1 - - K /K = > ------------

Bn Cf3 n+ l ' A n + A n- l) · Bn ((bn+ l + 1) · A n + A n-l) ' Bn

1 1

(A n+ l + AJ · En i .e . ,

(4') I ( �: - K) / K I > -(A_

n_+_l

-+ 1_A_J

_·_E_

n

These bounds (4'), ( 1 ') , (2 ') , and even (3') are indeed sometimes quite good, as Table 1 shows.

Table 1. Error bounds of Wallis's continued fraction for the calculation of "'

(4') relative error (1 ')

1 (:: -+KI 1 n (An+1 + A,) · Bn An+1 · Bn

0 4.00000 . 1 0-2 4.50703 . 1 o-2 4.54545 · 1 o-2

4.024 1 4 . 1 0-4 4.02499 . 1 o-4 4.29000 . 1 0-4

2 1 .371 22 . 1 0-5 2 .64896 . 1 0-5 2.65746 . 1 0-5

3 8.48081 · 1 o-8 8.49137 . 1 0-8 8.50976 . 1 o-8

4 1 .45001 . 1 0- 1 0 1 .83948 . 1 0-1 0 2.89509 . 1 0-1 0

5 0.96284 . 1 0-10 1 .05560 . 1 0-10 1 .44508 . 1 0-10

In our example, we have for the fourth approximation 355 ,

1 1 3

the true relative error 8 .49137 · w-8, and the upper bounds

(1 ') 103993 . 1 1 3

= 8.50976 · w-s,

(2') -1- • -1- = 8.53710 . w-s. 292 355 . 1 13

(3') 1

355 . 1 1 3 = 2.49283 · w-5.

(2') (3') 1 1

bn+1 · An · Bn An · Bn

4.76190 . 1 0-2 3. 33333 . 1 0- 1

4.32900 · 1 o - 4 6.49351 . 1 o-3

2.83302 · 1 o-5 2.83302 . 1 o-5

8.5371 0 . 1 0-8 2.49283 . 1 o-5

2.90497 . 1 0- 1 0 2.90497 . 10-1 0

2.88529 . 1 0-1 0 2.88529 . 1 0- 1 0

Clearly, the upper bounds (2') are better, the bigger the partial denominator bn+l is . Thus, two rather near-by situated bounds ( 1 ') , (2') are obtained for n = 3 (b4 = 292), and for n = 1 (bz = 1 5) . On the other hand, the bound (3') is weaker, the bigger bn+ l is. Nevertheless, this bound is useful, if one does not have at hand more than the denominator and numerator of the approximating quotient.

© 2006 Springer Science+ Business Media, Inc . . Volume 28, Number 3, 2006 41

Since An+1 > An holds, another lower bound, often much weaker, is obtained when in the lower bound (4') An is replaced by An+1 . Thus

� • An+� · Bn < I ( �: - K) I K I < An+� • Bn

may be a useful relative error inclusion, if binary number representation is used.

The Lore Explained Writing in (3') -

1- = 10-0og An+Iog s,;;, we see the connection of the weak bound -1- with the sum

An · Bn An · Bn of the numbers of digits of the numerator An and of the denominator Bn, which is llog Anl + [log B,]: from (3') I (�: - K )/K I < 10-Clog An+Iog Bn) ::::; 10-Cllog AnJ+Llog BnJ) ::::; 10-<:l.log AnJ+Llog s,J-2).

Example: The 50th approximation of Wallis's continued fraction for 1T reads

1639 76053 94050 96444 37461 06649

521 95199 06667 07447 72628 22481 '

it gives with 66 decimal digits

3.14159265358979323846264338327950288419716939937510582099887318696 .

while 1r results in

3. 141 59265358979323846264338327950288419716939937510582097494459230 . . . ' they agree in 56 (underlined) places; the absolute error is 2.39285947 . . . . 10-56,

the relative error is 10-56·1 1823267 . . = 7.61670824 . . . . 10-57 ,

the sum of the number of digits of the numerator and denominator is 29+28= 57;

the bound 10-(log An+ log B,;) amounts tO

10-(28.21478043 . . . +27.71763056 . . . ) = 10-55.93241099 . = 1 1 .68393171 . . . . 10- 57;

the bound 10-Cf"log Anl+f"log Bnl-2) amounts tO

10-(29+28-2) = 10-55 = 100 · 10-57.

Speed of Convergence The bounds (3) and (3') show that in the large the growth of B� is responsible for the quality of the con­vergence of an infinite ordinary continued fraction. In 1935, Alexander Khintchine obtained4 the rather deep-lying result that there exists a positive constant y such that\YBn has the same limit y for almost all regular continued fractions. In 1937, Paul Levy5 showed that y = e1T2112·tn 2 = 3.275822921 . . . . Thus, B� grows asymptotically with ( y2)n, where y2 = 10.73101 580, log y2 = 1 .030640834, meaning that roughly one decimal place is gained for every step of the continued fraction. This can be checked empirically for the continued fraction for 1r in Wallis's Tractatus of 1685:

- logiAw!Bw - 1rl = 1 1 . 7930

- log !Awo/ Bwo - 1rl = 102.605

- logiAwool Bwoo - 1rl = 1022.86

llog Awl + llog Bwl = 13

llog Awol + llog Bwol = 104

I log A wool + I log Bwoo l = 1024.

However, not all regular continued fractions follow Khintchine's doctrine; for the periodic continued frac­tion [1 ; 1 , 1 , 1 , . . . ] ,�� CVs + 1)/2.

4A. Khintchine, Zur metrischen Kettenbruchtheorie, Compositio mathematica 3, part 2, 1 935, p. 275. 5Paul Levy, Theorie de /'addition des variables a/eatoires, Paris 1 937, p. 320.

42 THE MATHEMATICAL INTELLIGENCER

And for Lambert's continued fraction6 of 1770 for f = arctan 1 ,

7T

4 1

1 1 + ---------------------------------------------

1 3 + --

------------------------------------------1 5/4 + -------------------------

28/9 +

81/64 +

704/225 +

1

1

1

325/256 + 1

768/245 + -1-

the relative error is decreased asymptotically by a factor � = 1/5.828427125 . . . per step--but 3+V8 Khintchine's result holds for regular continued fractions and Lambert's continued fraction is not regular, since the partial denominators are not integers.

6See F. L. Bauer, Historische Notizen: Lamberts Kettenbruch. lnformatik-Spektrum 28, p. 303-309 (2005).

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© 2006 Springer Science+ Business Media, Inc., Vt>ume 28, Number 3, 2006 43

Gode l 's Vienna JOHN W. DAWSON JR. , AND KARL SIGMUND

Does your hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the cap?

where the famous conjecture was

made, the desk where the famous

initials are scratched, birthplaces,

houses, or memorials? Have you

encountered a mathematical sight on

your travels? If so, we invite you to

submit to this column a picture, a

description of its mathematical

significance, and either a map or

directions so that others may follow

in your tracks.

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium

e-mail: huylebrouck@gmail.com

y August 17, 1939, a European war was imminent. Two weeks before Hitler invaded Poland, Dr.

Kurt Godel received a letter from his tailor: 'Sending repaired trousers. As I heard, you will journey to America again. You will certainly need a suit. . . . With German greetings, Decker.'

Godel ordered the suit. His journey back to Princeton seemed to offer no problems. On August 30, 1939, a few days after the Stalin-Hitler pact, Godel blissfully announced to his friend Karl Menger his intention of returning to Princeton forthwith, in a letter which, in Menger's eyes, 'may well represent a record for unconcern on the threshold of world-shaking events' (Menger 1994).

Two days later, Hitler informed a wildly cheering German Reichstag that 'since 5:45, the fire has been returned. ' Godel's outlook changed drastically. He had to write Oswald Veblen in No­vember: 'It now seems likely that I will not be able to come to Princeton this academic year, because it will probably be impossible to obtain a German visa during the war-time. ' Godel was trapped in Vienna. He would spend the next few months in desperate attempts to leave for the US. Against all odds, he finally succeeded. But after the war, Godel would never return to Vienna again. He was through with it.

Vienna: A Logical Choice In 1924, when he arrived in Vienna as an eighteen-year-old from provincial Brno to study at the university, things had looked very different. Vienna had overcome years of hunger and misery, and the economy was picking up. The intellectual and cultural life underwent an amazing flowering. Very soon, Kurt Godel would contribute to it. His work may one day well be viewed as the most lasting achievement of that epoch.

Today's tourists to Vienna follow the traces of Habsburg, visit the imperial

44 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Business Media, Inc.

museums, and are shown the many dwellings of Beethoven and Mozart, or the churches where Haydn and Schu­bert performed. Increasingly, tours in­clude aspects of Vienna between the two world wars, most notably the re­cent Leopold museum, with its paint­ings by Klimt, Schiele, and Kokoschka, or the architectural monuments of Red Vienna, or the art deco villas built by Hoffmann and Loos. If, as a tourist, you relax in a coffee-house between visits to these sights, you will already be very close to Godel. Let us pick up his trail , a map of which appears at the end of this article. (For more details, see Daw­son 1997.)

Kurt Godel's parents were well off­his father was manager, and part owner, of a textile firm in Brunn, a charming little town which used to be called 'the Czech Manchester', less than two hours by train north from Vienna. In 1919, the treaty of St. Germain had established a border between Austria and Czechoslo­vakia, but for the large German-speak­ing segment of what was by then Brno, Vienna as the former capital was still the focus, and obviously the place to go to study. At that time, young Godel could probably not have chosen a site more tailor-made to his talents any­where in the world.

Settling Down Kurt Godel moved in with his brother Rudolf, four years his elder, who stud­ied medicine under the illustrious fac­ulty to which Freud had often dreamt of belonging. Kurt first enrolled for physics, but switched to mathematics under the spell of superb introductory lectures on calculus by Furtwangler and a 'survey of the major problems in phi­losophy' by Gomperz (Sigmund 2006).

In his fifteen years in Vienna, Godel lived in seven different apartments. Tourists will be reminded of Beethoven or Mozart, who also moved a lot. The Godel brothers obviously had a well-

''lta't'. VJI'6t, "TR,\SSB OBK JUI.l�A.)tPfJ-:H S!

oh boren H rrn • l.l'v GOdcl

... . , i 1 tr. .

r ochrtor II rrl

J:, • "811 t 1

u

rdon cwisa tw&. br lchen

Letter from Godel's tailor, August 1939.

defined image in mind in their apart­ment hunts: the seven houses look remarkably alike. All are massive four­story buildings erected at the turn of the century, staid and stately. If you have seen one, you have seen them all. Only one of the houses has a plaque com­memorating Kurt Godel, but this may change with the 2006 centenary-Gc)del was born on April 28, 1906, and Vienna is set to celebrate.

Most of the houses are close to the university, and especially close to the building on Boltzmanngasse where the institutes of physics and (in Godel's time) the Mathematische Seminar were located. On one occasion Godel lived just across the street, two floors above the Josephinum, one of the most dec­orative cafes favoured by Viennese aca- Cafe Josephinum.

n.

demics and students (but brutally dis­figured today).

Godel was a very quiet young man, but not always the hermit he later be­came. He studied diligently, and was soon invited to join the Vienna Circle, a brilliant group of positivists gathered around the mathematician Hans Hahn (famed for his work on functional analy­sis) and the philosopher Moritz Schlick. Both were professors at the university. The young German philosopher Rudolf Carnap had also just moved to Vienna and joined the Circle. Gbdel's closest stu­dent friends were Marcel Natkin and Her­bert Feigl, who studied mathematics and philosophy and were disciples of Schlick. They all met, every second Thursday, in a small lecture room of the Mathematis­che Seminar (Stadler 2002). Informally, most of them also met at the Josephinum, or the Cafe Reichsrat, Cafe Central, and Cafe Arkadenhof, among other places filled with journals and tobacco smoke. These cafes were crowded with intellec­tuals and world-reformers nurturing delusions of grandeur and talking phi­losophy, literature, psychoanalysis, eco­nomics, or politics late into the night.

A Nervous Splendor Austrian politics was dominated by a fierce struggle between the Social Dem­ocrats and the Conservatives. The for­mer had the majority in Red Vienna, and were engaged in a sweeping pro­gram of social reforms. The Conserva-

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 3, 2006 45

II. XeU 9Jlaffage �illrct � . . III . 'Dietrid,g. 55. �ttenauer �., VIII. �Iinben· w•

gaffe 36 . .(J abacd .furl u ID?atie, XVIII.

IDlid;aelt rftr. 15. �ctf�er 3. , IX. �iif}ringer 1 � .. nn""n �iirte( 112. • ;n. ,.. � " .Saufmnnn Beopolbine, IX.

SRttflborfet <Str. 4. H . .

�nnuer ��. ,XIII�J �afangatt!n· e DfJC gaffe 76 · . XVII Elt I . �ral �rete , VI. IDlatia�iifer . er emp �tra�e 47.

·�)hi �ner Sl)oiit) , I. �dJtoang. l . ·'9limbur�ft) �beie , VIII. Bange

®affe 67. �inn�fd � . . XII.<;cJ,onbrunn er

15tra j3e 291 . iReiner SJRaria , I. �[utg . 7. ed)im!o 9tid)., XVII. i!eopolb

<¥-rnft·�. 8 . ��U,;er i!iiii:J, I. <;ingetftr. 8.

T R-20-3-55. steibl m: . , IX. �iibringer ®iide�

XVIII. (�ere 6d,

�r. 166. T A-18-4-97. --�===-�l)omnnn imma, XVI. <;d)eii·

b a mmerg. 6. !Ju 3 m

Plaque on Godel's apartment house in Himmelsstrasse (Himmel means "heaven"). A divorced woman working for a living, Adele Nimbursky.

tives held the Catholic countryside. The strict fiscal policy of their chancellor, the prelate Seipel, had stabilised the cur­rency but exacerbated social unrest. The tensions exploded in a wild riot in 1927. After its bloody suppression, which claimed 80 deaths, the little republic seemed doomed to civil war.

In 1929, Godel struck up an ac­quaintance with a young woman living across the street, in Lange Gasse. Adele Nimbursky, nee Porkert, had divorced and returned to live with her parents. A trained ballet dancer, she had worked for a spell at the Nachtfalter (literally, 'nocturnal moth'), a night-dub in down­town Vienna. Thirty years old, she worked as a masseuse. According to Kurt's brother Rudolf (by then a young MD), Kurt had during his school-boy years experienced an 'escapade' with a mature woman, until his parents put a stop to it (cf. Kohler 2002) . This time, he kept his new friendship under wraps. It was meant to last: Ten years later, Adele would become Mrs. Godel.

In February 1929, Godel's father died

46 THE MATHEMATICAL INTELLIGENCER

unexpectedly. In the following weeks Godel obtained Austrian citizenship, and shortly afterward his PhD thesis was approved by Hans Hahn. The dis-

sertation was a remarkable piece of work: he had proved that the axioms for first-order logic given in Russell and Whitehead's Principia Mathematica

Authors on the Strudelhof­steige

JOHN W. DAWSON, JR., is Professor of Mathematics, Emeritus, at Penn State York

He received his S.B. in mathematics from M.l.T. and his Ph.D. from the University of

Michigan. Since 1 979 his work has focused on Gi:idel's life and work During the years

1 982- 1 984 he catalogued Godel's Nachlass at the lnsitute for Advanced Study in

Princeton. His biography of Gi:idel, Logical Dilemmas, was published in 1 997.

lnfotTnation Sciences and Technology Center Penn State York

I 03 1 Edgecomb Ave. York PA 1 7403

USA e-mail: jwd7@psu.edu

Kurt Godel, (1906-2000).

were independent and complete. 1 This solved two problems that Hilbert and Ackermann had posed in their recent book on mathematical logic. (In that same year, incidentally, Godel's sixty­year-old professor, Philipp Furtwangler, succeeded in proving Hilbert's Haup­tidealsatz [principal ideal theorem]).

Godel studied at the Mathematische Seminar (a small institute with only three professors).

