Logic and Mathematics Hintikka

7/26/2019 Logic and Mathematics Hintikka

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History of Logic vs. History of Mathematics. Jaakko Hintikka. 032012 1

Jaakko Hintikka

WHICH MATHEMATICAL LOGIC IS THE LOGIC OF MATHEMATICS?

1.Logization of mathematics

One of the banes of current scholarship is overspecialization that leads to ignorance of developments in other

fields different from ones own even when they are directly relevant to it. Often ignorance nevertheless is not

the right word. Rather, what is involved is a failure to understand and to appreciate that relevance. A striking

example is offered on the one hand by the histories of mathematics and its foundations as they are dealt with by

working mathematicians as a part of their professional work and on the other hand by the history of logic as it has

been cultivated by philosophers and some mathematicians as a separate subject for philosophical and foundational

purposes. Here certain especially interesting aspects of the respective histories of mathematic and logic since the

early nineteenth century are examined. The overall development of mathematics in this period is well known, at

least in its broad outline. Around 1800 mathematics consisted of the study of two or three subjects. Geometry

was the study of space, and arithmetic and algebra were parts of the study of numbers and functions of numbers.

Analysis and analytic geometry combined ideas from both directions.

The changes in the nature of mathematics since early nineteenth century have been described in many

different ways, emphasizing different aspects of the mathematical enterprise. These characterizations include

among others an increase of rigor, especially the avoidance of appeals to intuition; greater abstractness, especially

the genesis of set theory and the increasing use of set theory as a medium of mathematical theorizing and

mathematical reasoning; the use of axiomatization, and the arithmetization of analysis. As a consequence,

mathematics has changed from the study of space and number to a study of all and sundry structures, not only

those structures that are exhibited in traditional arithmetic, analysis and geometry. In some projects, such as the


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Bourbaki program and the New Math movement, set theory is thought of as the lingua franca of all

mathematics. (Cf. Bourbaki 1938)

It is not badly controversial to suggest that the common theme in these developments has been a greater

and greater reliance on logic in mathematical concept formation, in the analysis of mathematical concepts, inmathematical theorizing in general. For instance, the way in which the enhanced rigor is implemented is usually

an analysis of mathematical concepts and mathematical modes of reasoning in purely logical terms. The extreme

doctrine of logicism claims that all mathematical concepts and rules of reasoning can be reduced to logic. Even if

such a complete reduction is not possible, the less radical but historically more prominent reductions for

mathematical theories to arithmetic or to set theory mean defining logically the concepts and modes of reasoning

needed in these theories in terms of natural number or sets, respectively. This enterprise is essentially logical

analysis, and accordingly it is a challenge to the logic that is (usually implicitly) employed in these reductions, but

it need not involve a formalization of the logic that is being used.

The first stages of these developments included that analysis of geometrical and semi-geometrical

concepts in analytical terms. Developments like the Gauss-Riemann theory of surfaces are emblematic steps in

this direction. The notion of space itself was analyzed as a structure of a certain kind. Once this was done to

what intuitively seems to be the actual space, analysis automatically shows what alternatives are mathematically

possible, thus opening the door to non-Euclidean geometries. What was involved was not only the deductive

structure of geometry, but a conceptual analysis of the basic geometrical concepts.

The deductive independence of Euclids fifth postulate showed only that non-Euclidean geometries areself consistent mathematical structures. An analysis of the structure of different geometries in metric terms was

needed to show what it means for out actual observable space to instantiate some particular geometry, Euclidean

or not. In a foundational perspective, these developments meant a gradual elimination of geometry from

analysis, which virtually automatically meant the disappearance of appeals to intuition in analysis. In this

analytization of geometry, one of the most critical bunch of concepts were those pertaining to continuity. In the

early twentieth century, Hilbert was still struggling to express them in purely logical and axiomatic terms. (See

e.g. Hilbert 1899, 1918.)

