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‘To Infinity and Beyond!’
Bernhard Bolzano and the rigors of analysis
Paper 2
History of Mathematics
December 6, 2001
Problem 14.30
ABSTRACT
Every college math student despairs during the required first course in analysis. Professors everywhere hear the popular complaint, “Why do we have to prove this again?” from students, the author of this work included, who can’t see the mathematical forest for the trees. Analysis is more than the rigor of complicated proofs that beginning mathematicians take for granted. Analysis is philosophy – mathematics philosophy. By studying not only the proofs but the ideas behind them, it is much easier to come to an appreciation of the thought put into the development of this rigor by Bernhard Bolzano and other mathematicians of the 18th and 19th centuries
It’s 5:25, Monday afternoon, and I’m sitting in Dr. Goldberg’s Advanced Calc I class.
Like any good math major, I’m thinking about numbers. I’m exactly one-third of the way
through the 75-minute class. In five minutes, I’ll be two-fifths of the way through, and in 10
minutes…
At least when I look at the clock, it looks like I might be looking at the board. It seems
more polite than looking the other way and staring out the window.
At the top of my notes, I’ve written the word continuity. I draw a curvy line all around it.
My line is continuous. But it’s not a function, because I’ve got more than one y for each x. We
learned that in high school. So why am I sitting here, proving the intermediate value theorem,
after six semesters of college math? I look back at the clock. Three-fifths done. I hear the word
“quiz” and sit up straighter. We have a quiz? Oh, well. I wouldn’t understand the proofs any
better if I had studied anyway.
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On the board, Dr. Goldberg writes “Let, for two reals a and b, a<b, a function f be
continuous on a closed interval [a,b] such that f(a) and f(b) are of opposite signs. Then there
exists a number x0 in the interval [a,b] with f(x0)=0. Prove.”
Well, he always says, when in doubt, draw a pretty picture. So I draw this:
It’s a function. F(a) and f(b) are of opposite signs.
Then I have x0 in [a,b] such that f(x0)=0. Fine. Nobody else
has handed in their quiz yet, so I start daydreaming. The
funniest thing is that what comes to mind is Dr. Troutman’s
math history presentation from the previous class. He was talking about Bernhard Bolzano and
the development of analysis, and the intermediate value theorem. I remember that the theorem
Dr. Goldberg is asking us to prove is the Bolzano theorem, and it’s a special case of the
intermediate value theorem. If I can remember all this, why can’t I remember this proof?
I give up and look out the window. Leave these proofs to Bolzano… I don’t need any of
this analysis stuff. I can just look it up if it ever becomes essential that I prove the continuity of a
real-valued function, which seems doubtful at best. My friend Jessi looks over at me and
whispers, “Joan!” But it’s too late. I’ve already dozed off and begun to dream. Again, I’m back
in Math History, and it looks like I’m giving a presentation on one of my papers.
“Bernhard Bolzano was born in 1781 in Prague, which was then part of Bohemia, now
Czechoslovakia,” I hear myself say. I remember reading that in Eves. “On page 486,” I
continue, “Eves claims that Bolzano was so fully aware of the need for rigorization in analysis
that Felix Klein later referred to him as ‘The Father of Arithmetization.’”
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f(a)
f(b)
I walk into the class and sit down, all the while listening to myself present. No one else
seems to be aware of me. This might be useful, I think to myself, so I get out a pen and paper
and begin to write, taking notes from my own presentation:
Bernhard Bolzano, born Oct. 5, 1781, in Prague, was one of the greatest
mathematicians to produce work in the 19th century. Bolzano was one of the first
mathematicians to delve into the concept of infinity, working with infinite sets and describing the
boundedness of functions (Eves 486). In fact, a special case of the intermediate value theorem so
important to calculus is known as the Bolzano theorem in his honor (Bogomolny).
The saddest aspect of Bolzano’s life is that most of his great contributions to
mathematics were not recognized until years after his death. Most of his peers ignored his work,
and many proofs that were rightly his were claimed by Cantor, Weierstrauss and others. Even
today, most books on math history seem to gloss over Bolzano’s contributions, although the
Internet provides more information, kept by those mathematicians who do understand the
importance of Bolzano’s work.
One problem leading to Bolzano’s lack of recognition was his name – Bernardus
Placidus Johann Nepomuk Bolzano. In 1805, when he was ordained as a priest, Bolzano took
the more modern Bernhard as his first name, and texts today often modernize it further to
Bernard. However, even these simple changes were confusing in the inaccurate system of
records kept by the Catholic church. It was not until 1871, 30 years after Bolzano’s death, that
mathematicians realized that Bernhard Bolzano the Catholic priest was the same man as
Bernardus Bolzano the mathematician (Wachsmuth).