After he received his doctorate, life for Gi:idel did not change much. His mother had moved to Vienna after her husband's death, and she now lived with her two

cious, adjacent apartments in the Josef­stadt. The Gi:idels often went to plays and concerts but otherwise kept pretty well to themselves. Kurt used to work at night, sleep late, and hang around the mathe­matics library, occasionally helping Hahn

sons and an aged relative in two spa- prepare a seminar or correct exams.

1The names "completeness" and "incompleteness'' might wrongly be thought to imply the incompatiblity of

those two theorems. The confusion arises from the two different senses in which the English term "com­

plete" is used in logic. In the former sense it means "capable of proving every statement that holds in all

models of the axioms"; in the latter sense it means "capable of proving or refuting every sentence of the

language." The corresponding German terms are vollstandig and entscheidungsdefinit. The former is prop­

erly glossed as "complete," but the latter is better rendered by "decisive," which captures the meaning of

the German root entscheid- and suggests that the theory makes a definite determination (that is, yields a

proof or refutation) for each sentence.

KARL SIGMUND is professor at the faculty of mathematics at the University of Vienna for longer than he cares to think, and lives mostly in the Thirties of the previous century. Somewhat after that time, he worl<ed in ergodic theory and dynamical systems, and even later in biomathemathics and game theory. Together with John Daw­

son and Kurt Muhlberger, he wrote: Kurt Godel-The Album" (Vieweg, Wiesbaden, 2006)

Facu� for Mathematics University of Vienna

Nordbergstrasse I S I 090 Vienna, Austria

and Institute for Applied Systems Analysis Laxenburg, Austria e-mail: karl.sigmunc@univie.ac.at

On weekends and holidays, he made many excursions with his mother and brother to the scenic mountainside at Semmering or Mariazell, close to Vi­enna. Later, in letters to his mother, he would often hark back to those times. 'Our excursion on the Leopoldsberg must have been in 1932 because I re­member thinking all the time of my coming lecture in Hahn's seminar.' Or: 'Of course I remember the sausages in

Adele Nimbursky, 1932.

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 47

Kurt Godel outside his workplace on the Strudlhofgasse.

to 'go public' by founding the Ernst Mach Verein. But they kept attending the sessions, and they often met with individual members.

Josefstadterstrasse 43, complete with in-house movie theatre. Incompleteness "No Small Matter"

Annaberg but if I am not mistaken the digestive tonic from the apothecary soon made me feel better.'

Godel had become friends with Karl Menger, who in 1927 had returned from postdoctoral years in Amsterdam to take up a position as associate professor in geometry (Galland and Sigmund, 2002).

Godel on the road.

48 THE MATHEMATICAL INTELLIGENCER

Although Menger was barely four years older than Godel, he acted as his men­tor. Together, the two gradually drifted away from the positions held by the Vienna Circle. They shared neither Schlick's infatuation with Wittgenstein nor Hahn's political activism, and they had little interest in the Circle's attempts

In the summer of 1930, while meeting in the Cafe Reichsrat (today's Konditorei Sluka) with Carnap and others to plan a joint journey to Konigsberg, Godel first dropped the news that he had succeeded in proving, not the completeness, as ex­pected, but-rather startlingly-the in­completeness of formalized arithmetic.

1 ersp e1. c:s torma Mat e-matik nad:�gewiesen hlitte. Denn man kann von keinem formalen System mit Simerheit behaupten, daB aile inhaltlimen Oberlegungen in ihm darstellbar sind.

v. NEUMANN: Es ist nimt ausgemamt, dal! a1le SdJ!uBweisen, die intuitionistisd:J erlaubt sind, sim formalistisd:J wiederholen lassen.

GODEL: Man kann {tmter Voraussetzung der Widersprucbs­freiheit der klassismen Mathematik) sogar Beispiele fiir Satze (und zwar solme von der Art des G o I d b a c h smen oder F e r m a t ­sd:Jen) angeben, die zwar inhaltlim rimtig, aber im formalen System der klassismen Mathematik unbeweisbar sind. Filgt man daher die Negation eines solmen Satzes zu den Axiomen der klassismen Mathematik hinzu, so erhalt man ein widersprud:Jsfreies System, in dem ein inhaltlim falsd:Jer Satz beweisbar ist.

REIDEMEISTER: Ich m&htc die Diskussion mit einigen Berner-

How to drop a time-bomb.

Ticket to fame. Godel's standing-room ticket to Karl Menger's lecture on 'The New Logic' .

The axioms in Principia Mathematica did not suffice to derive all true state­ments about natural numbers.

Apparently the members of the Vi­enna Circle did not immediately grasp what this momentous news meant to Hilbert's program for proving the con­sistency of mathematics. They had planned to hold a workshop in Konigs­berg, a satellite meeting to the ritual

Jahrestre.ffen [annual gathering] of the German Mathematical Society. The pur­pose of their meeting was to discuss the foundations of mathematics. On the first day of that meeting, Carnap and Hahn gave their lectures as if nothing had happened, and Godel presented his completeness theorem from the year before. It was only during the fi­nal discussion that he mentioned, al­most casually, his incompleteness re­sult. The protocol of the meeting bears witness to a double take: at first, the discussion flowed peacefully on, but then Gbdel was invited to add an ap­pendix to the proceedings, and explain what he meant. John von Neumann, of

few weeks later, he wrote to Gbdel say­ing that the proof of the incomplete­ness theorem showed that Hilbert's program must fail : consistency cannot

be formally established within the sys­tem. But by that time, Gbdel had al­ready submitted the same result to the Morzatshefte.

course, had understood immediately. A Sanatorium Purkersdorf, designed by the architect Hoffmann.

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 3, 2006 49

The impact was tremendous. Hahn hailed the work as an 'achievement of the first order . . . that will take its place in the history of mathematics'. And Mar­cel Natkin wrote to Godel: 'So you have proved that Hilbert's system of axioms contains unsolvable problems-why, this is no small matter. '

But Kurt Godel kept on, as before, with his quiet life, making himself useful at the Mathematische Seminar in many ways. Young Olga Taussky would later write about his kindness in helping students and colleagues. With some malicious amusement she

also noticed his encounters with mem­bers of the opposite sex. Godel was discreet, to be sure, but not unwilling to be seen with a good-looking girl (see Taussky 1 987) .

Standing up for The New Logic Some more bright young people had collected around Menger: Nobeling, Wald, and Alt. Their informal group, the Kolloquium, became a hotbed of results on topology, mathematical economy, and of course, logic. Godel gave a num­ber of lectures, over the years, and most of them were published in the yearly

0 2 2

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lHigoruj�·n· llllb

(i}r£tblh3it1X.:n s . g

21. )�. , q .; 1 . J et � U;. � c-t ? t 1.r� � 1 o/,V.. V 01.- ­

U1 � II 1 X kih\ J tA o( £-1 h\ e,� � Ulvt £. I

� �� ) � t;) tit � d v. "',f,. � p�

y� �· . �,(;(

proceedings, the Ergebnisse eines math­ematischen Kolloquiums, of which he was co-editor. The sessions also at­tracted philosophers from beyond the Vienna Circle, such as Karl Popper, and economists such as Oskar Morgenstern. The latter, at the time, attempted to prove that predictions in economics were in principle impossible, in loose analogy to Godel's incompleteness re­sults in logic.

() () () Godel applied to become Privatdozent (which would give him the right to lec­ture at the University, but implied no appointment there). In due course, he obtained his Habilitation, and thence the title. Fifty-year old Eduard Helly was also Privatdozent at the time; so was the topologist Walter Mayer, before Ein­stein called him to his side in Berlin (and right afterwards took him along into exile at Princeton). Privatdozenten got pitifully small fees for their lectures (in one term, Godel would earn enough to have bought three beers). But hap­pily Godel did not need a salary from the university. He still had his own means, and more importantly, he was invited to the newly founded Institute for Advanced Study in Princeton! John von Neumann and Karl Menger had pointed him out to Oswald Veblen, who was talent-scouting for the lAS. Veblen listened to a Kolloquium lecture by Godel and was suitably impressed.

By that time, Gbdel was obviously the star of the institute in Vienna. Al­fred Tarski, John von Neumann, and Willard Quine visited him. Karl Menger, especially, sang his praises wherever he went. Menger and Hahn had launched a very well-attended series of public lec­tures, and they used the substantial ad­mission fees to pay the salary of young Olga Taussky, and to erect a funeral monument for Boltzmann. When Menger lectured on 'The New Logic' , Godel's discoveries were for the first time pre­sented to a wider audience. Godel at­tended, but he only purchased a stand­ing-room ticket; after all, he was well acquainted with the topic!

Private Rest Homes and Public Bedlam Godel paid a heavy price for his genius. From his childhood, when he suffered a

Godel was often forced to cancel his lectures. The fees were modest, to say the least. bout of rheumatic fever, he had been a

50 THE MATHEMATICAL INTELLIGENCER

Vienna's new Mayor inspects the University of Vienna in 1938.

Dor./ •• 0980/6/:59 llnlaMIU18 a "Dottent oeuer Or4nuog.

.4c 4 L

Pa t der pb1loaoph1aobou

J'lli<Uli&t

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Per b1ebar1ga Dozent Pr• � 0 0 4 o 1 iat

wtaaeoaobattliob Silt beaohriaboo. e1Da Bab1l1t1aruas wurde YOD

4 ... �ll.dhGIIen �nuor liMn durqo:l:llbrto 'Sa wird U... Yorgewor­

ton i�ar in l1boraL-�Ud1oobon rei .. o Yartabrt cu habon. K DB

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gag n den •at1oo.1leoz1al1saua a1n4 air n1obt bekaont aoworden. se1De

raobkollege bon 1hn niobt nlher l<enoeng lerot, aodd ••1tare

ADaliiU>tte tiber 1hn niob.t au erhCil.too iod. Ja iot air d.ohor uob

Dinht atlgliob aa1De J:rneooun& 111111 Poaeoton oeuer Or<lnoog auadrt1olo-

11nh r;u botll.rworteo,ebolliiO ani& babe 1oh ber 41.a Gruadl po ainh

4 sosen uuuapreob.eu.

Bell llitler l

;Jv �l# 'f-' Pr. rohat, Pbdt.

A Nazi leader informs: Gi:idel had studied with a jewish professor.

On the other hand, all of mathematics had formerly been ' verjudet' .

hypochondriac. The stress of his work led to episodes of nervous breakdown, and eventually, paranoia. His second visit to Princeton had to be aborted af­ter two months, and Rudolf was called upon to escort his brother from Paris back to Vienna. During the next few years, Kurt Godel spent much of his time in sanatoria and rest homes around Vienna, in Rekawinkel, Purkersdorf, and Aflenz.

During his time in Rekawinkel, he was so plagued by fears of poisoning that he would only eat what Adele pre­pared for him, in his presence. She had to eat from the same plate, with the same spoon. Later, she recorded that Godel's mother was so frightened by her son's psychosis that she locked her­self in her bedroom at night. Eventu­ally, Godel would recover (for a time) and return to the mathematics institute. But altogether, between his three visits to Princeton and his stays in sanatoria, he lectured for only two semesters at the University of Vienna, once on logic and once on set theory.

The political situation in Austria in the 1930s deteriorated rapidly. Under Hitler's greedy eyes, the Austrian chancellor Dollfuss had established a fascist regime in an ill-guided attempt to obtain Mus­soHni's protection. In 1934, a short burst of civil war was followed by an abortive Nazi putsch. Dollfuss was murdered. His successor Schuschnigg was unable to build up a national consensus. In March 1938, Hitler's troops entered Austria, greeted by a jubilant mob.

The Nazis lost no time in "purifying" the university. Godel had to prove that he was Aryan. His rank as Privatdozent was too low to demand a loyalty oath to the Fuhrer, but all professors had to take the oath. Non-Aryans were literally sent packing, and Adolph Eichmann set up offices in Vienna to organise the exodus.

Intellectually, Godel was very iso­lated: Hahn had died in 1934; Schlick had been murdered in 1936 by a para­noiac former student; Menger had em­igrated in 1937 to the USA; Feigl and Natkin were gone, as were Carnap, Mayer, and Taussky; Helly, Alt, and Wald had all managed, with great pains, to flee to the United States.

But Godel was not alone . Adele was at his side. She had helped him while he was in the sanatorium, and she had helped him to get out of it. Now she helped him on the stairs of the Strudl-

© 2006 Springer Science+Business Media, Inc .. Volume 28, Number 3, 2006 51

flen, am 27 . lfove.aber 19:59.

All se. '<agnifisou don Herrn Role tor d:er Uh1Tere1 tl1t

in W 1 o n . Ubor AUtford•l'WI6 orotatte ioh hi..Ut lloriobt

Ubor die Pera�nlicbkoit doo Priv.Doa.nr. turt G � 4 e l in faobli­

oher, pOlitiooher und obarolcterltcher Jt1rut1ohtt

nnoh .Aoerika verea.gt warden aollte, erhebt aich allardings die rmae doe Lebonaunte.rhllltea � ihn. otS4el Tertugt bier Uber keinerlei H:ln­

ko ... ncP'\ wtd will die Binladung n.aoh OSA !lliZ o.nnehmen, WI eeinen unter­

hc." t �ol'tro1tcn zu ldSnnen. Die go.nze l'ra.ae der A.uartiee ware h1ntU.llia,

wenn ea gelH.nge , �del 1Merhalb dee Reioheo aine antsproohen4 bozahlta

St�ll.ung Zll bietcn.

OHd.el genieaat in Jaohltrl1een, wie ich oua den

Orteilen der be1den o.Protcsooren dar Ka:thematik an u.naerer 7akul.Ut,

t:. a.trlyerhoter und A. Bubar, entneh.ae, 1n aeinea Arbai teberoioh, dae

dae TOft otS4ela Lohrer, d.OJ:I. �tl41eohen Pro:feeaor Hahn baeondero soptloa­

te Granmcebiet dor llathe.aatilc Wld Loglk w:afaoot, booon4oreo AllOobonJ

� in USA, wo 41eoo GrWldl.a;!enfraa-n der IIAthe:.atilc woi tero

�1•• 1ntereaa1eren, wird: Q6d.el eehr seaohlltzt.

wogen dor politieobon Dourte1lung �dolo bnbe

ioh den lh\1Tere1Ute-Dosentenbl.Uldatuhrer Prot. Marohet cu R.nte a:e&ogen

doooon urte11 oioh llit meineo pereHnl1obon Bindruok ..Ul.l.l.s doolct.G'Idel

d.er in der Zeit heramru.oha, d.a die KathematUceraohat't wtene &flnSl.1oh

untor �U41eobem Binfl.Wie otand, beo1 ht lco\Ul oin innereo vcrhlll tnia

sua llational.eozialiamu.a. !r Moht d.en B::l.ndru.ck einee 4urchaua Wlpol1-

t1oob.e.n Menooben. EB wird daher auob aller vorauaeioht naoh echw1er1-

s:erc LaB•n, wie aie aioh tur einen vertrotor dee neuen Do11tcchl.aM

in USA. eioherlioh ersebon werden, kawa gewa.obaen et1n •

.Uo Charnlcter Mcht �del. einon guten Bindruolq

ioh babe in d1ooor Jlinaicbt auoh n1o eine 046• Ubor ibn gebHrt, S1"

bat gut• Uat&Jl8oforaaen und wird goael.loobaftl1ob gowioe koino ,.bl.er

begehln, cUe dae .t.neohen aeinor Reimat 1l:l J.u.alande herabeetzen ki:Jnntea

Jnllo �dol 8WI pol1 Uoobon GrUnden dio �;· j, 4 s rt r-·: -� o,) •

�·

The dean writes that Godel has hardly any inner feeling for National Socialism. He will hardly be up to the difficulties awaiting him abroad. On the other hand, his manners are good.

hofstiege next to the Seminar, when he was stopped by SA-rowdies who took him to be Jewish and knocked his glasses off. She chased them away, with her handbag or with her umbrella-in any case with great spirit.