This elimination of geometry from analysis naturally took the form of an analysis in logical and

arithmetical terms of the basic concepts of analysis, such as limit, continuity, convergence, differentiation, and so

on. The first great figure in this work was Cauchy, but the fundamental results were achieved by Weierstrass.

(See here Grattan-Guinness 1970, Bottazzini 1986, Grabiner 1981 and the references given there.)


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These tendencies typically reflect, and are reflected by, the use of axiomatic method whose nature was

spelled out especially forcefully by Hilbert (See e.g. Hilbert 1899, 1918). It is part and parcel of the axiomatic

method that all the theorems are strict consequences of the axioms alone, so that new information that is not

contained in the axioms is not smuggled in the derivation of the theorems. And this implies, as Hilbert saw

especially clearly that the theorems must be purely logical (formal) consequences of the axioms, independently of

what the axioms are talking about. This precludes of course all appeals to intuitions in the deductive structure of

an axiomatic system, although it does not restrict their role in the choice of the axioms.

The story of these changes is an important part of the history of mathematics in the nineteenth century.

This increasing logization naturally meant that mathematicians had to develop ways of handling logical concepts

themselves. That they did, but they did not systematize, let alone formalize, their logical techniques. They

expressed their conceptualizations and differences in ordinary language, trusting that their readers master the tacit

logic that our ordinary language relies upon. As a consequence neither historians and historiographers of

mathematics nor historians and philosophers of logic have inquired with any real depth into the mathematical

logic that was used in the mathematical practice of the time. Both have in effect trusted Frege and early modern

logicians whose project was to formalize the general logic that all our conceptual thinking relies on including

mathematicians reasoning. What these logicians claimed to have done is to free our ordinary language from

unclarities and ambiguities. Thus they in effect claimed that they had captured fully the informal modes of

reasoning that mathematicians had been using. This universality is reflected for instance in Freges term

Begriffsschrift.

The core area of philosophers logic and all logic is what in our day and age is called the received first-

order logic, in brief RFO logic. This is the logic that has been generally considered to be the basic part of our

actual working logic also in mathematics. It is the logic that is relied on for instance in set theory.

But were these universality claims right? This historically and theoretically fundamental question has not

been seriously attended to in the earlier discussion. Does the implicit logic of nineteenth century mathematicians

resemble RFO logic? If not, what is it and how is it related to logicians logics?

2. The epsilon-delta treatment of quantifiers

In tacitly practicing logic, nineteenth century mathematicians in quest of rigor had to deal with the most central

concepts of all nontrivial logic, the two quantifiers, the existential quantifier and the universal one. How did they


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do so? Quantifiers taken one by one in isolation are easy. They express the nonemptyness or exceptionlessness

of some (usually complex) predicate. The interesting case is that of dependent quantifiers. Their job description

is not only class theoretical. They are the only way of expressing the dependencies of variables (viz. variables

bound to them) on each other on the first-order level. But the most basic concepts of analysis involve dependent

quantifiers. So how did Cauchy and his followers handle dependent quantifiers in defining notions like limit and

convergence?

The answer is known to everybody who has taken a rigorous introductory calculus course. They used

what is known as the epsilon-delta method, sometimes referred to as epsilontics. This method is a logical

theory of dependent quantities expressed in ordinary language. (plus the usual mathematical notation) For

instance, the continuity of a function f(x) at x is expressed as follows:

(1) For any given one can choose such that for any

whenever y

Here are reals with

The definition of differentiability says likewise that one can choose, for any given such that

(2)

whenever Here d is the derivate of f(x) and reals o

The definition of the convergence of a sequence of functions f1(x),f2(x),to fo(x) was likewise expressed

somewhat as follows:

(3) Given any one shall choose k such that for any n> k

Whenever n

Here is a real number and k, n are natural numbers.


