dependence-handicapped logic. IF logic is not axiomatizable, and therefore

there cannot be any set of inferential relationships that fully characterize our

basic logic. In brief, the web of logic itself cannot be held together by inferential

relations. If so, it is a fortiori futile to expect that a set of inferential relationships

could hold together a scienti¬c theory and its different concomitants.

Another way of seeing that Quine™s idea of a web of knowledge held

together by relations of logical implication is not only misleading but grund-

falsch is to note that these relations of logical implication are tautological. If

F logically implies G, the information that G can yield is merely a part of the

information content of F. How such empty relations can create the kind of

connecting links required by Quine™s metaphor is a mystery, and is bound to

remain one.

This may sound too general and too abstract to be convincing, and perhaps

it is. But it can be particularized in a way that has a direct impact. Quine™s web

analogy is but an expanded form of the old idea that theoretical concepts, even

though their objects are not observable, can receive their meaning from the

logical connections between the propositions that contain them and observa-

tion statements. Indeed, Quine compares in so many words the postulation of

mathematical object to the postulation of theoretical entities in physics. (Quine

1981a, pp. 149“150.) But if so, then in view of the crucial role of the “vertical”

connections between propositions and reality, it might be expected that we

could separate what a proposition says about observables from the rest of it,

and replace the proposition by the outcome of that reduction. The result would

have the same observable consequences as the original proposition, leaving

the theoretical concepts of the original proposition unemployed without any

work to do. Indeed, this is what it was hoped that the so-called Ramsey reduc-

tion would accomplish. A Ramsey reduct is obtained from the sentence”say,

S = S[T1 , T2 , . . . ]”where T1 , T2 , . . . are all the theoretical terms of S”by

replacing T1 , T2 , . . . by replacing them by variables”say, X1 ,X2 , . . . ,”and then

binding those variables to sentence-initial existential quanti¬ers. We can call

this reduct

(∃X1 )(∃X2 ) . . . S[X1 , X2 , . . .] (1)

In brief, (1) is the Ramsey reduct of S[T1 ,T2 , . . . ]. The reducer™s hope was

that (1) would have the same observable consequences as S[T1 , T2 , . . . ] and

thereby show that theoretical concepts are unnecessary in science. In terms

of Quine™s metaphor, even the propositions in the middle of the web, away

from the observational rim, could be said to receive their empirical meaning

directly from observable consequences”namely, from the consequences of

(1), without having to consider the logical relations of S[T1 , T2 , . . . ] to any

other propositions. This would have pulled the rug out from under Quine™s

The a priori in Epistemology 111

idea that the empirical meaning of a proposition is constituted by means of its

relations to other propositions. And this would have in turn vitiated Quine™s

argument for the indispensability of mathematics in science.

At ¬rst, this attempted refutation of Quine could not be carried out, how-

ever. For if S[T1 , T2 , . . . ] is a proposition of a ¬rst-order language using the

usual logic, (1) does not always have an equivalent in the same language.

In other words, the Ramsey reductions cannot be carried out in the same lan-

guage. Hence, we apparently cannot get rid of the existential quanti¬ers (∃X1 ),

(∃X2 ) . . . But these range over the kinds of theoretical entities we were trying

to get rid of.

However, this counterattack in defense of Quine fails. (See Hintikka 1998.)

For it turns out that the failure (1) to have an equivalent on the ¬rst-order

level is merely a result of the unnecessary restrictions that Frege and Russell

imposed on ¬rst-order logic and that have only recently been lifted. If we

make it possible to have arbitrary patterns of dependence and independence

between quanti¬ers represented on the ¬rst-order level, we obtain a stronger

logic, which is known as the IF ¬rst-order logic and which is still a ¬rst-order

logic. By its means, (1) can be expressed on the ¬rst-order level, hence elimi-

nating all quanti¬cation over theoretical entities in the sense of references of

theoretical terms. For reasons indicated earlier, this deprives Quine™s line of

thought its raison d™ˆ tre. The indispensability of mathematics may be a fact,

e

but it cannot be defended a la Quine. According to Quine™s own lights, a theory

`

is committed to the existence of the entities over which its quanti¬ed variables

range. Accordingly, for Quine, (1) involves a commitment to the existence

of theoretical entities, but its translation to the corresponding IF ¬rst-order

language does not, for there we quantify only over individuals.