It was time to get married, even if his bride-by now almost forty, a di­vorcee, a former dancer-was not ex­actly up to the expectations of his fam­ily. Forty years later, Rudolf Godel would write icily 'I do not presume to form an opinion on my brother's mar­riage' (Kohler 2002). Godel's mother re­turned to her villa in Brno, and Kurt and Adele left the Josefstadt, moving to the outskirts of Vienna.

The civil ceremony took place in September 1938, at the height of the Czechoslovakia crisis. Two weeks later, Godel left for another stay in the United States-without Adele. When he boarded the steamer, the Munich treaty was just a week old.

52 THE MATHEMATICAL INTELLIGENCEA

Godel had achieved another break­through by proving the consistency of the continuum hypothesis. One half of Hilbert's famous first problem was thus solved. During his residency at lAS, news of the pogroms of the Kristallnacht shook public opinion in the US. Godel spent the following spring with Karl Menger in South Bend, Indiana, and during that term, Hitler dismembered what was left of Czechoslovakia. In spite of these dire developments and the urgent warn­ings of his former mentor, Godel in­sisted on returning to Europe the fol­lowing summer.

Trapped in Vienna Godel returned to Vienna at the climax of a political nightmare, which soon be­came a personal nightmare as well. He had hoped to return to the Institute for Advanced Study in the fall, this time with Adele, but now he encountered a

fiercely adverse bureaucracy. The min­istry of finance, for example, looked askance at his earnings in America. A confused exchange of letters between the ministry of education and the au­thorities at the University of Vienna re­sulted in the loss of his title of Privat­dozent, when it emerged that the previous fall he had left without proper permission. Worst of all, Godel was ex­amined for the Wehrmacht and found fit for garrison duty. In addition, the American Consulate, flooded as it was with demands for immigration visas, took a very dilatory attitude towards his application. Then, Hitler's troops rushed into Poland, and war was declared in the west. German troops on one side, and French and British on the other, dug themselves in, although the west­ern front, in this early stage, saw action only on the waters of the Atlantic.

Although there seemed to be no hope that Godel would get back to

Jmllamen des fiil)rers ernenne ich

unter 13erufung in das 13eamtelll)erl)altnis d(n priNta�un Dr.vhil.J{urt q�.kl�l

�um D�nt�n. Jd, l'OII3iehe diefe Url'unde in der £rwartung, da� der £rnannte getreufeinem Dienfteide (eine Cmtspflid,ten gewiffen�aft erfiillt und das Dertrauen red,ifertigt, das il}m durd, diefe frnennung bewiefen wird . 3ugleic� darf er des befond.eren 5cl)uf3es des ,fiil)rers ficher fein.

Berlin, den ! S. Juni 19't0.

Det .Reicl}sminifter fiir IDiffenfc�aft, £t3ie�ung

und Dolfs'bildung :Jm <luftr«qe

�

nu o. t !\ � 1 ;�:

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lo Dokwlte-nte Deeld tcrlaO 'I" P G�de.l 2 a v.2S.Vl.l940 und /o.rll heut:1,;en To.ce O.bDX'n.Oc::tc-!:l Zil :::U:.ben.

(Kurt lltiOel)

Towards new shores. Godel and his wife, Adele, on shipboard, 1940. A receipt that never was signed, and a University of Vienna

diploma that was never used.

Princeton, the lAS persevered. In a mas­terful analysis, John von Neumann ex­plained to the director exactly how and where to intervene with the State De­partment. It worked. Shortly before Christmas 1939, Godel went to Berlin to pick up his visa. Since the German authorities could not provide a job for him, they decided to Jet him earn dol­lars in the States. Some worthies still de­bated whether he was man enough to

master the difficult situations he was bound to encounter in the States as a representative of the New Germany, but they let him go. His field of mathemat­ical logic, obviously 'infested by Jewish thinking' and aloof from Deutsche Mathematik, surely meant nothing for the German war effort, and he had not yet been called to serve garrison duty. In early january 1940, Godel received his emigration permit, and shortly af-

terwards the necessary transit visa; this time he travelled to America with his wife. They had to take the long route, via the Soviet Union, Japan, and the Pa­

cific. After a gruelling winter journey through Siberia and a three-week delay in Yokohama, Godel and Adele reached New York in mid-March. Nine months earlier, he had started from there for his return to Vienna and what would be­come a nerve-racking world tour. He

© 2006 Springer Science+ Business Media, Inc., Volurne 28, Nurnber 3, 2006 53

lkflln TP .:-on• Unt�r ii'I"'I(IBOnl.,.

fr�: IUOIO floli_,.«Nwwto; Clntl11 IHM "-".tiN.ulll�o 1l�$• .......

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Stipendiu• von 4000 i engobo ten wordon i.et. De. dae Generol­konaulat ih11 erldlrt hat. daB die Annab.lle dioeea Angobote un­e-rwUneoht &d , dne &nde.re geeisnete 11lti&keit licb aber nieht

tinden ltlBt, bat Dr . Go&d(ll dll.rauthin ttir den Pall, Cla8 �1n" wettere Beurlnubuns at:,selehnt wU.rde, ih• mitzuteilon, ob er naeh tiner evtl . ReloaehD!:funs: einen bewhlten POiJten nn el.ner deu tacnen Hochschule erhelten kOnne. !AI •etze all Clrdinge bintu, dt�-B ihra intolse einea nculeldene anat.:rongendc BUro- ocier L�hrtlt1gke1t nioht �:��6glicb aei .

Ieh dart uta die dorttg• St.Hlungn.tlhme bitten.

Ia Au!tl'aa;

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The German consulate wanted Godel to return to Germany, but the Fatherland could offer him no job. And as Godel pointed out, his health was delicate.

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In a letter to his mother, Godel reports his nightmares about returning to Vienna.

� •t .(/-(. j) M t k .,...__ ,f I �

would never travel far again. And when his Viennese friend Oskar Morgenstern, also exiled in Princeton, asked him about Vienna, he remarked pithily, 'The coffee is wretched!'

Meanwhile, Back in Vienna �� c-h.t . lv'"a b..... o/ fA 75 � d � Jf-�-v- .0 ;,. � �p ?......

fk" O. ot. ;._ ' � � d OV'v"f- !fA- . Wit ]),. �·� /k.d-_t ol�: r w- .·"""""" r �·r;t.;r g b&... � . }) e,.. Alk ;.-

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1-1 � t1.r-fAU 1rt{. ....�r i vt-r-t "'"' N � vel) ,q I

"Decker's suit looks as good as new."

54 THE MATHEMATICAL INTELLIGENCER

The bureaucratic machinery Godel had left behind continued milling for a time. He had applied to become Dozent neuer Ordnung (the new German form of Habilitation), and although he was considered politically lukewarm, even suspect, he eventually received an em­blazoned diploma, c/o University of Vi­enna. That is where it still waits to be collected, together with a receipt Godel was meant to sign.

For years, there would be official in­quiries about the whereabouts of Dozent Godel. His brother Rudolf replied testily that as long as Germany

1. 11.11.37-9-11-39 Hirnmelstrasse 43/3/5, A-1190

2. 8.10.24-8.4.27 Florianigasse 42i2i16, A-108o 3· 8.4-27-20.7.27 Frankgasse 101>/n,A-1090

4· 6.10.27-1.].28 Wahringerstrasse 33/4/22, A-1090

5· 4-7-28-s.n.>g Lange Gasse 7213114, A-1080

6· 5.11.29-16.11-37 Josefstadterstrasse 431112112a, A-108·

1· Universitat wien,A-1010

8· Mathematisches lnstitut, Strudelhofgasse, A-1090

9· 9-"·39->4-2-48 Hegelgasse si>isa. A-1010

The Godel trail. Godel heard his philosophy lectures at the main building of the university (7), and his lectures on mathemat­ics and physics in the complex building (8) between Boltzmanngasse, Strudlhofgasse, and Wahringerstrasse, which had opened in 1915 . The Strudlhofstiege is close by.

could not offer him a job, he was forced to remain in Princeton. Moreover, as Rudolf could honestly point out, the German consul in New York had ex­plicitly advised against an Atlantic cross­ing; and the option of a return journey via Siberia was not open for long.

After the war, every other Sunday Godel wrote ro his mother (who had moved back to Vienna in 1 944, and thus avoided the expulsion of the German-speaking population from Czechoslovakia) . Although Adele re­turned to Vienna a number of times, GC'>del never joined her. In one letter, he confessed to his mother that he had had nightmares about being trapped in Vienna, unable to leave. Fi­nally, his mother, by then almost eighty, flew ro New York to visit her famous son in Princeton. Her visit was such a success that it was repeated every second year until her death in 1 966.

Both Morgenstern and Menger re-

sumed their contacts with Austria after the war, hut not Godel. He adamantly refused all honors conferred upon him by the University of Vienna and the Aus­trian Academy of Sciences; and when he wrote, 'It seems that Vienna is chang­ing only very slowly, ' he did not mean

it as a compliment. And, you may ask, what about the

suit Godel had ordered from his tailor Decker� In 1952 GC'>clel wrote to his mother: 'Decker's suit looks as good as new. · Made of pre-war quality cloth, that was to be expected.

REFERENCES

Alt. Franz (1998) Afterword to Karl Menger,

Ergebnisse eines mathematischen Kol/oqui­

ums, ed . by E. Dierker and K. Sigmund,

Springer-Verlag, Wien

Dawson, John (1997) Logical Dilemmas: The

Life and Work of Kurt G6del, A K. Peters,

Wellesley, MA

Menger, Karl (1994) Reminiscences of the Vi­

enna Circle and the Mathematical Collo­

quium, Kluwer, Dordrecht

Stadler, Friedrich (2001 ) The Vienna Circle,

Springer-Verlag, Wien, New York

Dawson John (2002), Max Dehn, Kurt Godel,

and the Trans-Siberian Escape Route, No­

tices AMS 49, 1068-1 075

Sigmund, Karl (2006) Pictures at an exhibition,

Notices AMS 53, 426--430 Taussky-Todd, Olga (1987) Remembrances of

Kurt G6del, in Gddel remembered: Salzburg,

10-12 July 1983 (ed. P. Weingartner and L. Schmetterer), Bibliopolis, Naples, 2 9-41

Galland, Louise and Sigmund Karl (2000)

Exact Thought in a Demented Time- Karl

Menger and his Viennese Mathematical Col­

loqium, Math lntelligencer 22(3) , 34-45

Sigmund, Karl A (1995) Philosopher's Mathe­

matician - Hans Hahn and the Vienna Circle,

Math lntelligencer, 1 7(4), 16-2 9

Kohler, Eckehard, et a/. (eds) (2002) Wahrheit

und Beweisbarkeit, Hi:ilder-Pichler-Tempsky,

Vienna

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 55

i;i§lh§i.lfj Osmo Pekonen , Editor I

Feel like writing a review for The

Mathematical Intelligencer? You are

welcome to submit an unsolicited

review of a book of your choice; or,

if you would welcome being assigned

a book to review, please write us,

telling us your expertise and your

predilections.

Column Editor: Osmo Pekonen,

Agora Centre, 40014 University of Jyvaskylii, Finland

e-mail: pekonen@mit.jyu.fi

�� ir Roger Penrose, a leading mathe-matical physicist and a renowned

� expositor of science, has taken on an enormous task in writing The Road to Reality, a vast survey of contempo­rary mathematical physics. The "the" in the title is perhaps misleading, though, as a single path does not emerge from this book. Rather, the author leads the reader along many avenues in territory that is yet to be mapped. Starting from the birth of Greek science and funda­mental definitions of numbers, symme­tries, and manifolds, he proceeds to ex­plain Relativity, Quantum Mechanics, Gauge Theory, the Standard Model, Quantum Field Theory, the Big Bang, Black Holes, Supersymmetry, String Theory, M-Theory, Loop Quantum Gravity, Twistor Theory, and you name it. The final chapter is tantalizingly en-

The Road to Real ity:

A Complete Gu ide

to the Laws of the

U niverse by Roger Penrose NEW YORK. ALFRED A. KNOPF, INC. , 2005, xxvi i i +

1099 PP. $30.00, ISBN: 0-679-45443-8

REVIEWED BY PALLE E. T. JORGENSEN

I t's always a delicate balance for a sci­ence book: encyclopedic versus well focused on a unifying theme. Pen­

rose succeeds admirably in striking this balance: the book offers a panorama of science, and its goal is quite ambitious.

Yet the presentation is for the lay reader; it is engaging, and certainly not boring. Indeed, there are preciously few authors who manage to guide begin­ning students into serious scientific top­ics, especially when the aim is the panoramic picture of science. The nar­rative flows well; Penrose captured my imagination and held my attention

titled "Where lies the road to reality?" In a poetic epilogue, an imaginary Ital­ian (female) scientist is brought to the scene to settle it all, some day, in per­haps a not-so-distant future.

In other magazine reviews, and in the blogosphere, the book has gener­ated divided opinion. While everyone praises the insightful presentation of some fundamental issues and concepts of mathematical physics, many readers feel unhappy with a certain editorial sloppiness which has marred their reading experience.

We exceptionally publish two re­views of this exceptional book, reflect­ing the extremes of the spectrum of opinion.

-Osmo Pekonen, Reviews Editor

through more than 1000 pages. I found the book inspiring, informative, and ex­citing. Penrose's writing is calm and composed. It is also honest about what mathematics and physics can accom­plish. I count Roger Penrose among the most outstanding scientists and exposi­tors of his generation.

Aim and Scope I admire authors who succeed in com­municating math to the man and woman "on the street." You can't argue with success: this book managed to hit a top spot on the Amazon.com best­seller ranking, so Penrose must surely be doing something right. Achieving popular appeal with a serious science book is impressive. What author of math or of physics books would not envy this degree of circulation, or even a small fraction of it?

While this is a popular book, its goal is not modest: a search for the under­lying principles which govern the be­havior of our universe. It raises the ex­pectation level, yet the presentation is modest when modesty is called for. It shouldn't surprise us that the principles Penrose has in mind take the shape of

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 59

mathematics (roughly the first half of the book) and physics (the good sec­ond half).

Despite the grand and ambitious goal, I didn't feel cheated. But I was first skeptical as, browsing in my book­store, I took the book from the shelf. As I read, I was pleased to find all the equations (not just hand waving). In­deed, the reader is first gently prepared with explanations for the technical sec­tions. And when the formulas come, the reader is ready and will then want the mathematical equations. They aren't just dumped on you! Penrose's book is likely to help high school students get­ting started in science, and to inspire and inform us all. There is something for everyone: for the beginning student in math or in physics, for the educated layman/woman (perhaps the students' parents), for graduate students, for teachers, for scientists, for researchers; the list goes on. I believe Penrose proves that it is possible for one group of readers to be respectful of the needs of another.

What Is the Book all About? It is both a big idea and, especially, a unifying vision. What laws govern our universe? How may we know them? How will this help us understand? Yet despite its vastness, the subject is well organized, and it is fleshed out in the language of science.

A small sample of topics from the contents will give you a taste. This is only a sample, as there are 34 substan­tial and wide-ranging chapters in all. The roots of science. An ancient theo­rem and a modern question. Geometry of logarithms, powers, and roots. Real and complex numbers. Calculus (a re­freshing approach, I might add). Func­tions and Fourier's vision. Surfaces and manifolds (plus calculus revisited). Sym­metry groups. Fiber bundles. Tensor bundles and tensor calculus. Cantor's infinity, Turing machines, and Godel's theorem. The physics topics range from classical (Minkowski, Maxwell, La­grange, and Hamilton) to modern, start­ing with Einstein's theory of relativity and the pioneers of quantum mechan­ics, Bohr, Heisenberg, Dirac, and Bohm. The modern topics further span quan­tum field theory, the Big Bang, cos­mology, the early universe, gravity, su­persymmetry, and they all merge into

60 THE MATHEMATICAL INTELLIGENCER

the final chapter, "Where lies the road to reality?" Save the Epilog for art!