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What these examples illustrate is a perfectly viable way of handling quantifiers in mathematical concept

formation and mathematical reasoning. It does not need any formalism to be understandable and applicable, as is

in fact done in innumerable textbooks. What is going on logically is not difficult to understand. Universal

quantifiers are expressed by speaking of what is given and existential quantifiers are expressed by speaking of

what one can choose.

Following this interpretation, what most philosophers of mathematics say here is that the real logical

structure of this largely informal method is shown by its representation in the RFO logic formalism that in effect

goes back to Frege. In the current notation of RFO logic a definition of the three sample mathematical concepts

could be expressed as follows:

(4)

(5)

(6)

This explication of mathematicians definitions is often considered a great achievement. Philosophers like Quine

typically present as a virtue of the logic that Frege founded that it can thus capture in precise formal terms the

epsilon-delta technique. In contrast, many historians of mathematics fail to appreciate the generality of the

technique or its logical nature. (See e.g. Alexander 2010, p. 142 and p. 287, note 21.)

3.Formal quantifiers vs choice terms

But who is capturing what here? There is an obvious connection between (1)-(3) and (4)-(6) and they can

admittedly be said to be pairwise equivalent. But there are deeper differences here than perhaps first meet the

eye. The informal logic of Cauchy and Weierstrass and our RFO logic obviously rely on altogether different

semantics. For Frege quantifiers are higher-order predicates that express the nonemptyness and exceptionlessness

of the (usually complex) predicates that follow them in the correlated brackets. The conditions of their doing so

can be formulated in a Tarski-style semantics.

In the epsilon-delta technique we consider quantifiers as proxies for certain choice functions. What a

quantificational proposition expresses is the claim that certain choices can alwaysbe made (one can choose), in

other words that the functions that implement the choices actually exist.


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Such a treatment of the semantics of quantifiers is possible and it not an unknown idea. In effect, a

treatment of quantifiers as choice functions in disguise was attempted by Hilbert and Bernays 1933-39). Their

attempt was not fully successful, however, largely because they did not spell out explicitly in their notation what

the choice in question depends on. The complications in Hilbert and Bernays are caused by the use of an

apparently free-standing choice term F(x) instead of an explicitly context-dependent Skolem function term

where the dependence of other terms is explicitly indicated by its argument. For the force of Hilberts and

Bernays epsilon term depends often on its context, but without any explicit rule of how it so depends. (As it will

be seen, they were not the only mathematicians who failed to appreciated this crucial question.) This defect has

been corrected in what is known as game-theoretical semantics, but only more than a hundred years after Frege.

It is based on the natural idea of thinking of the choices associated with quantifiers as moves in a game. This

natural idea was already relied on by C.S. Peirce in his interpretation of quantifiers. He was prevented from fully

implementing the game idea by not having the notion of strategy (in the von Neumann-Borel sense) at his

disposal. (See here Pietarinen 2006.)

From the point of view of game-theoretical semantics it is seen that ordinary language locutions like one

can choose are ambiguous in that they do not tell what the choice in question depends on. For instance, in (1)

the choice of obviously depends on , but does it also depend on x? A satisfactory notation should allow the

expression of either reading. In (4)-(6) this question is tacitly answered by the convention that a quantifier

depends on free variables in its scope, Formally speaking, these variables can be considered as being bound to

sentence-initial universal quantifiers. But this leaves the other possibility in a limbo. Can the choice of be

independent of x? How can such a reading be expressed? It will be shown here that that simple logical question

has played a significant role in actual mathematical practice.

The two semantics give the same results in the special case of RFO logic. However, they represent

entirely different approaches and facilitate radically different extensions. For instance, in the most natural way of

implementing a game-theoretical semantics the axiom of choice turns out to be a first order logical principle,

even though in the prevalent RFO tradition it has to be as a separate set-theoretical or higher-order axiom.