Many of Bolzano’s contributions to mathematics were actually produced from his
study of philosophy and religion. When Bolzano first entered the University of Prague in 1796,
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he studied philosophy, religion and mathematics (Golba). Because his parents were both very
pious Christians, Bernhard grew up with a very high moral code and felt strongly about sticking
to his principles (Golba). The Catholic Encyclopedia claims that Bolzano’s mathematical studies
were “notably successful,” but that his strongest work came in philosophical theory. Later, in
1805, Bernhard went against the wishes of his father (who hoped that he would continue his
study in mathematics), joined the ecclesiatic order and was ordained as a priest (Knight).
So great were Bolzano’s scholarly achievements that he was invited to be a
professor of the philosophy of religion at the University of Prague the same year that he was
ordained, a great honor (Knight). Strangely enough, Bolzano battled doubts about his religion
for much of his early life. It was not until after his ordination that Bolzano adopted the view that
a religious doctrine is justified “if it can be shown that believing it promises greater moral good
than not believing it” (Simons). This sort of rationalism was very evident throughout much of
Bolzano’s work, both in religion and in mathematics, and it was not well-accepted by the church
(Knight).
One of Bolzano’s duties as chair of the religious department at the university was to deliver an “exhortatory sermon” every Sunday to all the students (Simons). These lectures, although popular with the students, soon led to trouble for Bolzano. He openly criticized the prescribed theology textbook of the church, and was very progressive in his religious and philosophical ideas. According to Simons, this “brought him to the unwelcome notice of conservatives in the church.” In 1819, after several investigations, Bolzano was removed from his post and forbidden to publish any of his works (Golba).
According to Simons’ Web site, the restrictions were later loosened, so that Bolzano could
publish works of mathematics and other writings that had no relationship to the church. He also
was allowed to continue to lead Mass as a priest, but was not allowed to hear confessions.
It was during this time that Bolzano once again had time to focus on mathematics, which
he had not worked seriously on for perhaps 15 years. According to Simons and Hieke, from
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1823 until 1830, Bolzano spent every summer in the town of Techobuz at the home of his friends
Josef and Anna Hoffmann. Because Bolzano had always been in somewhat poor health, Anna
Hoffmann spent much time trying to care for Bolzano. From 1830 until 1841, Bolzano lived
permanently with the Hoffmann family. During this period, Bolzano wrote much of his major
work, Grossenlehre. It was a studious attempt on his part to “put the whole of mathematics on a
logical foundation,” as O’Connor and Robertson’s Web site says. This work was published in
parts, and Bolzano always had hoped that his students would finish and publish the complete
work after his death.
In this work, Bolzano expanded upon the idea that had made up his doctoral dissertation
at the University of Prague many years before. In the preface to his doctoral thesis, Bolzano
wrote: “I could not be satisfied with a completely strict proof if it were not derived from
concepts which the thesis to be proved contained, but rather made use of some fortuitous, alien,
ntermediate concept, which is always an erroneous transition to another kind” (O’Connor).
His later works were an attempt to derive a new system of mathematics that was in
keeping with this philosophy. In fact, in Der binomische Lehrsatz… and Rein analytischer
Beweis…, “Pure Analytical Proof,” Bolzano makes an attempt to free calculus from the concept
of the infinitesimal. In the preface of the first book, Bolzano clearly states his intention: to
provide “a sample of a new way of developing analysis.” (O’Connor). A later mathematician,
when analyzing Bolzano’s goals, said the following:
“In this work, Bolzano did not wish only to purge the concepts of limit, convergence and
derivative of geometrical components and replace them by purely arithmetical concepts. He was
aware of a deeper problem: the need to refine and enrich the concept of number itself.”
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In Bolzano’s writings, one of his major contributions was the proof of the intermediate
value theorem using this new approach. He also wrote a book called Paradoxien des
Unendlichen, a study of “paradoxes of the infinite,” according to O’Connor. In this work,
published posthumously in 1851, the first use of the word “set” appears, and Bolzano gives
examples of one-to-one correspondences between the elements of an infinte set and the elements
of a proper subset, using a concept that is now known as 0א , or aleph-null, for the cardinality of
certain infinite sets.