I believe that this book does a great job in (apparently effortlessly) moving the presentation from high school math to advanced topics (like Riemann sur­faces, manifolds, and Hilbert space), and in physics (quantum theory, rela­tivity, and cosmology) . In fact, I am hard pressed to come up with a book that is even a close second in this way. It is one of the very few science books of ambitious scope that is not viewed by students as intimidating. Penrose's clever use of Prologue and Epilogue en­gaged me as an uninitiated reader.

As the book has now become a best­seller, I expect that it worked well for other readers too. In fact, Penrose adds an element of suspense, and he man­ages to give the book the flavor of a novel. I can't begin to do justice to this book. Get it, and judge for yourself. I will also not give away the ending, other than by saying that the title of the book offers a hint. And you will be able to form your own opinion-your own take-and to shape your own ideas and draw your own conclusions. (You won't be spoon fed!) As with all good and subtle endings to novels, this one can be understood and appreciated on sev­eral levels.

It is no surprise that one of Penrose's unifying themes is attractive and pleas­ing geometric images. They underlie both the mathematics (roughly one third of the book: modern geometry, Riemann surfaces, complex functions, Fourier analysis, visions of infinity), and the physics: cosmology (the big bang, black holes), gravity, thermodynamics, relativity (classical and modern: loop groups, quantum gravity, twisters), and quantum theory (wave-particle duality, atomic spectra, coherence, measure­ments).

In the case of this book, a line-by­line overview from the table of contents is misleading. A compelling feature of the presentation of topics from mathe­matics is that it is sprinkled with ex­amples from physics. I wish this were done more in standard mathematics texts. Not only does this motivate and illuminate the concepts from mathe­matics, it also serves to introduce ideas from physics. For Penrose's grand am­bition, this is essential. And as a peda­gogical principle it works: the student

will already have seen the areas of physics and cosmology that will be re­visited in the second part of the book.

Of the author's earlier research pa­pers likely to have influenced the theme of the book, I would mention [Pe65] in which Penrose proved that, under con­ditions which he called the existence of a trapped surface, a singularity in global space-time must necessarily occur at a gravitational collapse. Roughly, this is when space-time cannot be continued and classical general relativity breaks down. The present book leads the reader on a search for a unified theory combining relativity and quantum me­chanics, since quantum effects become dominant at singularities.

A second influential paper is [PeMa73l, in which Penrose introduced his twistor theory; again an attempt at uniting relativity and quantum theory. Not surprisingly, this grand mathemati­cal scheme is directed at unification, combining powerful algebraic and geo­metric tools!

While it is true that the book is about the laws of the universe, the reader fa­miliar with other Penrose books will probably detect the contours of the au­thor's prolific scientific activities span­ning several decades, including what is often called "recreational" mathematics. Reflecting on the versatility of Penrose's activities, it is worth remembering some of them: Roger Penrose, a professor of mathematics at the University of Ox­ford, is known for his outstanding con­tributions to mathematics, to physics (relativity theory and quantum me­chanics), and to cosmology. In addition, he has for decades pursued his inter­ests in writing and in recreational math, for example geometric tessellations. Penrose tiles-tile systems covering a surface with prescribed shapes, say kites and darts-at first glance seem to repeat regularly, but on closer exami­nation do not: they are quasi-periodic. A "hobby" of apparently frivolous geo­metrical puzzles, Penrose tiles are actu­ally studied by solid-state physicists: some chemical substances are now known to form crystals in a quasi-peri­odic manner.

Readers interested in math illustra­tion might find it intriguing that the au­thor and his father are the creators of the so-called Penrose staircase and the impossible triangle known as the

"tribar." Both of these "impossible" fig­ures have been used in the work of Maurits Cornelis Escher in his creation of structures such as the waterfall, where the water appears to flow uphill, and the building with impossible stair­cases that rise or fall endlessly, yet re­turn to the same level.

The Pictures In fact, I taught a geometry course this semester, and had a hard time present­ing Riemann surfaces in an attractive way. It's a subject that typically comes across as intimidating in many of the classical books: take Herman Weyl's, for example. I found the graphics in Pen­rose refreshing: his many illustrations are full of his own artistic touch. They are done with flair and are an antidote to the flashy computer-generated color graphics and special effects that are typ­ical in textbooks. Readers will probably relate better to illustrations with a per­sonal touch: his clever use of shading presents core ideas much better and appeals to imagination. And they are less intimidating: We sense that we our­selves would have been able to make similar pencil sketches. Or at least we are encouraged to try!

A key to books for the classroom is student involvement. The choice of exercises is essential: they help stu­dents-and other readers-become part of the discovery. The pictures and the projects serve to bring to life the underlying ideas. Beginners might oth­erwise get lost in the math and the equations, or in the encyclopedic panorama of topics.

Is There Anything for a Reviewer to Complain About? Yes, the copy-editing of the book is sloppy. But by now there are websites with endless lists of tiny errors and omissions, misspelled names, etc. This will probably bother a few mathemati­cians and other specialists. Given the length of the book, and the realities of science publishing, it didn't bother me. I don't really think that the various cor­rection lists are alarmingly long. But it is a sad fact that modern-day book pub­lishers tend to skimp on copy editing. The publisher was probably reluctant to spend big bucks on a book with for­mulas, perhaps not expecting that it

would be a commercial hit. I hope that Penrose's book will now encourage book publishers to give our subject the attention it so richly deserves.

Postscript I discovered The Road to Reality in my bookstore by accident, and I was at first apprehensive: the more than 1000 pages and the 3.3 pounds are enough to intimidate anyone. But when I started to read, I found myself unable to put it down. And I didn't. I bought it and had several days of enjoyable reading. I am not likely to put it away to collect dust, either. It is the kind of book you will want to keep using, and to return to. Books like this are few and far between.

I expect that readers will react dif­ferently to the title, to the Prologue, to the math, and to the very ambitious scope. The choice of title gave me as­sociations (perhaps intended), bringing to mind Douglas Adams's amusing lit­tle book series, Hitchhiker's Guide to the Galaxy. Here's a sample of sub-ti­tles in those hilarious books: Life, the universe, and everything and Mostly harmless. Another association was a fa­vorite series of mine of popular science books by George Gamow dating back to my childhood. Several of Gamow's books were recently reprinted, for ex­ample, Mr. Tompkins in Paperback. I only mention my associations with these more lighthearted books to en­courage readers to be realistic in their expectations. Judging by its top rank­ing at Amazon.com on release, Pen­rose's book gained a rare entrance (for a science book) to the short list of pop­ular best-sellers. This is impressive for

a math book which is prolix, deep, thor­ough-going, and which at the same time tackles serious philosophical questions. While Penrose's book is indeed serious, I didn't mind a personal flair and the warm sense of humor he brings to the subject. Only rarely do science books make me smile.

REFERENCES

[I ] Penrose, Roger; Gravitational collapse and

space-time singularities. Phys. Rev. Lett. 1 4 (1 965) 57-59.

[2] Penrose, R. ; MacCallum, M . A. H . ; Twistor

theory: an approach to the quantisation of

fields and space-time. Phys. Rep. 6C (1 973), no. 4, 241 -3 1 5 .

Department of Mathematics

The University of Iowa

Iowa City, lA 52242- 1 4 1 9

USA

e-mail: jorgen@math.uiowa.edu

The Road to Real ity:

A Complete Gu ide

to the Laws of the

U niverse by Roger Penrose NEW YORK, ALFRED A. KNOPF, INC., 2005,

1136 PP. $40.00, ISBN 0-679-45443-8

REVIEWED BY ROY LISKER

more accurate title might be "Many Roads to Reality . " Only a

hidebound physicist (a class to which Roger Penrose assuredly does not belong) or religious fundamentalist would maintain that there is a unique road to reality.

Roger Penrose has gained interna­tional recognition for his research into cosmology, general relativity, geomet­ric combinatorics, quantum field the­ory, differential geometry, and related disciplines. He is passionately commit­ted to the advancement of knowledge, with particular emphasis on the pro­motion and defense of his own ideas. By turns provocative and profound, a number of them are re-introduced in this book.

Penrose's ambitious overview con­firms my assessment that the "road to reality" constructed by the theoretical physicists of the last century is a riddle within a quagmire inside the realms of Chaos and Old Night. This is particu­larly notable in the final chapters of the book, which deal with quantum field theory, supersymmetry, the elec­troweak standard model, loop gravity, string theory, twistors, and related top­ics. However, the many subjects cov­ered do not gain in transparency through the reading of this mastodonic treatise. In addition, the book suffers from a lackadaisical approach toward disciplines outside his fundamental re­search interests. The reader wanders through a wilderness of confusion, pop

© 2006 Springer Science+ Business Media, Inc . . Volume 28, Number 3, 2006 61

science, tortuous prose, yet also (for those with a command of the basics treated in the first 16 chapters) a host of brilliant insights worthy of a Pen­rose.

The writing is mediocre at best. It is painfully obvious that lbe Road to Re­ality was scantily edited, by both Pen­rose and the publisher, Alfred A. Knopf. Promoting it to a scientifically unedu­cated and ill-informed general public is at best disingenuous, at worst less than honest. The teacher mode is vintage "blackboard": breezy, redundant, with important and inessential observations mixed together with little attention to degrees of relevance. One finds fun­damental ideas (such as "orthogonality" or "open set") hastily inserted as after­thoughts following exposltlons in which knowledge of them has been as­sumed. Penrose employs Feynman di­agrams for two chapters before ex­plaining what they are.

My negative verdict on the haste, confusion, and sloppiness that charac­terizes lbe Road to Reality is reinforced by the fact that Roger Penrose is not confused in the least with respect to its subject matter. Any knowledgeable reader will recognize the powerful command he brings to bear on com­plex issues. Very few expositions of the ideas of modern theoretical physics present such a comprehensive vision of the whole picture. A summer's study of it is a valuable experience for someone like myself, that is to say someone with a solid grounding in the basics who, at

one time or another and in a sporadic manner, has had some exposure to the rest of its ingredients.

For the select audience with the background for understanding lbe Road to Reality there is much to be learned from it. We therefore examine its positive qualities first. Among them are:

(i) Shrewd observations only a Pen­rose is likely to make;

(ii) unique insights into subjects pre­viously mastered by the reader;

(iii) searching critiques of the deficien­cies of most of the "unifying" the­ories, or "theories of everything" of modern physics.

(i) For the informed reader Penrose will, from time to time, nonchalantly drop one of his amazingly cogent in­sights. For example, on page 153, after

62 THE MATHEMATICAL INTELLIGENCER

discussing the unique role of analytic functions in modeling causation, he ob­serves that this implies that "informa­tion" must be transmitted in discrete packages. An insight from causal de­terminism points directly toward quan­tum theory.

(ii) He wonders, somewhat rhetori­cally, if quaternions might be useful to physics, thereby leading us to a num­ber of observations that show why they aren't. His treatment of parallel trans­port is one of the best I've come across anywhere; his discussions of octonions, Clifford algebras, Grassmann algebras, and Clifford bundles are first-rate. (How useful they can be to anyone who doesn't even know the technical definition of an algebr�not given anywhere in the book-is open to question.) His discussions of gauge connections, covariant derivatives, bun­dles, and curvature unequivocally re­veal his understanding of these notions. Whether he can convey this under­standing to others who don't is another matter. In Chapter 2 Penrose does a great job of presenting a diversity of geometric models (Beltrami, Poincare, Klein, Minding) for hyperbolic geome­try. Their relationship to M. C. Escher's woodcuts is cleverly indicated. Com­puter transformations of the Escher pic­tures allow one to see the special ad­vantages of differing representations. The lack of a similar treatment of el­liptic geometry weakens the effective­ness of presentations, further on, of the properties of spinors, rotations in 3-space, and projective geometry. Chap­ter 17 on space-time is filled with orig­inal ideas. Observing that the natural setting for Galilean relativity is a fiber bundle, and that the space-time of Newtonian gravitation is best under­stood as an affine space, Penrose shifts the usual emphasis on the metric (which is analytic) to the null-cones (essentially topological) as the funda­mental determinant of space-time struc­ture. The treatment of Einsteinian space-time that follows is vintage Pen­rose at its best. Clear presentations of the Mach-Sehnder experiment (p. 5 13) and the Elitzur-Vaidman experiment (p. 545) provide convincing evidence for two mutually incompatible proce­dures at work in our quantum universe: the deterministic "U" process, and the statistical state reduction "R" process.

Penrose's perspective is particularly clear-sighted when he portrays the state of a highly confused science! His cele­brated arguments for a "low entropy" Big Bang appear on pages 690 to 712 .

(iii) A series of critiques of the "uni­fying" theories of modern physics, those which attempt to bring together relativity, quantum theory, and ele­mentary particle theory, begins in Chapter 28. They reflect the views and expertise of a mathematician who has thought long and hard about such mat­ters. Briefly: On page 755 Penrose di­rects the reader's attention to inconsis­tencies in inflationary models. On page 758 he shows up the circularity of ar­guments based on the Anthropic Prin­ciple. The pitfalls in the so-called Eu­clidization technique are noted on page 770. A truly impressive analysis of the six basic viewpoints on "quantum on­tology" begins on page 786. Inadequa­cies in ]. A. Wheeler's quantum foam hypothesis are pointed out on page 861 ; Penrose cogently observes that "quantum fluctuations" are insufficient to account for the formation of the galaxies. Throughout the book, Pen­rose expresses his dissatisfaction with theories that require the introduction of new dimensions, such as supersymme­try theories and string theory.

So the book has many strengths. What's wrong with it? A great deal, un­fortunately. In my opinion the negative features of lbe Road to Reality out­weigh its merits. These are:

(i) The fiction that it can be under­stood by someone not familiar with the disciplines it deals with;

(ii) the sub-standard writing; (iii) the incongruous manner in which

popular cliches about science and scientists are mixed in together with insights at the forefront of the­oretical physics;

(iv) The bad pedagogy in expositions of unfamiliar mathematics.

(i) Penrose claims that lbe Road to Reality can be profitably read by four audiences: The first class consists of people with so little aptitude for math­ematics that they have "difficulty in coming to terms with fractions." Yet Penrose assumes quite a lot of prereq­uisite knowledge, even erudition, from his readership. This may turn out, how­ever, to not be much of an impediment to sales. It has ever been the case that

owning a Bible will be deemed more important than taking the trouble to read it.

Next come those who are willing to peruse mathematical formulae but who lack the inclination to verify them. Pen­rose claims that the problems presented in the footnotes will help them do this. I did not find them helpful. Many are absurdly easy; others, forbiddingly dif­ficult.

Third are those readers who do have a mathematics background and wish to use Tbe Road to Reality as a textbook on the applications of mathematics to modern theoretical physics. Unfortu­nately all too many of the mathemati­cal expositions are flawed. Ideas and demonstrations are presented in the wrong order. The author has a ten­dency to correct himself as he goes along. Many treatments are truncated, while a confusing rhetoric contributes to obscure even the best of them.

The fourth group of readers consists of professionals working in areas of modern theoretical physics. Penrose is quite correct when he says: "You may find that there is something to be gained from my own perspective on a number of topics." Yet why should an expert spend $40 on a popularizer of a subject he already knows? Either he's already encountered his perspective through Penrose's many lectures, arti­cles, and books, or he would probably prefer to visit the library, where he can devote a pleasant afternoon browsing the pages between chapters 26 and 34.

(ii) Clear writing is essential to cor­rect thinking. It really does matter when a scientific idea is poorly expressed. Readers come away with the sense that they understand certain ideas, only to discover, once they try to use them, the extent to which they have been short­changed. This sentence on page 48 sets the tone: "For Hamilton found that ij =

-ji, jk = - kj, ki = - ik, which is in gross violation of the standard com­mutative law." What is a "gross viola­tion" of the commutative law? Is it worse than a "genteel violation" of the same law? What is the "non-standard" commutative law? Sentences like this sug­gest that little editing was done on a man­uscript that Penrose claims took him 8 years to write. In the Acknowledgments, he generously credits Eddie Mizzi and Richard Lawrence for help with editing.

How could they have missed so much? Evidently the staff editors at Alfred A. Knopf did virtually no editing, and a half­hearted job at proofreading.