This is indicative of the general situation. Game theoretical semantics can serve as a basis of much

stronger logics than Freges RFO logics. Moreover, the semantics that late nineteenth century mathematicianswere tacitly using was obviously GTS. Hence the epsilon-delta logic relying on GTS as it was already in Freges

time used by Weierstrass was much stronger than Freges logic or the currently and RFO logic.

For this reason, it is historically incorrect to assimilate the two kinds of logic to each other. Further

systematic and historical analysis only deepens the differences between the two. It is seriously misleading to


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think of Freges logic merely as a formalization of the episilon-delta technique or for that matter to think of the

epsilon-delta talk merely as a verbalization of Freges formal logic. It would have been a feather in Freges cap if

he could have presented his logic as doing the same job as mathematiciansinformal methods. But as a brute

historical fact, Frege never as much as mentions the epsilon-delta technique. And this is not simple oversight or

an unexploited possibility. For deep reasons, he could not have done so.

4 Cauchys theorem as a case study

These reasons can be seen by having a closer look at the history of mathematics, especially at the

development of the epsilon-delta technique. The first major steps in that development and in the entire

rigorization (logization) of analysis were taken by Cauchy. (In saying that, we must make a significant allowance

to the earlier role of Lagrange.) Cauchy formulated most of the modern definitions of the crucial notions like

continuity, limit and convergence. But the path of progress was not smooth. In exploring the role of the newly

defined concepts, Cauchy presented an important theorem. It says that the limit of a converging sequence of the

continuous functions is itself continuous.

This was no mean theorem. Cauchy gave it a prominent pace in his influencial text Course danalyse

(1821), as its apparent significance seemed to motivate. Systematically speaking it would have had huge

consequences. For one thing, it seemed to make the entire Fouorier analysis impossible in that one could not

represent a discontinuous function as a limit of a Fourier series of continuous functions.

Luckily for Fourier and luckily for mathematical physics, Cauchys theorem turned out to be fallacious.

Of course it was not literally a matter of luck. Cauchy had made a mistake. The way this mistake was overcome

was one of the most important progressive steps in the history of analysis. It is an instructive example of how

mathematics advances conceptually

It was not hard to see that something was amiss with Cauchys proof. It contradicted some of Dirichlets

results. The first one not only to suspect that something was wrong with Cauchys theorems but to see where

counter-examples might be found was Abel. But the precise nature of Cauchys mistake was far from obvious.

The first one to pull the emergency brake was P.L. Seidel (1848), but even he could at first say only that its proof

must basically rest on some hidden supposition.

But what was this hidden assumption? What Cauchy assumed was that the members of a sequence of

functions f1(x), f2(x), are all continuous and that they converge to fo(x). His definition of convergence was

correct and so was his definition of continuity. They were essentially (3) and (1) above. But it turned out that he


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should have assumed something more of his sequence of functions than ordinary convergence. But what? The

great progress that Cauchys mistake unwittingly prompted was brought about by mathematicians efforts to

answer this question. In our contemporary terminology, the progress was essentially the acknowledgment and

definition of uniform convergence as distinguished from ordinary convergence. Analogous to uniform

convergence, mathematicians came to define a host of other uniformity concepts: uniform continuity, uniform

differentiability and so on.

But what precisely is this new concept? What was wrong with Cauchys proof? Thejoker here was an

additional factor that Cauchy had overlooked. It was the role of the variable x. For any one value of x, the only

choice one apparently has is between convergence and non-convergence and non-convergence. In later usage,

uniformity concepts are in fact often defined so as to be relative to a range of values of a variable analogous to x.

For instance, uniform continuity is defined as in (4), but relative to a range of values x1 x x2.

But this is not a full diagnosis of the problem, for the sought-for stronger convergence is after all a local

phenomenon. It could be characterized by speaking of what happens in the arbitrarily small neighborhood of x.

One had to introduce distinctions between different modes of convergence relative to [a single value of] the

variable x, as Grattan-Guinness puts it (1970, p. 118). Seidel (1948) tried to do this by defining what he called

arbitrarily slow convergence. (See Grattan-Guinness, op.cit.) Stokes did the same with a different notion he

referred to as infinitely slow convergence. These terms should already warn you. These notions are very messy.