The first of these concepts, the proof of the intermediate value theorem, may seem
simple, but Bolzano’s method of direct proof posed one problem: The notion of continuity was
necessary for the first time, and only Bolzano recognized its importance (Stillwell 196). The
intermediate value theorem is as follows:
Suppose f:[a,b] into the reals is continuous. Suppose f(a)<0 and f(b)>0. Then there
exists some x0 in the interval (a,b) such that f(x0)=0 (Fitzpatrick 49). Bogomolny’s Web site
gives a thorough proof of this theorem by a mathematician named Scott Brodie. It is as follows:
Let f(x) be a continuous function on the closed interval [a,b], with f(a) < 0 < f(b). We are
to show that there is a real number c, between a and b, such that f(c) = 0.
We first need to understand what is meant by a continuous function. This is usually taken to
mean that for real numbers near the number x, the function values lie near f(x). This is usually
stated in terms of a "challenge" and "answer" dialogue:
We say "f is continuous at x" if, for each open interval J containing f(x), we can find an open
interval I containing x so that for each point y in I, f(y) lies in the interval J. We say "f(x) is
continuous" if it is continuous in the sense given above at every point x of its domain.
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As a simple application of this notion of continuity, we may note the "Sign-Preserving property
of Continuous Functions": If f is continuous at a, and f(a) < 0, then there is an open interval I
containing a such that f(x) < 0 for every x in I. It is also necessary to have what is sometimes
called the Axiom of Completeness. One common version of that axiom which is suitable for this
purpose is the "Least Upper Bound" axiom: Every non-empty set of real numbers which is
bounded above has a least upper bound.
Now, the proof: Let S be the set of numbers x within the closed interval from a to b where f(x) <
0.
Since S is not empty (it contains a) and S is bounded (it is a subset of [a,b]), the Least Upper
Bound axiom asserts the existence of a least upper bound, say c, for S. We show that this number
c satisfies the requirements of Bolzano's theorem:
There are three possibilities for f(c): either f(c) < 0, f(c) > 0, or f(c) = 0. We show the first two
choices lead to contradictions.
Suppose f(c) < 0, so that c is a member of S. Then the Sign Preserving Property of Continuous
Functions asserts the existence of an open interval I containing c where f takes only negative
values -- that is, I is a subset of S. But I (and thus S) contains points greater than c, so c cannot be
an upper bound (let alone the least upper bound) for S. This contradiction forces us to reject the
possibility that f(c) < 0.
Suppose instead that f(c) > 0. The argument is similar but not identical to the previous case.
Since f is continuous, there is an open interval H containing c where f takes only positive values.
But H contains points less than c, which must also be upper bounds for S, since no point of H can
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lie in S. This contradicts our choice of c as the least upper bound for S, so we must reject the
possibility that f(c) > 0.
We conclude that f(c) = 0. (From Bogomolny, http://www.cut-the-knot.com/fta/brodie.html).
Using this proof, many further theorems of analysis can be developed, such as those dealing
with the concept of boundedness for sets of real numbers.
Eves’ text states: “A real number a is called an upper bound of a nonempty set M of real
numbers if for each number m of M we have m is less than or equal to a, and a is called a least
upper bound of M if a<b whenever b is any other upper bound of M. A basic and important
property of the real number system asserts that if a nonempty set of real numbers has an upper
bound, then it has a least upper bound.” This is what was stated above in Brodie’s proof as the
Least Upper Bound, or supremum, axiom.
14.30 (a): Give a definition of a lower bound and a greatest lower bound of a nonempty set of
real numbers.
In this case, most of the question can be answered by “flipping around” the definition of an upper
bound and least upper bound. A lower bound of a set M, denoted a, means that, for all m in M, a
is less than or equal to m. Also, a is the greatest lower bound, or infimum, if b<a whenever b is
any other upper bound of M.
14.30 (b): Prove that a nonempty set of real numbers can have at most one least upper bound
and at most one greatest lower bound.
For this proof, we will prove that the least upper bound is unique and then use the same ideas in
reverse to show that the same is true for the greatest lower bound. First, assume that there exist
two upper bounds for a set M, call them a and b, with a and b unequal, and assume that they are
actually both LEAST upper bounds. Without loss of generality, assume a>b. Then, from our
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definition of least upper bound, b is a least upper bound of M, but a is not. However, if a>b and
a is a least upper bound of M, which is our assumption, then for some >0, a->b is an element
of M, and then, by the definition of being an upper bound, b is not an upper bound for M because
there is an element of M greater than b. Then, by definition, there is only one least upper bound
of M. The same will hold for greatest lower bounds; if the greatest lower bound is not unique,
then one of the two greatest lower bounds must be greater than the other, and there will exist
some number between the two bounds that is less than the smaller of the two bounds. In that
case, the smaller one really is not a greatest lower bound or even a lower bound at all, because
some element of the set M is less than it.