Publishers of books and magazines tend to choose one of two approaches to scientific texts. The first, of which Scientific American is a notorious practitioner, is to systematically rewrite every manuscript accepted for publica­tion in a dull-as-dust, predictable house style. Everything looks and feels as if it came from the same assembly line, and only people interested in a partic­ular subject will take the time to read articles about it. The other is to assume that the ideas embedded in the turgid prose are so arcane, that to change even a single word might risk con­demnation by the entire scientific es­tablishment. This appears to have been the policy of the editors at Knopf. From the many examples of substandard writing I select two:

Yet, remarkably, according to the highly successful physical theories of the 20th century, all physical in­teractions (including gravity) act in accordance with an idea which, strictly speaking, depends crucially upon certain physical structures pos­sessing a symmetry that, at a funda­mental level of description, is indeed necessarily exact! (page 247)

I shall certainly not be able to go into great detail in my description of this magnificent profound difficult sometimes phenomenologically ac­curate yet often tantalizingly incon­sistent scheme of things. (page 657)

An expositor of current science need not be a prose master like Bertrand Russell or D'Arcy Thompson. Prose like this, however, is beneath any accept­able standard.

(iii) Tbe Road to Reality combines an overly scrupulous concern for pri­ority recognition with narratives of the life and thought of major figures, yet it shows little respect for historical schol­arship. On page 81 Penrose feels the need to remind us that the properties of what is known variously as the "Ar­gand diagram" or the "Gaussian plane" were first discovered by Caspar Wessel (unknown for anything else) in 1 797. To learn more about Wessel, I con-

suited Florian Cajori's History of Math­ematics (1980). Paraphrasing page 295: "Caspar Wessel. Employed as surveyor by the Danish Academy of Sciences. Es­say on the Analytic Representation of Direction. Published in Vol. V of the memoirs of the Danish Academy . . . article buried for a century. French translation published in 1897 . . . . " Hav­ing thus restored an unjustly neglected reputation, one might assume that Pen­rose would behave with equal concern for such distinguished figures as Pythagoras, Plato, Leibniz, and Aristo­tle. Such is not the case. Granted, only the most fastidious math historian would be outraged rather than amused by Roger Penrose's popular science ex­humation of the life and works of Pythagoras. From page 5 of the Pro­logue one learns that the "sage" Pythagoras maintained a "brotherhood" of 571 wise men and 28 wise women at Croton in southern Italy. On page 10 he gives "dates" for "Pythagoras of Samos" as 572-497 B.C.E. Yet in a foot­note he admits that "almost nothing re­liable is known about Pythagoras, his life, his followers or their work. " Ig­noring his own cautionary note, he then attributes to Pythagoras the dis­covery of the idea of mathematical proof! The rest of page 10 is filled with a long list of accomplishments and dis­coveries by these anonymous figures.

Coming to Plato, the astounding reach of one of the greatest minds in history is encapsulated in a simple thumbnail phrase: Page 1 1 : " . . . Plato made it clear that the mathematical propositions [ . . . ] inhabited a differ­ent world distinct from the physical world. Today, we might refer to this world as the Platonic world of mathe­matical forms. " My objections to Pen­rose's one-line description of Platonism are not so much directed at his super­ficial version of the "story of philoso­phy" as to the obvious disdain that he and many research scientists manifest to­ward the kindred disciplines of philos­ophy and the history of science. Some­how it never seems worth their time and trouble to "get it right. " Despite this, Penrose then devotes all of section 1 . 4 to a travesty o f serious philosophy i n a series of meditations on "the three deep mysteries": the connections between the physical, the mental, and the Platonic mathematical. The shift from arid Pia-

© 2006 Springer Science+ Business Media. Inc . • Volume 28, Number 3, 2006 63

tonism to penny-ante neo-Platonism has been made without missing a beat. It's threadbare pop philosophy from the pen of a brilliant mathematician and ma­jor scientific figure.

Even worse is his portrayal of "Aris­totle" and "Aristotelian thought" in Chapter 17. On page 382 he writes: "In Aristotelian physics, there is a notion of Euclidean 3-space £3 to represent phys­ical space." Aristotle's dates are 384-322 B.C.E. He directed the Lyceum in Athens, from 335 to 323. Euclid's Ele­ments were published in Alexandria around 320, where he had passed 10 years working on them. In fact, Aristo­tle's physical universe has very little to do with Euclidean 3-space, and less with any notion of a geometrized "space-time."

Penrose's perspectives on Galilean, Newtonian, and Einsteinian space-time are less problematic. Not only does he know what he's talking about, he be­longs among the handful of modern thinkers with the best understanding of them. I would have liked him to men­tion the equally important universe models of Descartes and Leibniz. Des­cartes's "horror of the vacuum" has resurfaced in our own day in Dirac's "electron sea," and Leibniz's brilliant critique of Newton's absolute space and time has delivered its delayed fruits in General Relativity.

(iv) Many of Penrose's explanations are mystifying to all four classes of read­ers, including the experts. Here is a par­ticularly glaring example (p. 642): "A general U(2) transformation of the Her­mitian matrix (which we must bear in mind involves both pre-multiplication by the U(2) matrix and post-multiplica­tion by the inverse of that matrix) does 'churn around' the elements of this Her­mitian matrix, in very specific ways, but its Hermitian character is always pre­served. In fact this analogy is very close to the way in which U(l) indeed acts in electroweak theory (the only com­plication being that we must allow for a linear combination of the diagonal el­ements with the trace, in this identifi­cation, related to the 'Weinberg angle' that we shall be coming to in §25 .7) . " This paragraph is incomprehensible to someone who doesn't already know the subject. (What "analogy"?)

There are also notational inconsis­tencies, but they are quite minor in

64 THE MATHEMATICAL INTELLIGENCER

comparison with his masterstroke of obfuscation: a frequent appeal to ten­sor diagrams, starting on page 241 . These are tiny drawings that must look peculiar to anyone not working in knot theory, topological quantum field the­ory or the mathematics of "q-deformed matrices." Totally mystifying to out­siders, they reduce the readership to the tiny elite for whom they sometimes simplify the computational tedium as­sociated with tensor analysis. Yet on page 260 Penrose allows that his read­ers may not even know what a "deter­minant" is! After a 4-page crash course on determinants, he proceeds to "prove" the fundamental theorem of determinants (det(AB) = detAdetB, A, B matrices) by pointing to a particu­larly incomprehensible tensor diagram on page 264! The worst failing in this domain is, to my mind, the absence of any introductory material on topology.

In summary, Tbe Road to Reality is really only written for persons who've previously studied most of its contents. It is virtually unedited. It is very com­plete; indeed it tries to do too much. It can be used as a reference book at a rudimentary level for professional theo­retical/mathematical physicists. For all other classes of readers, unfortunately, it is likely to be a disappointment.

8 Liberty Street #306

Middletown, CT 06457

USA

e-mail: rlisker@yahoo.com

A I'Origine de Ia recherche

scientifique:

Mersenne by Jean-Pierre Maury edited by Sylvie Taussig PARIS, VUIBERT, 2003, 311 PP.,

ISBN: 2-711-75291-7, €33.00

REVIEWED BY JEANNE PEIFFER

· · · .. •

.•.

... , Li he title of this book clearly reveals

.··

..

· its author's aim: to write a biogra-; phy of Marin Mersenne (1588-

1648), a seventeenth-century monk trained by the Jesuits at the famous col­lege in La Fleche.

The author, Jean-Pierre Maury, pre­sents Father Mersenne as a founding fa­ther of scientific research. But what does he understand by "scientific re­search", given that no famous concepts or crucial inventions (except the Mersenne numbers) are named for his biographee? The phrase instead stands for scientific communication: Mersenne was one of the most prolific letter-writ­ers of his period. Except for some trav­els in France and abroad, he lived in his cell at the Parisian convent of his Order, the Minims, receiving visitors, or­ganizing meetings between the most fa­mous scientists of his time, experi­menting, and writing. His role in the history of science can be best described as that of an intermediary, questioning through letters the savants from Leiden to Gdansk, and thus establishing con­tacts among the members of a vast com­munity of scientists. It has even been said that he originated the seventeenth­century scientific community, whose boundaries coincide with those of his network of correspondents. His pub­lished correspondence fills seventeen (not eleven, as the readers of the book are wrongly told) volumes, edited in the twentieth century by several genera­tions of researchers. Thus Maury offers a highly modern understanding of sci­entific research, putting emphasis less on the actors and their results than on their practices: traveling, visiting each

other on recommendation of interme­diaries, meeting in private academies, experimenting, exchanging informa­tion, debating, writing journals, letters, and books, some of them published. Research is pictured as a collective en­terprise, a common effort toward the advancement of science. Mersenne's role was pivotal: he can be considered as one of the first mathematical intelli­gencers in history.

Maury has given his book an origi­nal structure. The first part is devoted to three of Mersenne's friends: Pierre Gassendi, Nicolas Claude Fabri de Peiresc, and Rene Descartes, with each of whom he initiated his role as a great communicator in the 1620s. The second part describes Mersenne's activities dur­ing the crucial year of 1634. Finally, in the last part, Maury discusses a prob-

!em that played an important role in the creation of modern science: the void.

The first part presents young Mersenne's efforts to build his network of acquaintances and correspondents. Each of his friends, Peiresc, Gassendi, and Descartes, enables him to make contact with further savants. Through Peiresc, he is introduced to musicians like Jacques Mauduit, an actor of Ba.ifs academy, to Jean Titelouze, an organ player from Rouen, and to Italian con­noisseurs of ancient music, like Gio­vanni Battista Doni in Florence or Jacques Gaffarel in Venice. It is espe­cially with Descartes, after he left for the Netherlands, that Mersenne achieved his apprenticeship as an "in­telligencer". He aimed to be the only link between the philosopher and his homeland. This part of the book also gives the reader a flavor of the scien­tific atmosphere in Paris when Mersenne arrived there in 1619. The monk took part in the heated debates on Paracelsian alchemy, and on the phi­losophy of the Rosecrucians. In a com­mentary on Genesis (1623), he con­demns all kinds of heresy in the strongest terms. Maury interprets Mersenne's attacks against the English Paracelsian Robert Fludd as a first sign of Mersenne's future orientation toward science put in the service of religion.

The second part, focusing on the year 1634, shows Mersenne's network devel­oping, especially in southwest France, where he discussed the problems of the fall of a body or of the pendulum with correspondents like the lawyer Jean Trichet and the physicians Jean Rey and Christophe de Villiers . Maury's book gives a lively account of the outburst of scientific research during the first part of the seventeenth century, and we en­counter a number of lesser-known ac­tors whose biographies are fortunately provided by Sylvie Taussig in an ap­pendix. In two of the five treatises he wrote in 1634, Questions inouyes and Questions theologiques, physiques, etc., Mersenne extended the method of ques­tioning he had developed with corre­spondents like Descartes, Van Belmont, and Beeckman. He was looking for an­swers to unsolved problems from his readers. One year after Galileo's con­demnation, Mersenne published the Me­chanicques de Galilee, Florentin and sig­nificantly contributed to the circu-

lation of Galileo's ideas, especially in France. Also in 1634, Tommaso Cam­panella escaped from the Roman Inqui­sition's prison and came to Paris. Maury is excellent in characterizing the relations between the two men, and especially Mersenne's disappointment when he fi­nally met Campanella. In his prison, Campanella had hardly been able to come to grips with the quick advance­ment of science. Mersenne saw him as a man of the past, while he himself was gathering in his circle the most modern representatives of science and especially of mathematics: Etienne Pascal, Gilles Personne de Roberval, Claude Hardy, Claude Mydorge, etc. He was thus a fore­runner of the modern academy.

The third part is a study of Mersenne's contributions to a well-known problem in the history of science, the problem of the void: is it possible for any part of the universe to be absolutely empty? The ac­cepted view at that time was that there could be no such thing, because of "na­ture's abhorrence of a vacuum". Blaise Pascal confirmed the existence of a vac­uum in 1648, thus bringing, in the au­thor's eyes, the new science born with Galileo to maturity. Mersenne, traveling in Italy in 1644, brought news of Torri­celli's barometric experiment to France. He served as an intermediary between Italian experimenters, like Gasparo Berti and Emmanuel Maignan in Rome, for in­stance, and his French correspondents, especially the engineer Pierre Petit, who was able to repeat Torricelli's experiment with Etienne and Blaise Pascal in Rouen. It was left to the young Pascal to give an interpretation of the experiment. In 1647, he set a glass tube sealed at one end in a bowl of mercury and showed that the space not occupied by the ris­ing liquid was empty. He then designed his famous experiment at Puy-de-Dome, in which he showed that, on a moun­taintop, the height of the column of mer­cury decreases due to the pressure of the air outside. This story is well known. Maury's narrative takes in a number of lesser known actors, for instance Valeri­ano Magni experimenting in Poland, and describes in detail the contacts between the different actors, their travels, en­counters, academies, written reports, and letters exchanged.

It is a pity that Maury, who excels in giving an accurate picture of seven­teenth-century science as a collective

enterprise, has not extended this con­cept to modern history of science. He hardly makes any use of the important secondary bibliography; he does not even make the slightest reference to Ar­mand Beaulieu, the editor of the last volumes of Mersenne's correspondence and also the author of a biography of Mersenne. Professional historians will be upset by the lack of references, some unfortunate references to "the dark mid­dle ages", and the ignoring of their own work. But this book will be read with profit by all those whose interest in sev­enteenth-century science is new and who want to know how modern sci­ence took its present form.

Jean-Pierre Maury, who died in 2001 , was not able to finish his book; it has been edited by Sylvie Taussig, a philoso­pher and Gassendi scholar. She added footnotes and valuable appendices to the core of the work: a chronology, short bi­ographies of all the actors of Maury's story, a bibliography of Mersenne's writ­ings, and a substantial afterword putting emphasis on the links between science and religion in Mersenne's work and in the seventeenth century.

Centre Alexandre Koyre (CNRS)

27 rue Damesme

F-7501 3 Paris France

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Irresistible Integrals:

Symbol ics, Analysis and Experiments in

the Evaluation of Integrals by George Boros and Victor H. Moll NEW YORK, CAMBRIDGE UNIVERSITY PRESS,

xiv+306 PP., US $29.20 ISBN 0-521-79636-9 (pbk)

REVIEWED BY J. J. FONCANNON

he physicist Richard Feynman once claimed that he acquired his initial professional reputation not

as a physicist, but as a redoubtable eval­uator of integrals. A colleague would

© 2006 Springer Science+Business Media, Inc., Volume 28, Number 3, 2006 65

approach him with a knotty integral dis­covered during the investigation of some physical problem. A couple of hours later Feynman would return the integral, fully evaluated. The donor, typ­ically, would react with bemused rev­erence. Feynman, who had a mischie­vous nature, wa� secretive about how he accomplished what he did, but he later revealed that he had developed an armamentarium of techniques. Promi­nent among these was differentiation of the integral with respect to a parame­ter to produce a differential equation, which he subsequently solved. Further­more, Feynman had acquired an intu­itive feel for how to introduce a pa­rameter artificially when the original integral lacked one, and then he would work with the modified integral. This approach, of course, constitutes a sub­case of a widely recognized mathemat­ical ploy: if you cannot solve the orig­inal problem, solve a more general problem. Differentiation with respect to a parameter is a simple technique the authors of the present book use often. Other techniques they use are to gen­erate a recurrence for the integral and then solve the recurrence, or to employ partial fractions. Many current victories in the evaluation of challenging inte­grals have been obtained through the use of modern mathematical develop­ments, like group representation theory or arguments based on combinatorial reasoning. However, these accomplish­ments lie far beyond the scope of the present book.

Like all good things, skill at the eval­uation of integrals increases with expe­rience, and expertise does not depend, necessarily, on one's being a practicing mathematician. One of the most skillful practitioners I have ever known was an aeronautical engineer, who was a wiz­ard at evaluating integrals around branch points in the complex plane.