They help to expunge Cauchys mistake, but they do not yield an insight into what the logical (conceptual) gist of

the problem is.

5. Uniformity concepts

The crucial distinction can be seen from the definition of any uniformity concept. The problem comes

down to the same conceptual unclarity as was seen to have bothered Hilbert and Bernays. When it is said in (l)

that one can chose , it is left open what the choice depends on. Does it depend on alone, or does it also

depend on x? The latter answer yields the usual definition of plain pointwise continuity, the former a definition of

uniform continuity. In this precisely analogous way we can distinguish differentiabilitysimpliciterand uniform

differentiability by spelling out whether the choice of in (2) depends on x or not. Likewise, in the similar

definition of the convergence of a sequence of functions fi(x) we can distinguish uniform convergence from the

ordinary variety by making the choice of independent of x.

Thus the informal but accurate definition for uniform continuity is obtained from (1) by stipulating simply

that the choice of must be made independently of x, and a definition of uniform differentiability is obtained


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similarly from (2). In the definition of convergence (3) uniformity is obtained by making the choice of k

independent of x where


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(7)

But this is not a correct definition. To see this, consider the negation of (7)

(8)

When you unpack these, you can see that what (8) essentially says is that for some the function f(x) has a

discontinuity of the order somewhere in the interval x1


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about the work of Weierstrass and his followers. Its gist was an enrichment of the epsilon-delta logic of

quantifiers by the use of the notion of quantifier independence.

But this independence cannot be expressed by means of our everyday RFO logic. For the independence

of variables has to be expressed in RFO by the independence of the quantifiers they are bound to. Now in (3) asreproduced by (6) the quantifier (k) must depend on x, for the definition should of course apply for all its values.

(There is an implicit quantifier (x) fronting the definition.) Likewise for (1)(2) and (4)(5). Hence the enriched

epsilon-delta logic used by Weierstrass and his ilk was much richer than received logic of quantifiers, i.e. RFO

logic.

Since the first-order part of Fregeslogic is essentially equivalent to RFO logic, this shows that Freges

project failed abjectly. Far from being a universal notation for our concepts, it fails even to capture the modes of

reasoning of his fellow mathematicians at the time.

7.IF logic to the rescue

Needless to say, the flaws in RFO logics are reparable. The first main step is to introduce a notation to exempt an

existential (existential-force) quantifier (y) from its dependence on a universal (universal-force) quantifier (x)

within whose formal scope it occurs by writing it (y/x). Likewise, the independence of (x) of the variable z

can be expressed by (x/z). By means of this notation we can express the missing independence relation in (4)-

(6) by writing the critical quantifiers.

Hence, no criticism of RFO, as far as it goes, is intended here. However, this innovation was introduced

only in the nineteen-nineties. It took more than one hundred years for the symbolic logic tradition to catch up with

the Cauchy-Weierstrass tradition. It was not only Frege who failed to capture fully the epsilon-delta technique in

his logic. For a hundred years, other logicians did not do any better. Thus Freges failure to deal with dependent

quantifiers slowed down the development of logic by more than a century.

Furthermore, the improvement just mentioned meant replacing RFO by another richer logic. The first step

takes us to what is ill named as independence-friendly (IF) first-order logic. Thus IF first-order logic is not

(Stanford Philosophical Dictionarynotwithstanding) a further development of RFO logic. It replaces RFO. It is

naturally based on game-theoretical semantics, and as such is an implementation (among other things) of the

epsilon-delta technique. Far from being a superstructure on RFO, IF logic was in effect used before RFO existed,


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however informally. In a very real sense, IF logic is not a novelty. It is simply the logic that mathematicians like

Weierstrass were already using in the nineteenth century.