3. Are Mathematical Objects Dispensable?

The origins of the indispensability problem in Quine™s philosophy show up

here in another way, too. Since Quine has no sharp distinction between ana-

lytic and synthetic truths, he cannot distinguish mathematical and empirical

knowledge from each other, either. Hence he cannot very well speak, much

less argue for, the indispensability of mathematical knowledge in science, even

though this is arguably the crucial question here. Quine can distinguish math-

ematical propositions from experimental and observational propositions only

in terms of the objects they pertain to. In this way, the problem has come to

be construed as the problem of the indispensability of mathematical objects in

science. In particular, the critics of indispensability have typically formulated

their arguments as showing how mathematical objects can be dispensed with

in science.

These efforts are not much more convincing than a Quinean defense of

indispensability. One problem here is the relevance of the ontological problem

of the dispensability of mathematical objects to mathematical practice, and in

Socratic Epistemology

112

particular to the usefulness (reasonable or not) of mathematical knowledge in

science. This is obviously a highly complex matter. Does the use of mathemat-

ical reasoning in science commit a mathematician to the existence of mathe-

matical objects? No simple answer is in the of¬ng. According to a quip by Tom

Lehrer, doing arithmetic in base eight is just like doing it in base ten”if you

are missing two ¬ngers. Perhaps we can take his quip to illustrate the fact that,

if the missing two ¬ngers were to make a difference to the preferential base,

their existence or non-existence does not affect the actual arithmetic. Would

the similar absence of two entities called numbers from the number sequence

make a difference to the uses of arithmetic in science? Obviously we have a

most complex problem here. It is made more dif¬cult by the common failure

to distinguish our knowledge of mathematical truths from our knowledge of

the identity of mathematical objects. (For this distinction, see Hintikka 1996

(b).) I will return to these problems later in this chapter. (See Sections 11“12)

Thus I ¬nd some of the typical recent arguments for the dispensability

of mathematics in science shallow. By and large, what they are calculated to

establish is that we need not assume in scienti¬c theories any individual objects

called numbers. A characteristic example is offered by what might be called

“reverse representation theorems.” (See Hartry Field 1980, 1989 and Colyvan

2001, chapter 4.) A typical ordinary representation theorem might show that

a certain purely relational structure”for instance, the system of Euclidean

geometry axiomatized by Hilbert”is equivalent to a structure characterized

by metric notions such as distance. The values of such metric functions are,

of course, numbers. A reverse representation theorem might show that such

numerical-valued functions as distance can be eliminated in favor of purely

relational concepts.

Such results can be interesting and illuminating. However, to proffer them

as arguments for the dispensability of mathematics in science presupposes a

narrow conception what mathematics is. Admittedly, mathematics used to be

considered as the “science of numbers,” but that conception was superseded

more than a hundred years ago by the rise of what has been called “conceptual

mathematics.” (See, e.g., Laugwitz 1996.) If there is a notion that epitomizes

this more abstract, and at the same time more comprehensive, idea of mathe-

matics, it is not the concept of number but the concept of function, naturally

understood in the wide sense, that is not restricted to numerical functions.

The dispensability of mathematics in science will then mean the possibility of

managing without functional concepts. Such a possibility is totally unrealistic.

If a historical perspective is needed to illustrate this irreducibility, an early one

was provided by Ernst Cassirer in 1910. The history of the concept of func-

tion is in fact one of the most crucial aspects of the development of modern

mathematics. (See Youschkevich 1977; Hintikka 2000.)

In logical theory, much of the metatheory of ¬rst-order logic can be devel-

oped in the format of a theory of the Skolem functions. This is natural, because

the existence of Skolem functions can be considered as an implementation of

The a priori in Epistemology 113

our pre-theoretical ideas about the truth of quanti¬cational sentences. (The

Skolem functions of a sentence S produce as their values the “witness individ-

uals” that must exist for S to be true.) The perspective provided by Skolem

functions is also especially illuminating, when more patterns of dependence

and independence between quanti¬ers are considered than are allowed in the

needlessly restricted “ordinary” Frege-Russell treatment of the logic of quan-

ti¬ers. Thus, functions turn out to be the most important part of the ontology of

logic and mathematics. And, hence, the entire discussion of the indispensability

or dispensability of mathematics in science should be addressed to the ques-

tion of the role of functions in the language of logic and mathematics, rather

than to the question of the existence of objects (individuals) called numbers.

One of the ¬rst results that follows when this is done is that a reduction of some