The present book is replete with riches. However, in my view, it suffers from two major shortcomings. One is in the paucity of material from the theory of special functions. The other is the lack of function-theoretical methods. Most integrals arising in mathematical physics call for the special functions of mathematical physics, either in the def-

inition of the integral or in the evalua­tion of the integral: Bessel functions, hy­pergeometric functions, the error func­tion, the exponential integral, sine and cosine integrals, etc . . Hypergeometric functions are mentioned in the present book, but not presented in enough de­tail to give the reader a feel for them. All mathematical software has the ca­pability of handling such functions, and I find it strange that the authors, who rely heavily on Mathematica in their ex­position, have declined to define and provide the properties of such func­tions. They utilize only the gamma and beta functions and some related func­tions.

I fully understand the authors' rea­son for not including function-theoretic arguments or more material from spe­cial functions, namely, the lack of ex­perience among the undergraduates for whom the book was primarily written. But the authors have ended up with a book that will be of only provisional appeal to its chosen audience, and it certainly will not provide enough math­ematical substance to interest those who will go on to become practicing physi­cists or mathematicians. As a conse­quence the book has a fussy, ad hoc, chase-your-own-tail quality, and too rapidly degenerates into a bricolage of formulas. I suspect that some of the ma­terial was included not because the au­thors considered it truly germane, but because they couldn't stop. (How well I understand this affliction.)

All the integrals the authors treat that are of the form

r X' (a + x)f3 (b + x)'Y dx, 0

and the investigation of these com­prises a substantial part of the text, are actually hypergeometric functions1 , 2F1's . To speak a little more of these functions would have saved countless pages of busywork. Much of the rig­marole in the book could have been avoided, leaving more room for so­phisticated matters.

As a number of current texts have shown, the tyro can rapidly be given the knowledge necessary to apply the residue theorem-precisely what one needs to evaluate integrals. The book

1 Erdelyi, A., et a/., Higher Transcendental Functions, v. 1 , p 1 1 5, (5), New York, McGraw-Hill, 1 953.

2Seaborn, James B., Hypergeometric Functions and Their Applications, New York, Springer-Verlag, 1 991 .

66 THE MATHEMATICAL INTELLIGENCER

of Seaborn2 does this brilliantly, in a scant 30 pages. A student who can con­ceptualize power series (and the au­thors rely on these heavily) can deal with the rudiments of complex analy­sis. The authors talk at great length about the evaluation of

Lm(a) = Ioo dx o CXZ + 2ax + l)m+l

m = 0, 1 , 2 ,

A s previously pointed out, these are hypergeometric functions, but a more elegant way of proceeding is to evalu­ate by residues the integral

J x' dx c CXZ + 2ax + l)m+ l

along a Mellin contour C-a circle with radius R centered at (0) cut along the positive real axis, R -7 eo--and then to let s -7 = 0.

In Chapter 2, the authors show that Lm(a) satisfies the recursion

2m - 1 Lm(a) = ( 2)

Lm-l(a) 2m 1 - a a

but they state that obtaining a precise form for Lm(a) from this relationship "seems to be difficult. " No-it is easy. The equation is a linear first-order non­homogeneous difference equation and can be solved by the standard tech­nique: obtain the solution of the related homogeneous equation, and then a

particular solution by variation of pa­rameters. The authors should have in­cluded this procedure in their intro­ductory material.

A minor misgiving I have is the em­phasis on "conjecturing closed formu­las . " The authors are devoted to this arithmancy, and they invoke it re­peatedly. Typically, they will compute several selected values of a number­theoretic sequence Xn and, providing hints, ask the reader to conjecture a general formula for Xn. This approach reminds me of the hoary query I was posed on moving to what is now my hometown: What is the next entry in the sequence

30, 34, 40, 46, 52 , 60, 63, 0 0 0 ?

The answer is Millboume, the next stop on the westbound Market Street El. Al­though the answer will resonate only with a Philadelphian, the vaunted tech­nique of conjecturing an answer, along with $2, will certainly entitle you to a ride on the El.

Before I move to more positive things, I must remark on the numerous typos in the book-"cosideres," "disc­tinct, " etc. And then there are exposi­tory blunders. Wallis's formula is de­clared to be proved, but Wallis's formula is not clearly limned. However, who am I to complain? I sometimes think that my first book, Sequence Traniformations, could have been des­ignated more accurately, Mistakes in Se­quence Traniformations. At least my culpability is less; at that time, we did­n't have spell checkers.

The first chapter of this book dis­cusses some basics: prime numbers, the binomial theorem, the ascending facto­rial symbol:

(a)n = a(a + 1)(a + 2) . . . (a + n - 1 ), n = 1 , 2, 3, . . . , (a)o = 1 .

Chapter 2 discusses the integration of rational functions. Chapter 3 is entirely allocated to the integral

Loo X' I(m,n) = ( + )m+ 1 , o q1x qo m > n,

m, n, integers. (More hypergeometric functions!) The authors first show that

1 I(m,n) = m-n n+ 1 qo q1

I (- l)n�j (!'?), J�o m- 1 1

and then proceed, after many pages, to find a formula for the sum. If they didn't want to use 2F1 's, I don't under­stand why they didn't at least make use of the easily proved formula

(*)

The value of the integral I(m, 0) is im­mediate. Using (*) , one differentiates this formula n times with respect to q1 , then replaces m by m-n to give the au­thors' final formula. Of course, there is no reason to do any of this if one uses residues. The integral can be evaluated

for general n by the Mellin contour technique mentioned previously.

Chapter 4 talks about power series. For a power series with coefficients Cn the authors give for the radius of con­vergence R of the series the formulas

and

These formulas are both wrong, the first more so, since it isn't rescued by re­placing the lim by a limsup, although the second formula is. A power series that illustrates the perils is furnished by the one with coefficients { 1 , 1, 1,4, 1,42,1,43, 1,44, . . . } , having radius of convergence 1/2, though neither of the above limits exists.

The authors next define the diloga­rithm function and display one of my favorite-and one of the most arcane­formulas in analysis, taken from the book by Lewin3:

Jc\15-n;2 ln(l - x) o X dx =

]n2 ( Vs- 1 ) - �. 2 10

A treatment of Eulerian polynomials

An(X) = (1- x)n+ 1 X - -- , ( d )n 1 dx 1 - x

which have applications in the sum­mation of series, closes the chapter.

Chapter 5, on the exponential and log­arithmic functions, is rudimentary, but there are interesting nuggets sprinkled throughout: the irrationality of e, a su­perb section on Stirling's formula, and an extended treatment of Bernoulli num­bers. Chapter 6 discusses the basic trig functions, 'TT, Wallis's product, more power series expansions for elementary functions, and the Riemann zeta function.

In Chapter 7, the authors treat the quartic integral

r (x4 + 2a�� + nm+ 1

another 2F1, with parameters {1/2, m + 1 , 2m + 2} and argument 2VT-1""1 CVT-J"" - a), actually, a Legendre function. All of its salient properties re-

3Lewin, L. Dilogarithms and Associated Functions. London, Macdonald, 1 958.

suit from the known properties of Gauss's function, including the recur­rence relations it satisfies. In addition, because of the form of its parameters, the function admits a large number of quadratic transformations. One of these (Erdelyi, p . 1 13 , (30)) reveals that the integral is, essentially, a polynomial 2F1 of degree m in a.

Chapter 8 looks at some classical ma­terial with fresh eyes. The topic is the normal integral,

Loo v; I = o e-xz dx = -2- .

The authors have, apparently, made a hobby of collecting information on the evaluation of l The first method they give is the one known to all of us: ex­pressing I2 as an integral over the first quadrant in the x,y plane and switch­ing to polar coordinates. But they give an even simpler method:

I2 = r· e-x2 dx f"e_ uz du = 0 0 f"' C x2 f"' e-xzyz X dy dx 0 0 = J'"' r· xe-xZ(l+yZ)dy dx

0 0

1 f"' dy 7T = 2 0 1 + y = 4 • They next advance a profoundly clever number-theoretic argument that relates I to the number of representations of an integer n as the sum of two squares, rin). The derivation utilizes the identity ( 00 )2 00

n�oo xn2 = �0 r2(m)xm. I wish I could reproduce this soul-sat­isfying derivation in its entirety. The student is asked to prove the above identity as an exercise-no hints. This should separate the wheat from the chaff.

The next chapter provides a nice in­troduction to what is known about Euler's constant,

y = lim An, An = ( I -k1 - In n) , n--->OO

k� 1

y � .5772156649015328606065120900824024310421 59335939923598805767 .

y is one of the most inscrutable con­stants in mathematics. (Why couldn't the authors have tantalized the reader

© 2006 Springer Science+ Business Media. Inc . • Volume 28. Number 3. 2006 67

with just a few of its digits?) Its irra­tionality is still undecided. Basically, the difficulty is that we do not have repre­sentations of y that converge with suf­ficient rapidity. Perhaps the solution to the problem will eventually be found by developing such representations, in other words, with progress in algorith­mic mathematics. The above limit is pa­thetic, for, as the authors show,

1 1 1 - - -- < An - y < - . 2n 8n2 2 n

The authors derive the standard inte­gral representations for y and some ad­ditional series representations which, to my mind, are of limited interest.

Chapter 10 on the Gamma and Beta and related functions, has much stan­dard material. The authors don't prove the Bohr-M0llerup theorem-rex) is the only positive logarithmically convex continuous function that satisfies rex + 1) = xrex) , re1) = 1-which is a shame. The proof is elegant and sim­ple.4 However, they do provide Totik's proof that rex) satisfies no differential equation (of a certain type).

Chapter 1 1 , on the Riemann zeta function, is one of the most interesting in the book, including, as it does, some intriguing contemporary findings. Apery, in a celebrated paper, demon­strated the irrationality of �(3),

oc 1 �(3) = I 3 ·

n= l n The authors don't furnish the proof, de­cidedly non-trivial, but the background information they provide is interesting in its own right.

Chapter 1 2 deals with logarithmic in­tegrals, and Chapter 13 , "A Master For­mula, " studies the integral

M =

I""( x2 )r ( x2 + 1 ) dx

o .0 + 2ax2 + 1 x' + 1 x2 ·

M is, surprisingly, independent of s. Thus its evaluation can be effected by replac­ing s by 2, which gives an integral pre­viously studied. As I have noted, this is a Legendre function, and its properties follow from known results for that func­tion.

There is an appendix of great value, "The Revolutionary WZ Method." The ti­tle is not hyperbole. The method, due jointly to Herbert Wilf and Doron Zeil­berger, is demonstrably one of the supreme achievements of algorithmic mathematics. It continues, perplexingly, to be ignored, even though the algorithm is available online. I have used it fre­quently to obtain elegant and simple dosed-form representations for certain hypergeometric functions, for instance, the associated Legendre polynomials. I gave a lecture on this method 7 years ago at a Southern university. From the questions asked, I concluded that almost no one in the audience had ever heard of it. The experience reinforced my per­ception that purblind specialization, of which such ignorance is the inevitable issue, is the constant enemy of mathe­matical progress.

All in all, I think the present book, despite its problems and despite its stip­ulated audience of naifs, may appeal to many readers. Some of the results are astonishing, and others point directions for future research. (Is �(5) irrational? �(7)? �(9)? �(1 1)? Rivoal and Zudlin have shown in Uspekhi Mat. Nauk (56 (2001), no. 4, 1 49-150) that at least one of these is.)

Philadelphia, Pennsylvania

USA

e-mai l :bolu.bolu@verizon.ne

Bourbaki . U ne

societe secrete des mathematiciens by Maurice Mashaal COLLECTION LES GENIES DE LA SCIENCE, 2002,

BELIN-POUR LA SCIENCE, PARIS, 160 P.,

€16.00, ISBN 2-84245-046-9

REVIEWED BY OSMO PEKONEN

'7\ �· aurice Mashaal is a French sci-1 \/ ence journalist who has com­. piled this lovely, richly illus­

trated book about the collective mathematician Nicolas Bourbaki, born

4Conway, John B., Functions of One Complex Variable, New York, Springer-Verlag, 1 978.

68 THE MATHEMATICAL INTELLIGENCER

in 1934 and possibly still alive and kick­ing. It first appeared as a special issue of the French magazine Pour Ia Science and is sold at newsstands in the streets of Paris. Now it has been reissued as N° 1 of the book series 'Geniuses of Science' , which portrays Becquerel, Cu­vier, Einstein, Fermi, Galilei, Kepler, Leonardo, Pauling, and Poincare, among many forthcoming others. Some of these volumes are remarkably well­edited and certainly worth translating into other languages.

Nicolas Bourbaki is known as the au­thor of Elements de Mathematique, a treatise measuring so far some 7000 pages, whose ambition is a unified pre­sentation of all mathematics deemed worthy of attention. Mashaal's book is a well-balanced synthesis of mathemat­ics, biography, history of learning, and anecdotes and humor. It is aimed at the man in the street who understands-as many Frenchmen do-mathematics as an integral part of culture but doesn't necessarily have formal training in the field. Some case studies of formal math­ematics are developed in the axiomatic Bourbaki style for the benefit of those readers who remember the basics of their lycee curriculum, but the main text can be read independently of the boxed formulas.

The story of the "real" Bourbaki, a general of the French-Prussian war of 1870-1871 , has often been told. Inci­dentally, a street (rue Bourbaki) in Pau, southern France, is named after him. The general's forename was Charles, whereas the fictitious forename Nicolas was suggested by Andre Weil's wife Eveline. A picaresque aspect of the tale was the appearance in Paris in 1948 of a Greek diplomat carrying the ominous name Nicolaides Bourbaki. He was courteously invited to dine with the mathematicians. Many other legendary aspects of Bourbaki, and other folklore typical of Ecole Normale Superieure, are discussed by Mashaal who has had ac­cess to some old issues of La Tribu, the confidential and funny newsletter of the group.

The text includes minibiographies of the following five "archicubes" : Henri Cartan, Claude Chevalley, ]ean Delsarte, Jean Dieudonne, Andre Wei!. Rare pic-

tures and many stories of other celebri­ties such as Armand Borel, Pierre Cartier, Adrien Douady, Samuel Eilen­berg, Alexander Grothendieck, Laurent Schwartz, and Jean-Pierre Serre are in­cluded. Of particular documentary value are the two photographs, taken in 1937 and 1938, in which Andre Weil's famous sister, the philosopher and mys­tic Simone Wei! (1909-1943), is seen taking part in sessions of the Bourbaki group.

The notion of structures emerged simultaneously as a key concept of mathematics and literature. Some Bour­baki-inspired authors, like Raymond Queneau and Jacques Roubaud, are dis­cussed. Mashaal doesn't believe, though, that mathematics influenced much the birth of structuralism as a lit­erary theory; he rather views these phe­nomena as interesting parallel develop­ments.

The story of Bourbaki is the saga of a handful of determined young revolu­tionaries who wanted to reshape the way mathematics was taught in France. Their success was overwhelming, with worldwide implications. In due course, the archicubes themselves developed into clerics of a new orthodoxy which occupied all available academic space and, after a golden heyday, may have actually hampered the development of science in France. The New Math of the 1960s (to be compared with the New Economy of the 1990s . . . ) may have started with Dieudonne's exclamation in 1959: "Down with Euclid!" By the early 1970s, he rued his words and de­nounced the birth of a new form of scholasticism. Benoit Mandelbrot (whose uncle Szolem Mandelbrojt was one of Bourbaki's founders) claims that he fled to the United States to escape its "suffocating influence" in France.

Notable omissions in the Bourbaki program, which Mashaal doesn't fail to discuss, include probability, mathemat­ical physics, and, strangely enough, many foundational issues. Bourbaki also chose to ignore category theory. Mashaal suggests that its inclusion would have implied far too devastating modifications in the whole architecture of Elements de Mathematique, which had been rooted in set theory since the 1930s. Categories and functors crept into the text nonetheless, but without being named so.