The replacement of RFO logic by IF first-order logic is not only architectonic, a question about how to

best formulate and formalize our logic. It has potentially important foundational consequences. In IF logic, thelaw of excluded middle does not hold. We must allow in it predicates with truth-value gaps. For instance, it can

easily be shown (as in Hintikka 2011) that a mathematical induction works only for fully defined predicates (i.e.

predicates without truth-value gaps). In such a logic, unrestricted use of mathematical induction can in principle

lead to paradoxes. Since both IF logic and Weierstrasss implicit epsilon-delta logic are cases in point,

mathematicians must be on alert as to what kinds of predicates they apply mathematical induction to. Whether in

the actual history of mathematics negligence in this respect has led anyone into actual trouble does not seem to be

known.

The failure to catch up with the epsilon-delta tradition has not prevented the symbolic logic from being

developed and applied in other directions. It has nevertheless distorted symbolic logicians perspective on the

foundations of mathematics, especially on what can be done in mathematics by logical means, including the

famous incompleteness and impossibility results by Gdel, Tarski and Paul Cohen that are often considered as the

major results in logical theory in the twentieth century. We have to realize that what these results reveal are

merely limitations of RFO logic, a logic that was flawed from the start, and not a limitation of logic as such or of

axiomatization. (See here Hintikka forthcoming (a).)

We can consider as a test case the claim that elementary arithmetic is not completely axiomizeable. Ifthis claim had been made a hundred years ago to a mathematician in the epsilon-delta tradition, he or she might

very well have countered by claiming that such a complete axiomatization is easily accomplished. Most of the

Peano axioms are unproblematic. The problem is to express the principle of induction in purely logical terms.

This can be done if we can express that natural numbers are well-ordered with zero as the only number that does

not have an immediate predecessor. This well-ordering can be expressed by saying that there are no infinite

descending chains of natural numbers. Now the instance of such a claim can be expressed by the epsilon-delta

technique as follows:

(9) For any given natural numbers 1and 2, one can choose 1depending on 1 only and 2 depending on

2only, such that 1= 2if and only if 1= 2and that 1< 1and 2< 2.

Here 1, 2are also natural numbers.


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Technically, this does not amount to a counter-example to Gdels first incompleteness (meta)theorem,

for several reasons. One of them is that Gdel in effect requires that the logic elementary arithmetic uses is

axiomatizable in the sense that there exists a recursive enumeration of all logical truths. In fact, in IF logic

there is no such axiomatization. But apart from that, (9) it is not expressible by means of RFO logic, which is

the logic used in Gdels arithmetic. This is because of the independence of the choices of 1and 2of 2and 1,

respectively. But if this is the reason for the inexpressibility of a categorical axiomatization, Gdels

incompleteness results must be considered as showing the limitations of RFO as used in elementary arithmetic

than any limitations to the use of logical conceptualizations in arithmetic and elsewhere in mathematics. With the

rise of IF logic and its extensions, questions of complete axiomatizablility are put to a new light. For a

philosopher it is instructive to realize that in principle Weierstrass could have formulated such an axiomatization

in a perfectly natural sense.

8. Freges failure

Historically, this use of too poor a logic of quantifiers goes back to Frege. Was it merely an oversight on Freges

part, an historical accident? No, it was a mistake waiting to happen. It was based on an inadequate understanding

of the meaning (semantics) of quantifiers by Frege.

Frege never betrays any awarenessor at least any appreciationof the important development in

analysis that the uniformity concepts facilitated. The apparently only reference to them in his writing is a mention

of the arithmetization of analysis in his review of Hermann Cohen (See Frege 1885.) What is even more

striking, Frege never mentions epsilon-delta definitions. It would have been an impressive proof of thesignificance of Freges logic if he had pointed out how it enables us to formulate the epsilon-delta technique.

Freges logic is often presented as having accomplished that. Yet Frege nowhere as much as mentions the

epsilon-delta method. What is more, he never even comments on the phenomenon of quantifier ordering, let alone

its semantical meaning as an ordering of subsequent givenness and choices.