14.30 (c): Give an example of a nonempty set M of real numbers that has the following: Both an
upper and a lower bound. An example of a set like this is the set {(-1)n} or {-1,1,-1,1,-1,…}.
The upper bound of this set is 1, and the lower bound is –1. An upper bound but no lower
bound. An example of a set like this is the set {-1/n2} where the upper bound is 0 but the
function goes to negative infinity. A lower bound but no upper bound. An easy example of a set
like this will be {1/n2} which has a lower bound of 0 but goes off to positive infinity. Neither an
upper nor a lower bound. The set of integers has no upper nor lower bound; it goes to infinity in
both directions. A least upper bound that is in the set M. The previously given set of {(-1)n} has
as its least upper bound 1, which is an element of the set M={(-1)n}. A least upper bound that is
not in the set M. The previously given set of {-1/n2} has a least upper bound of 0, but 0 is not an
element fo the set M={-1/n2}.
14.30 (d): Prove that if a nonempty set M of real numbers has a lower bound, then it has a
greatest lower bound.
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Proof: For this proof, we will consider the set of “negatives” of a set S, where S is bounded
below. That is, let T={real numbers x such that –x is an element of S}. Then for any number x, b
is less than or equal to x if and only if –x is less than or equal to –b. Thus, a number b is a lower
bound for S if and only if the number –b is an upper bound for T. Since the set S has been
assumed to be bounded below, it follows that the set T is bounded above. What Brodie called
the Completeness Axiom, above, says that there must be a least upper bound for T, say c. Since
lower bounds of S occur as negatives of upper bounds of T, the number –c is the greatest lower
bound for S.
Suddenly, I sat straight up. Looking around the room, I realized that only Dr. Goldberg
was left. I hurriedly handed him my paper containing the doodle I had drawn of the graph.
Embarrassed to have been sleeping, I rushed out of the room.
The next class, when Dr. Goldberg gave our quizzes back, I sat with my usual
resignation, expecting my typical 7 out of 15 points. Imagine my surprise at the great 25 out of
20 I saw at the top of the page! Scrawled in the margins was this simple comment from Dr.
Goldberg: “Only Bolzano could have given this much background on this proof! Good job!”
I don’t think I’ll tell him that it was math history paper that showed me just how
important the rigors of analysis are to all of mathematics. I’ll let him think I knew that all along.
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BIBLIOGRAPHY
Bell, E.T. Men of Mathematics. New York: Simon and Schuster, 1937.
Bogomolny, Alexander. “Intermediate Value Theorem – Bolzano Theorem,” “Remarks on Proving the Fundamental Theorem of Algebra.” http://www.cut-the-knot.com/fta/brodie.html and http://www.cut-the-knot.com/Generalization/ivt.html. Copyright 1996-2001.
Eves, Howard Whitley. An Introduction to the History of Mathematics. With cultural connections by Jamie H. Eves. Philadelphia: Saunders College Publications, 1992.
Fitzpatrick, Patrick M. Advanced Calculus: A Course in Mathematical Analysis. Boston: PWS Publishing Co., 1996.
Golba, Paul. “Bolzano, Bernhard.” http://www.shu.edu/html/teaching/math/reals/history/bolzano.html. Copyright 1994-2000.
Knight, Kevin. “Bernhard Bolzano.” New Advent Catholic Encyclopedia. http://www.newadvent.org/cathen/02643c.htm. Copyright 1999.
Muir, Jane. Of Men and Numbers. New York: Dover Publications Inc., 1996.
O’Connor, J.J., and E.F. Robertson. “Bernhard Placidus Johann Nepomuk Bolzano.” http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Bolzano.html. December 1996.
Pappas, Theoni. Mathematical Scandals. San Carlos, California: Wide World Publishing/Tetra, 1997.
Simons, Peter, and Alexander Hieke. “The Bernard Bolzano Pages.” http://www.sbg.ac.at/fph/bolzano/bolzano-life.html. Copyright 2000.
Stillwell, John. Mathematics and its History. New York: Springer-Verlag Publishing, 1989.
Wachsmuth, Bert G. “Bolzano Theorem.” Interactive Real Analysis. http://www.shu.edu/html/teaching/math/reals/cont/proofs/bolzthm.html. Copyright 1994-2000.
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