According to the rules, one must re­tire from Bourbaki at the age of 50, and presumably this also corresponds to the life-cycle of the whole enterprise. When I was a student in Paris in the late 1980s, Bourbaki was already referred to as someone who had passed away but whose posthumous reputation kept lin­gering on. In October 1988, the French radio channel France Culture even broadcast a formal announcement of the death of Bourbaki, due to a "non­removable pathological singularity." Even so, a posthumous Chapter 10 of Commutative Algebra appeared in 1998.

Mashaal's account is a rewarding one also for the professional mathematician as a source of anecdotes and rare pic­tures, and as a glimpse into the French history of mathematics. It conveys very well an overall impression of the rise and fall of the poly cephalic scientist. For a more scholarly account, one should rely on works by Liliane Beaulieu and others.

Agora Centre

4001 4 University of Jyvaskyla

Finland

e-mail: pekonen@mit.jyu.fi

M usic and

Mathematics edited by]. Fauvel, R. Flood, and R. Wilson OXFORD, OXFORD UNIVERSITY PRESS 2003

ISBN 0-19-851187-6 £39.50

REVIEWED BY EHRHARD BEHRENDS

everal years ago, preparing activ­ities for the general public on the occasion of the ICM'98 in Berlin,

the Berlin universities organized several seminars to discuss the relations be­tween mathematics and music. Many different aspects were covered by the lectures, which were given by people working in these fields, among them a number of young composers. For me, it was a surprise to learn that mathe­matical ideas are rather influential in certain areas of contemporary music (which, however, are not very close to my personal musical interests: classical

music and jazz). Also I had not realized before how important physiological facts are to the possibility of using math­ematical structures successfully. For ex­ample, most of us are unable to recog­nize the absolute pitch of a tone. Only the relations between different pitches are perceptible: the interval E-G is "the same" as the interval A-C. Also, in many cases a tone may be replaced by its oc­tave without noticeably changing the character of a musical piece. As a con­sequence, much of the work concerned with scales can be reduced to finding the appropriate pitch ratios between one and two.

"Music and mathematics" is not a well-defined area, but most of what can be said concerns one of the following queries:

1 . What are the mathematical princi­ples that underlie the construction of musical scales? What are the defects

12;::: of the Pythagorean scale, why is v L. important?

2 . Are there other problems of interest to the "working musician" that can be solved by using (more or less ad­vanced) mathematics? For example, of what help is Fourier analysis for the artisan who wants to construct a guitar?

3. Is mathematics helpful in analyzing musical compositions? Did certain composers have mathematical struc­tures (like symmetry or remarkable relations between numbers associ­ated with the composition) in mind during their work?

4 . Can mathematics be used as a tool­box to produce interesting music?

All of these aspects are discussed in this book. The first contribution, "Tun­ing and temperament" by Neil Bibby, starts with a description of the early at­tempts of the Pythagoreans to relate harmony to mathematics. They discov­ered that intervals are considered to sound harmonic if the ratio of the fre­quencies is a rational number with "small" numerator and denominator. The ratio 2 : 1 is not very interesting since the octave is somehow identified with the original tone. The Pythagorean scale is based on the ratio 3 :2 , the per­fect fifth. If one starts with "C" , one first obtains "G", then "D" etc. Here, one has to apply the principle, mentioned earlier, that a note can be identified

© 2006 Spnnger Science+ Business Media, Inc., Volume 28, Number 3, 2006 69

with its octave. So the fifth of the fifth, the ratio 9 :4, is replaced by 9:8 to make the ratio lie between one and two. In this way the notes corresponding to the white keys of the keyboard are gener­ated. (However, to produce the "F" one has to go backwards: "C" is the fifth of "F": "F" is added to the scale for this reason.)

Soon it was realized that many of the intervals which occur in this scale are far from being simple. For example, the frequency ratio of the major third ("C" to "E", say) is 81 :64. Also, the Pythagorean scale is not well suited for modulation. If one considers a note al­ready constructed as the keynote of a new scale it will be necessary to include new notes. For example, one has to add "F sharp" when starting with "G". This process never stops. There is no finite scale which is closed under forming Pythagorean scales if one can select any note as a basic key note.

Bibby describes some of the pro­posals that have been made to over­come these difficulties. For example, we learn how the frequency ratios in the scale of the just intonation are defined and to what extent it is superior to the Pythagorean scale. And that Marin Mersenne designed a keyboard with 31 notes for each octave, which made it possible to distinguish between "F sharp" and "G flat", a distinction which is important in the Pythagorean scale and in the just intonation.

Most readers will know that now­adays nearly all instruments use the equal temperament. The frequency ra­tio between two adjacent notes of the 1 2 notes in an octave on a keyboard is always the same, and in this way the twelfth root of 2 comes into play. It is a really "democratic" scale, as each note plays the same role. Nevertheless it is a rather ironical aspect of the history of music that one started with the philos­ophy of small rational numbers and ended up with a scale where no inter­val (up to the octave) is rational.

But this is not the end of the story. Later (in Chapter 9) we learn why, be­sides our 12-tone system, certain n-tone systems play a role in music theory. Here in particular the cases n = 13, 19, 21, 31, and 53 are of some interest. Sim­ilar questions are mentioned also in some other contributions. For example, in Chapter 2-a chapter with a more

70 THE MATHEMATICAL INTELLIGENCER

historical than mathematical empha­sis-]. V. Field explains how Kepler tried to find musical proportions in var­ious quantities of the solar system.

For me, the most interesting chap­ters are those of Part Two, "The math­ematics of musical sound". First, in Chapter 3, Charles Taylor describes some experiments with real instruments to demonstrate how one can hear com­binations of notes. It is a strange fact that the ear sometimes "hears" the dif­ference note of, e.g., 80 Hz if two notes of 500 and 580 Hz are played simulta­neously. Taylor has no convincing ex­planation of this phenomenon; it is ar­gued that the effect is caused by a combination of physical and physio­logical reasons.

Next, in Chapter 4, Ian Stewart demonstrates that many interesting mathematical problems are touched upon if one wants to calculate the po­sitions of the frets of a guitar correctly. He explains the difference between "construction with circle and ruler" and "construction with circle and unmarked ruler" and how simple it is to trisect an angle if one is allowed to mark a dis­tinguished point on the edge of the ruler.

The main part of Stewart's article is the description of Strahle's construction of the position of the frets. Strahle, a Swedish craftsman, suggested his con­struction in 1743, but it was erroneously argued by Jacob Faggot, of the Swedish Academy, that the argument has a flaw. Strahle had in fact found an approxi­mation of VZ by simple geometric means, which in a sense, is optimal: the best approximation of 2x by a function of type (ax + b)!( ex + d) is

(2 - \12)x + \12 (1 - \12)x + \12 ,

and Strahle used this function; he ap­proximated \12 by 17/12, a ratio which appears when expanding v2 as a con­tinued fraction.

The third article in this part (Chap­ter 5) is David Fowler's essay on Helmholtz. For many readers it will be a surprise to learn that Helmholtz not only was a famous physicist but also a physiologist who worked on the phys­iological basis of the theory of music. Fowler starts with a description of Helmholtz's experiments with sound generators; they were used to demon-

strate combinational tones like the dif­ference notes mentioned earlier. More substantial and mathematically more in­teresting, however, is the solution Helmholtz proposed for the problem of consonance. A fifth and a fourth, for ex­ample, which are defined by the ratios 3 :2 and 4:3, are perceived as a harmo­nious sound, which is more pleasant for the ear than an interval selected at ran­dom.

What is the cause of this phenome­non? Some answers, among them those of Plato, Kepler, and Galileo, are sketched (in my opinion, Euler's gradus suavitatis should also have been men­tioned here).

The starting point of Helmholtz's ap­proach is his consonance curve. Imag­ine two instruments playing a note in unison. If one of the frequencies is slowly increased, one will hear a beat­ing. First it is slow, but it becomes quicker and "more unpleasant". Helm­holtz quantified this sensation by asso­ciating a "degree of unpleasentness": if the number of beats is x, then cp(x) mea­sures the "unpleasantness". Obviously one has cp(O) = 0, and qualitatively it is clear that cp will first increase up to an maximum (which is assumed to be around x = 30) and then decrease.

In order to have a mathematically simple representation of cp, Helmholtz chose cp(x) = Ax/(30 + r)2, a choice which of course is somewhat arbitrary. This cp is used to explain consonance as follows. If two instruments play an in­terval, one has to sum up the cp-values which belong to every pair of frequen­cies from the list of all pitches which oc­cur in the Fourier expansion of the two notes which constitute the interval.

The result is a rather rough curve with minimum zero at the frequency ra­tios 1 : 1 and 2 : 1 . But, remarkably, there are also some steep valleys in the graph at 3 :2 , 5 :4, and the other ratios which correspond to the Pythagorean scale.

Part Three concerns "The mathe­matical structure of music" . I am sure that the fascination one can feel when listening to a Schubert sonata or a Chopin mazurka will never be accessi­ble to a mathematical analysis. There are, however, many "intellectual" as­pects of music where a mathematical language can reasonably be applied. For example, group theory naturally comes into play when speaking about

musical symmetries: see Chapter 6, "The Geometry of Music", by Wilfrid Hodges. But most of these symmetries are only perceptible by optical inspection of the score. (As an experiment, I suggest playing the notes from the first two bars of a popular song in reverse order. It is rather unlikely that an untrained listener will recognize the original.)

Not only group theory plays a role here. In Chapter 7, in the article by Der­mot Roaf and Arthur White on "Bells and mathematics", the emphasis is on combinatorics. "Ringing the changes" is the art of ringing a collection of n bells sequentially such that, at the end, all n! permutations have been heard. In ad­dition, certain conditions must be satis­fied, for example, from one round to the next, only transpositions between adjacent bells are admissible. This is so because, otherwise, it would be difficult to perform the sequence with really ex­isting heavy bells. It is interesting to see how this problem can be solved by rather simple algorithms and how the solutions are visualized graphically.

In the last chapter of this part, "Com­posing with numbers" by Jonathan Cross, we are introduced to some math­ematical ideas which have found their way to being used as tools for com­posers. The story begins with the twelve­tone row of Arnold Schonberg; a num­ber of other examples are also discussed. The idea is always the same. First, one associates certain musical parameters, like pitch or duration, with numbers or more complicated mathematical objects, and then the structure of the mathemat­ical part is translated to a piece of mu­sic. For example, one could select a magic square and then use the rows (or the columns, or the diagonals) to define the pitches of the clarinet line or the du­rations of the bassoon line.

Similar ideas are found in Part Four, "The Composer Speaks" (Carlton Gamer and Robin Wilson on "Microtones and projective planes" and Robert Sherlaw Johnson on "Composing with fractals"). The titles indicate the mathematical source of the compositions. Finite pro­jective planes are used to identify cer­tain subsets of tones. For example, if one wants to select three notes out of seven in such a way that the selection generates a "cyclic design" , one finds everything that is needed in the geom­etry of the Fano plane. (A cyclic design

in this case is a pattern such that trans­lations modulo 7 give rise to subsets of (0, . . . ,6} in which each pair of num­bers is contained in precisely one of the translations.)

Dynamical systems are very com­mon in contemporary music. Here the well-known two-dimensional iterative patterns which lead to the Mandelbrot set generate the musical material. For example, if the channel of the synthe­sizer has to be determined where the next note will be generated, then a dis­cretization of the y-value of the pres­ent position of the system is important: e .g . , if it lies in [7,8[, then choose chan­nel 5 .

I t should be noted that the book is very carefully edited. It is a pleasure to read, and there are many interesting pictures and scores to illustrate the ma­terial. Readers who are particularly in­terested in the historical part of the sub­ject can consult the book Mathematics and Music (edited by Gerard Assayag, Hans-Georg Feichtinger, and Jose Rod­riguez, Springer 2002; reviewed in The Mathematical Intelligencer, vo!. 27, no. 3, p . 69) . There is, surprisingly, only a small overlap in the content of these two books. The generation of scales by mathematical principles naturally plays a prominent role in both of them.

For me, only two aspects are miss­ing. The first omission: I would have appreciated an article on Euler's work on music. He was one of the first to re­late mathematics to consonance, and it would be interesting to compare his work with that of Helmholtz. And I was surprised to see that one cannot find anything substantial on "probability and music". In the music of the last century there is an abundance of examples in which the building blocks of certain compositions are generated stochasti­cally, be it the pitches, the durations, or even the wave forms of the sounds.

But these objections are not essen­tial. Let's praise the editors that they have presented an attractive volume that covers almost all of the important aspects of the interplay between math­ematics and music.

Fachbereich Mathematik und lnformatik

Freie Universitiit Berlin

D-1 41 95 Berlin Germany e-mail: behrends@math.fu-berlin.de

The Pea and the

Sun: A

Mathematical

Paradox by Leonard M. Wapner WELLESLEY, MA, A. K. PETERS, 2005,

xiv + 218 PP. , US $34.00, ISBN 1-56881-213-2

REVIEWED BY JOHN J. WATKINS

here is nothing quite like a good paradox. In their great comic opera, The Pirates of Penzance,

W. S. Gilbert and Arthur Sullivan use a 'simple arithmetical process' to create "a paradox, a paradox, a most inge­nious paradox' that renders the entire cast of pirates, maidens, and policemen helpless with laughter and amusement on the rocky seacoast of Cornwall . The paradox in this instance is extraordi­narily silly, but Gilbert and Sullivan lightly hang the entire plot of Pirates upon it. Leonard Wapner manages a similar sleight-of-hand with the Banach­Tarski paradox in his immensely en­gaging book The Pea and the Sun: A Mathematical Paradox. This book may not be quite as much fun as a Gilbert and Sullivan opera, but it is pretty close.

To be fair, W. S . Gilbert and Arthur Sullivan should probably be ranked as geniuses, but Wapner does have one huge advantage over them. The rather flimsy basis of the Gilbert and Sullivan opera is the paradox that their naive young hero Frederic had the misfortune to be born on February 29, and so, by counting birthdays he is but "a little boy of five" and thus must continue on in his grossly unfair apprenticeship to a band of pirates until his twenty-first birthday. This is hardly a paradox wor­thy of the name. But Wapner has cho­sen as the basis for his book the Ba­nach-Tarski Paradox: quite simply, the finest paradox in all of mathematics.

At first glance, the Banach-Tarski paradox seems so nonsensical that it might well belong in the world of Gilbert and Sullivan, for it says that it is possible to dissect a ball-that is, a solid sphere-into a finite number of pieces and then rearrange these pieces to form two balls exactly the same size as the

© 2006 Springer ScJence+ Business Media, Inc., Volume 28, Number 3, 2006 71

original ball. Yet, nothing has been cre­ated or stretched in the process! All that has happened is that pieces of the orig­inal ball have been moved around in space like the pieces of a j igsaw puz­zle. Well, if you can form two balls, you can form three, or four, or any number you like; or, alternatively, you could re­arrange the pieces to form a single ball that is twice as big as the original ball, or three times as big, or as much big­ger as you like. Thus, a rather more dra­matic way to restate the Banach-Tarski paradox is to say that you can trans­form a ball the size of a pea into a ball the size of the sun.

The Banach-Tarski paradox is such a remarkable paradox precisely be­cause-as impossible as it seems-it is true. And so, as Stefan Banach and Al­fred Tarski proved in 1924, it is indeed a theorem, and we should now refer to it more properly as the Banach-Tarski theorem. Leonard Wapner's goal in his book is to make this famous theorem accessible to a general, perhaps even non-mathematical, audience, and in pursuit of this goal he wisely chooses to focus on the paradoxical nature of this famous theorem-after all, who would be amused for long by "a theo­rem, a theorem, a most ingenious theo­rem"?

Wapner begins his story in the first chapter with history and introduces not only his main cast of characters, Georg Cantor, Stefan Banach, Alfred Tarski, Kurt Godel, and Paul Cohen, but also many of the mathematical ideas that will be important later, such as countable and uncountable sets, the Axiom of Choice, Hausdorff's Paradox (though, strangely, Hausdorff himself is not ele­vated to the main cast), as well as a nice discussion of the questions of con­sistency and independence of the Ax­iom of Choice and the Continuum Hy­pothesis.