But perhaps we should not give much weight to such evidence from silence. Be that as it may, a

comparison with contemporary logicians should offer a fair perspective on the scope of Freges active knowledge

and interests. An obvious object of such comparison is Freges co-inventor of modern symbolic logic, Charles S.

Peirce. This comparison is unexpectedly stark. Even though Peirce, unlike Frege, was not a professional

mathematician, he shows a firm grasp of the conceptual development of mathematics from Cauchy to Weierstrass

and comments on it in considerable detail. In particular, he is aware of the distinction between pointwise and

uniform concepts in the foundations of analysis. Peirce not only wrote a review of Forsyths 1893 treatise of

function theory where uniformity concepts are used. (See Peirce (1894).) He pointed out that Forsyth in certain


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theorems commits the same mistake as Cauchy and used in his assumptions only pointwise convergence when

uniform convergence is needed. He was cognizant of Weierstrass (1886) and praised Weierstrass for improving

the logical clearness of mathematics.

In his own work, Peirce developed an explicitly game-theoretical account of quantifiers, complete withtwo players, order of moves etc., not only as one possible illustration of the logic of quantifiers, but as revealing

their meaning. Clearly, Peirce was a more perceptive and more creative logician than Frege. The evidence

suggests that he might also have been a better informed logician and philosopher of mathematics. (I am here

indebted to Ahti-Veikko Pietarinen for information about Peirces work.)

Freges mistake was in fact far deeper than an oversight. It is not only that he did not recognize quantifier

independence when it occurred. He did not understand quantifier dependence in the first place. For Frege,

quantifiers were higher-order predicates that expressed the nonemptyness or exceptionlessness of lower-order

predicates. (See Frege 1893, sec. 8, pp. 11-14.) As a consequence, quantified sentences had behaved according

to him like long (possibly infinite) disjunctions and conjunctions. In such a thinking, quantifier (in)dependence

becomes (in)dependence of propositional connectives on each other. This idea was unknown until our day and

age, and would have been totally incomprehensible to Frege.

Frege simply failed to understand fully the meaning of quantifiers. He understood their semantical role in

expressing the nonemptyness or exceptionlessness of certain (lower order) predicates. Indeed, he characterized

quantifiers as doing just that. But he never acknowledged the even more important semantical role of quantifiers

of expressing through their formal dependence on each other the actual dependence of the respective variablesbound to them. This is not a matter of semantical interpretation only. I have shown (Hintikka forthcoming (c))

that a neglect of dependence relations between quantifiers is what caused the paradoxes of set theory and thereby

the entire crisis of foundations

A way of putting Freges mistake in a historical perspective is to say that he restricted himself to the

symbolic logic tradition. (Yes, he helped to start it in the first place.) The choice function interpretation was

foreign to him, and it remained foreign to most mainstream logicians after him.

9.Freges anti-intuitionism as a source of his failure

The difference between the two traditions can be illustrated by a case study that is interesting in its own

right. The axiom of choice serves to illustrate the systematic and historical issues involved here. That it has not


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long ago been recognized as a first-order logical principle is due to a self-imposed restriction on the use of logical

operations. This restriction is merely notational, without any theoretical motivation other than a convenience in

manipulating symbols according to formal rules. The usual formulations the rules characterizing logical notions

like quantifiers can only be applied to them in a formula-initial position (or when they occur otherwise as the

principal connective or operator). In particular, existential instantiation is usually applicable only to a formula-

initial existential quantifier (x)F[x] allowing us to replace it by F[b], where b is a new individual constant. But

we could equally well apply existential instantiation to any existential (existential-force) quantifier in context

(10) S [(x)F[x] ]

Then we would have to replace x, not by a constant term but by a function term f(y 1, y2, ) Here f is a new

function constant and (Q1y1),(Q2y2), are all the quantifiers on which (x) depends on in S.

This obvious generalization brings the full force of the axiom of choice to bear on the first -order level.

This generalization is trivially easy to explain and to motivate, as was just done, to anyone who appreciates the

crucial role of dependence relations between quantifiers, in other words to anyone relying on the epsilon-delta

logic. The axiom of choice is an integral part of this logic. In contrast, in the symbolic logic tradition it has

never been integrated in logic itself, and has been treated as an optional axiom in a special mathematical theory.

This is not accidental, for the axiom of choice is closely related to the basic ideas of IF logic and indeed to

the entire epsilon-delta tradition. Furthermore, the implementation of the axiom of choice through liberated

instantiation rules is suggestive of the sources of Freges way of thinking, including his mistake.

Freges failure to understand the dependence-indicating role of quantifiers has in fact interesting

philosophical roots. His avowed purpose in logical theory was to dispense with the use of intuition. This was

thought by him as a refutation of Kant. Now what was this use? I have shown (Hintikka, forthcoming (b)) that

for Kant an appeal to intuition in mathematics meant (expressed in our contemporary jargon) an application of

instantiation rules. Hence a part of Freges project was to dispense with instantiation rules. This is not possible

absolutely, but in the ordinary first-order logic they can be limited to instantiations of sentence-initial quantifiers.

Such quantifiers can apparently be interpreted somehow as not involving intuition. Hence Frege could construe

his logic as being intuition-free.

However, independence friendly logic and accordingly in the use of the unrestricted epsilon-delta

technique, instantiation of dependent quantifiers is indispensable. Hence, Frege could not incorporate unrestricted


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instantiation rules in his logic and as a consequence this logic could not do justice to the unrestricted epsilon-delta

technique.

9. Formalization vs. logic of ordinary language

We are touching a great many important issues here. Among them is the rationale of formalization. Frege claims

as a merit of his formalizedBegriffsschriftthat it frees the study of logic from the fetters of ordinary language.

He says that in trying to achieve complete rigor in reasoning, he found an obstacle in the inadequacy of language

[] the more complex the relations became, the lessprecision [] could be obtained. (Frege 1879, preface).But

one can ask what it was that was difficult for him to understand, the propositions of ordinary language that occur

in reasoning or the subject matter itself, that is, the structures one is reasoning about. Using as a test case the

episode in the history of mathematics that has been discussed here, where was the source of difficulty in

mastering notions like uniform convergence? Was it in the vagueness of the informal or semi-formal language

that mathematicians like Cauchy used? Is the difficulty in understanding the definition (3) due to the use of

ordinary language in it? Was it perhaps an ambiguity or unclarity of the words choice and choose?One

might suggest that. But the very same problems come up in the formalized version (6) of (3). Does the quantifier

(k) depend on x or not? This is not any clearer or unclearer a question when asked about (3) or (6). The fact that

we have introduced a formalism for the logical concepts like connectives and quantifiers does not help to answer

the question. a formalization may result in fixing the meaning of an ambiguous expression, but without awareness

of the ambiguity of the original ordinary language expression, the same problem persists and it is only pushed to

another location. In the case of uniformity concepts, it became a problem of a meaning missing from the

formalization, which was solved only more than a century after Freges formalization.

Frege understood and formalized a wealth of logical concepts. But when it comes to his central concepts,

the two quantifiers, his difficulties in dealing with them are not problems of translation from ordinary language to

a formal notation. Rather, the formalism was for him a tool for trying to understand what it is that is expressed in

informal discourse. Admirable as Freges creation of a formal language in many ways is, a formalism is neither a

necessary nor sufficient precondition for mastering logical reasoning.

Acknowledgements

This paper was written when Jaakko Hintkkka was a Distinguished Visiting Fellow of the Collegium for

Advanced Studies of the University of Helsinki. He was assisted by Antti Kylnp. This support is gratefully

acknowledged.


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