Chapter 2 is an ill-conceived digres­sion on fallacies which Wapner con­fusingly calls Type 2 paradoxes; he dis­cusses and resolves several such paradoxes, but this gets rather far afield even though for example the jigsaw fal­lacies of Sam Loyd he presents are very amusing. I worry that while Wapner is actually trying in this chapter to alert the reader to the all-important distinc­tion between a surprising theorem (a Type 1 paradox) and a false statement

72 THE MATHEMATICAL INTELLIGENCER

(a Type 2 paradox), some readers will come away with the impression that Ba­nach-Tarski is nothing more than an­other trick very much like Sam Loyd's Get Off the Earth puzzle, in which a disk is rotated to make twelve Chinese war­riors magically turn into thirteen Chi­nese warriors.

The level of the book varies consid­erably. Chapter 3 begins at a very ele­mentary level clearly intended for ab­solute beginners and, therefore, says such things as that the proof of the im­portant result that the power set of a fi­nite set with n elements has cardinality 2n is by mathematical induction and will be omitted. This inductive argument is very easy to make clear to a non-math­ematical reader and it is even easier to make clear the real reason for this re­sult: namely, that each element of the original set of n elements is either in or not in a given subset, that is, there are two choices for each of the n elements. Wapner does an excellent job of intro­ducing the notion of an isometry and the all-important idea that isometries have a group structure, but having just explained what the associative property is, what an identity element is, and what an inverse element is, he then wants to show that

(plpz)- l = Pz-1PI-I and begins his proof with

(pipz) (pz - IPI -1) = P1 (pzpz - I) P1 -1 without any explanation of how asso­ciativity gives you this. (It requires an intermediate step.)

Continuing in this chapter on pre­liminaries, he then does some very nice scissor dissections, square to equilateral triangle, octahedron to square (though then wandering off topic a good bit), as a warm-up before transitioning beau­tifully to the vitally important notion of equidecomposability, showing, for ex­ample, that a circle and the same circle with a single point removed are equide­composable, that is, the first circle can be decomposed into a finite number of sets (in this case, two) and these sets can then be reassembled (using isome­tries) to form the second circle with a point missing. Since this example is so similar in spirit to the actual proof of the Banach-Tarski Theorem given in Chapter 5, I'd like to give the details here. Let the two circles each have ra-

dius 1 . On the first circle designate an arbitrary point as 0. This is the point to be removed. Then, from 0 moving counterclockwise at unit intervals mark off points 1 , 2 , 3, 4, and so on, one point for each natural number. Since the circumference is irrational, all of these points will be distinct. Let A be all of these points and let B be all of the re­maining points on the circle. This de­composes the first circle into two sets. The rest is easy. We reassemble the two pieces by leaving set B unchanged (the identity isometry) and by rotating set A one step counterclockwise (obviously, an isometry on the circle), thus form­ing the second circle with 0 missing as advertised.

In Chapter 4, called Baby BTs, Wap­ner presents several similar paradoxes leading up to the Banach-Tarski Para­dox, showing for example that a sphere and a sphere with one point removed (or even countably many points re­moved!) are equidecomposable, but also he gets somewhat sidetracked dis­cussing the cardinality of various one­and two-dimensional point sets. He gets back on track with beautiful presenta­tions of the Cantor set and of the ex­ample found in 1905 by Giuseppe Vitali of a bounded non-Lebesgue­measurable set. The statement and proof of the Banach-Tarski Theorem then forms the core of the book in Chapter 5, and Wapner then brings the paradox to a satisfying resolution in the following chapter.

One thing that gets lost a bit both in the book but also in the inherently geometric nature of the statement of the Banach-Tarski Theorem is that there is a fundamental algebraic truth at its core. In 1914 Felix Hausdorff dis­covered a remarkable way to decom­pose the free product Z2* Z3. This group is just all 'words' using two gen­erators, u and T, where u2

= r3 = i where i is the identity of the group. Hausdorff partitioned this group into three disjoint sets

A = { i, V'T, O'TV', O'TT, TTV', . . . } ,

C = {TV', TT, TTV'T, TO'TTV', . . . } , with the defining property that B = TA and C = T2 A, and so all three sets are congruent and A constitutes a third of the group. But also uA = B U C and

so A is congruent to its complement and thus constitutes half of the group. The remarkable fact that A can be si­multaneously both half and a third of the group is now known as Hausdorff's Paradox.

This purely algebraic paradox is at the very heart of the Banach-Tarski Paradox. You might say the rest is just window-dressing. It was Hausdorff who applied it to the surface of a sphere. Banach and Tarski then showed how to extend the paradox to a solid sphere, that is, a ball, but Hausdorff had done the heavy lifting. More recently, Jan My­cielski and Stan Wagon have applied this paradox to the hyperbolic plane where, using Mathematica, one can ac­tually see the set A filling both half and a third of the standard Poincare disk [ 1 , 2 ] . Wapner does mention this lovely in­terpretation, but rather strangely places it in his first chapter on history and the cast of characters.

While the Banach-Tarski Theorem clearly seems to rest solely within pure mathematics and offers no hope at all for practical applications, Wapner does show us, in Chapter 7, that at least a few physicists in the world take seri­ously such ideas as that the muon, a short-lived particle 200 times as massive as an electron, might in fact be a par­ticle that expanded in a Banach-Tarski­like way from an electron, or that cer­tain well-known reactions in particle physics have Banach-Tarski interpreta­tions, such as when a proton changes into a pi meson and a neutron. Such ideas have more than a whiff of fantasy about them, which Wapner freely ad­mits, yet when I pause to consider how little we actually understand about the real world, it is not beyond the realm of possibility that infinite sets and even non-measurable sets might exist in some real sense that we assume to be impossible today.

Wapner closes his book with a chap­ter in which he bravely takes a crack at explaining seven big problems in math­ematics today and also takes the op­portunity to offer some of his own mus­ings about the future of mathematics, guided by noted thinkers such as Yogi Berra and Niels Bohr who, as it turns out, had almost the exact same thing to say about the future.

Like Gilbert and Sullivan's Major-Gen­eral Stanley, Leonard Wapner is indeed

"teeming with a lot o' news with many cheerful facts" about matters mathemat­ical in this thoroughly delightful book. While it is certainly true that a non-math­ematical reader might well miss an im­portant preliminary or two or get side­tracked by one or more of the many interesting diversions along the way and, in the end, not be able to follow fully all of the details of the proof in Chap­ter 5, I would be very surprised if he or she hasn't understood the general flow of ideas, learned a lot about mathemat­ics, been amazed by the sheer reality of what unexpected things can be true, and had lots of fun in the process.

REFERENCES

1 . Bennett, Curtis, A Paradoxical View of Es­

cher's Angels and Devils, The Mathematical

lntelligencer 22(3) (2000), 39-46.

2. Wagon, Stan, A Hyperbolic Interpretation of

the Banach- Tarski Paradox, The Mathemat­

ica Journal 3 (1 993), 58-61 .

Department of Mathematics and

Computer Science Colorado College

Colorado Springs, CO 80903

USA e-mail: jwatkins@coloradocollege.edu

Divine Proportions:

Rational

Trigonometry to

U niversal Geometry by Norman Wildberger SYDNEY, WILD EGG, 2005 (http:/ jwildegg.com),

300 PP, AUD79.95 HARDCOVER,

ISBN 0-9757 492-0-X

REVIEWED BY JAMES FRANKLIN

nlike "lesser" disciplines, mathe­matics is not rent by disputes over what is true. What we have

proved true has stayed true, give or take rare exceptions. Our argumentative en­ergy has not gone to waste, however, and mathematicians debate vigorously questions on what topics are interest­ing, what conjectures credible, how classical fields can be better seen in the

light of new results, and of course, how to teach.

A heated debate a hundred years ago-one with close parallels to the revolution in trigonometry that Wild­berger urges in his new book-resulted in major changes to linear algebra. That is a branch of mathematics which the nai've student might expect to have de­veloped smoothly, rationally, and with­out controversy. Axioms, span, inde­pendence-what is there to become heated about in that? Yet the modern point of view is the end point of re­covering from several false starts, no­tably Hamilton's inept attempt to do vector geometry and physics with quaternions and Grassmann's barely in­telligible foundation of the subject on what we call "flags of subspaces" . There were also difficulties in moving beyond coordinates and matrices to the more abstract point of view of vectors and linear maps. It was only in the 1920s that British mathematicians and engineers swallowed their pride and admitted that the Germans had it right about vectors. (The story is told in Crowe's History of Vector Analysis.)

Trigonometry is a much older and more settled branch of mathematics than linear algebra. It comes much ear­lier in the syllabus, and every becom­ing-numerate generation invests enor­mous effort in the painful calculation of the lengths and angles of compli­cated figures. Surveying, navigation, and computer graphics are intensive users of the results. Much of that effort is wasted, Wildberger argues. The con­centration on angles, especially, is a re­sult of the historical accident that seri­ous study of the subject began with spherical trigonometry for astronomy and long-range navigation, which meant there was altogether too much attention given to circles.

Wildberger's alternative is simple. We should avoid the concepts of length and angle as far as possible, and so do without their complicated formulas in­volving square roots and transcenden­tal arcsines and the like. They should be replaced with two (algebraically) simpler concepts, "quadrance" and "spread". Quadrance is just the square of length, so its formula in terms of co­ordinates just involves the sum of squares of co-ordinates. Spread is a measure of separation of lines. It is (to

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006 73

slip into oldspeak for a moment, though the aim is to learn to think in the new language as fast as possible) the square of the sine of the angle(s) between the lines. The spread between the lines ax + by = 0 and ex + dy = 0 is a simple rational expression in a,b, c,d.

Let us take one elementary and one more mathematical example to show­case the point of doing things this way. Consider the problem, useful in such fields as railway engineering, of the re­lation between slopes when climbing a hill "at an angle" . For example, if a grade of one in fifty is the maximum a train can climb and the hill has a grade of one in thirty, in what direction across the hill must one build the railway? Stan­dard trigonometry would attack this problem using angles and their tangents, but the problem and its answer do not mention angles. The solution in terms of spreads (p. 231) is very simple.

Mathematicians may be more ex­cited by the way that the avoidance of square roots and transcendentals ren­ders the results independent of the real field, and hence a true "universal geom­etry". For example, at first sight the re­sult that the spread subtended by a chord of a circle is a constant (p. 178) seems much the same kind of result as the classical one that the angle sub­tended by a chord is constant. But there are subtle differences. With angles, one must consider on which side of the chord the angle lies. That is awkward in itself and prevents generalization be­yond the field of real numbers. For spreads, constant really means con­stant, and one may change the under­lying field and retain the theorem.

Wildberger develops his universal geometry at length, dealing for exam­ple with the replacements of the sine and cosine rules, an alternative to spherical and polar co-ordinates with

74 THE MATHEMATICAL INTELLIGENCER

applications to moments and centers of inertia, and simplified treatments of classical surveying problems like the Snellius-Pothenot and Hansen's prob­lems. Reform is intended not just of trigonometry but of the foundations of Euclidean geometry. The subject is de­veloped from first principles over a general field-one cannot have "on this side of the line" in fields other than the reals, but almost all other Euclidean geometrical properties remain available (including the inside and outside of cir­cles). Similarly, conics are treated from a point of view that resembles algebraic geometry but includes a metric.

It is true that there is a need to re­tain the "circular" or "harmonic" func­tions to deal with circular motion, Fourier analysis and the like, but those wave-like functions with no natural zero would be better not called "trigonomet­ric". They are not related to triangles.

It is certainly convincing that we would have been better off if trigonom­etry had developed this way instead of the way it did.

Now to the crunch. Is it feasible for the mathematical world to junk its im­mense investment in the old technol­ogy and move to a new one? It is a big ask, a very big ask, but there are a few reasons to think it might just be possi­ble. The first is that despite the dead hand of conservatism, it has happened before. Replacing co-ordinates, matri­ces, and quaternions with abstract vec­tors and linear transformations was an effort, but worthwhile in the end. The same was true of replacing sines and cosines in Fourier analysis by complex exponentials. Long before that, Arabic numerals replaced Roman because they were more rational. Revolutions are possible. One must regretfully call the author's attention to the fact that they usually take more than a single lifetime. It could be questioned also whether

launching the project from a small in­dependent publisher in Australia is a good idea, but in the twenty-first­century world of the Internet and blogs, perhaps that does not matter.

Second, a careful examination of 3D vector geometry will reveal that a cer­tain amount ofWildberger's philosophy is implicit in it already, suggesting that he is on the right track at a more ba­sic level. What makes geometry with vectors so successful is that all the in­formation about lengths and angles is contained in the scalar product, which is algebraically very simple. The stu­dent soon learns that the way to ap­proach typical problems, say on the closest distance between two non-in­tersecting lines, is to stay with vectors and their scalar products as long as pos­sible and only extract any needed lengths and angles at the last moment. Wildberger simply goes one step fur­ther: he recommends we do the same in two dimensions, and suggests that we hardly ever have any real need for lengths and angles in any case.

There has been considerable debate by interested amateurs on Internet fo­rums about this book. There needs to be more mature consideration in better informed mathematics and mathematics education circles. Having things done better is one major payoff, but equally important would be a removal of a substantial blockage to the education of young mathematicians, the waterless badlands of traditional trigonometry that youth eager to reach the delights of higher mathematics must spend painful years crossing. Wildberger's book de­serves very careful examination.

School of Mathematics University of New South Wales Sydney 2052 Australia e-mail : jfranklin@unsw.edu.au

w-1fii•i•l9•hri§l Robin Wilson I

The Ph i lamath' s Alphabet-M Mathematics teaching Many

stamps feature the teaching of

mathematics. This stamp, issued by

Guinea-Bissau for the International

Year of the Child, illustrates the teach­

ing of the geometry of a circle;

Maxwell's equations Using the most

advanced vector methods of his day,

James Clerk Maxwell (1831-1879) syn­

thesized Faraday's laws of electro­

magnetism into a coherent mathe­

matical theory, confirming Faraday's

intuition that light consists of electro­

magnetic waves. His celebrated Trea­tise on electricity and magnetism, containing the fundamental mathemat­

ical laws now known as 'Maxwell's

Math teaching

Please send all submissions to

the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes,

MK7 6AA, England

e-mail: r.j.wilson@open.ac.uk

equations', predicted the existence of

such phenomena as radio waves.

Mayan mathematics The Mayans of

Central America (300-1000 AD) used a

counting system based on the numbers

20 and 18. Most of their mathematical

calculations involved the construction

of calendars: a 260-day ritual one with

13 cycles of 20 days, and a 365-day one

with 18 months of 20 days and five ex­

tra days: combining these gave a 'cal­

endar-round' of 18980 days. Only a

handful of Mayan manuscripts have

survived, most notably the Dresden codex, painted in colour on fig-tree

bark and containing many examples of

Mayan numbers.

Metric system Throughout the cen­

turies various counting systems have

been used for weights and measures.

After the French Revolution, a com­

mission was set up to investigate the

desirability of a metric system for

France; the chairman of this commis­

sion was Joseph-Louis Lagrange. This

Maxwell's equation

Metric system Mobius strip

76 THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Business Media, Inc.

stamp shows an allegorical figure rep­

resenting the French metric system.

Mobius strip The Mobius strip was

named after the German mathemati­

cian and astronomer August Mobius in

1858. It has only one side and one

boundary edge, and is constructed

from a rectangular strip of paper by

identifying its ends in opposite direc­

tions. Mobius was not the first to dis­

cover it-Johann Benedict Listing beat

him by a few months.

Monge Gaspard Monge (1746-1818)

taught at the military school in Mez­

ieres. While investigating positionings

for gun emplacements in a fortress, he

improved the known methods for pro­

jecting three-dimensional objects on to

a plane; this subject became known as

'descriptive geometry'. Monge's other

interests included 'differential geome­

try', in which calculus is used to study

curves drawn on surfaces, and he

wrote the first important textbook on

the subject.

Mayan codex

Monge