mathematics in science is often thought of as consisting in relating different

propositions to each other. We encountered this view in the beginning of this

chapter as a foundation of Quine™s belief in the indispensability of mathemat-

ics in science. Now we have found another role of mathematics in experimental

science”namely, conclusiveness conditions for the answers to experimental

questions. In less technical terms, mathematics is needed in order to express

adequately the answers to empirical questions. This role of mathematics in

science is different from its deductive role. The knowledge that mathematics

provides is about the identity of mathematical objects, including functions, not

about deductive connections.

It is not obvious, however, what such an indispensability of mathematical

knowledge in experimental science implies about realism vis-a-vis mathemati-

`

cal objects. Indeed, our rejection of the Quinean picture of mathematics as

holding together of the fabric of science might seem to destroy all such impli-

cations. Quine™s idea seems to be that whatever is indispensable in science

must enjoy the same objectivity as science. This assumption can perhaps be

accepted. But in order to obtain an argument for ontological realism, one has

to assume also that whatever is indispensable for scienti¬c purposes must enjoy

the same kind of reality as the (other) objects of science. This assumption has

The a priori in Epistemology 127

been seen to be false. The kind of knowledge codi¬ed in the conclusiveness

conditions is indispensable, and it may be as objective as scienti¬c knowledge

in general. But it is not factual knowledge, but conceptual knowledge, and it

is not about the natural objects that science is normally thought of as deal-

ing with. The indispensability of mathematics in science is a fact, but it does

not imply that mathematical and scienti¬c knowledge are ontologically on a

par, and hence it does not imply ontological realism, either, with respect to

mathematical objects”or so it seems.

Furthermore, my reason for indispensability differs sharply from Quine™s.

The argument presented here does not depend on a denial of a distinction

between conceptual (analytical) and factual (synthetic) truths. It discour-

ages any such denial. The conclusiveness conditions (18)“(21) are mathe-

matical and a fortiori conceptual in an even clearer sense than the conclu-

siveness conditions (10) of simple wh-questions. In contrast, an unidenti¬ed

function-in-extension is merely a summary of the raw data yielded by the

experiment. Hence there is an unmistakable consequential difference between

the two.

But even though the argument so far presented for the indispensability of

mathematics in science does not yield reasons for ordinary ontological realism,

it shows something else. First, it shows that mathematical knowledge must have

the same objectivity as experimental knowledge. Indeed, this follows a fortiori,

for it was seen that a certain type of mathematical knowledge is a component of

experimental knowledge”that is, knowledge needed to answer experimental

questions conclusively. Hence, if experimental knowledge is objective, then so

is mathematical knowledge.

Moreover, a review of the argument outlined earlier reveals an interesting

feature of the conceptual situation. What kind of mathematical knowledge is it

that was found indispensable in experimental science? A glance at such exam-

ples as (18)“(21) provides an answer. The knowledge in question pertains to the

identity of mathematical objects, typically functions, rather than to the math-

ematical truths. In the classi¬cation of Hintikka (1996(b)), it is knowledge of

mathematical objects. But there clearly cannot be objective knowledge of the

identity of mathematical objects unless such objects enjoy objective existence.

In epistemology, we can invert Quine™s dictum and say: No identity without

entity. More fully expressed, there can be no knowledge of identities without

existing entities that that knowledge is about. In order to know which func-

tions a controlled experiment (experimental question) has produced, there

must be such a function, just as much there must exist the next governor of

my state in order for anyone to claim truly to know who he or she will be. If

the constitution of my state has been changed, and the next head of that state

will be a prime minister rather than a governor, it cannot be true to say that I

know the identity of the next governor”there just ain™t no such person. Hence

the line of thought outlined in this chapter provides a proof of the ontological

realism of mathematics after all.

Socratic Epistemology

128

However, the argument presented here does not show what kind of objec-

tive existence mathematical objects enjoy (or suffer). This question calls for a

separate investigation. Suf¬ce it here to point out only that for such an inves-

tigation we need sharper distinctions between kinds of objective existence

than are found in the standard literature. Objective existence does not mean

existence independently of human thought and human action. We can have

perfectly objective knowledge of such man-made objects as bridges, highways,

buildings, and books. Non-physical existence does not have to mean existence

in some supersensible “Platonic” region of the actual world, either. Concep-

tual objects may enjoy a reality that is not relative to a particular scenario

(state of affairs, course of events, a.k.a. “possible world”). For instance, such

objects may pertain to relations between different scenarios, such as identities

between denizens of different “worlds.”

One has to be careful here in other respects, too. One of them is the precise

sense in which “there is no identity without entity.” The entities relevant here

are functions. As (18)“(20) illustrate, the identity of such functions can be

expressed without quantifying over entities other than individuals, provided

that we have an independence indicator (slash) at our disposal. We do not

have to quantify over higher-order entities in the literal sense.

Moreover, I have shown in Hintikka 1998 that certain conceptual objects”

namely, functions”are being tacitly presupposed already by our logic of quan-

ti¬ers. This logic is not fully explainable by the feeble metaphor of quanti-

¬ed variables “ranging over” a certain class of values. Quanti¬ers are proxies

for certain choice functions, known in the trade as Skolem functions, where-

fore these functions are presupposed by all use of (nested) quanti¬ers. Hence

a clever philosopher may try to eliminate numbers in favor of geometrical

(space-time) objects (See Field 1980), but the very theory of such geometrical

objects inevitably presupposes other kinds of mathematical objects”namely,

functions.

12. Knowing What a Mathematical Object Is

A comment is in order here to avoid unnecessary misunderstandings. It might

be objected to that the idea of knowing a function (knowing which mathe-

matical function is represented by a function-in-extension) is a vague notion.

(Strictly speaking, as indicated earlier, we should be speaking of knowing what

a function is, rather than knowing it, for the latter expression is normally used

of perspectival identi¬cation, which is not the issue here.) What precisely are

the criteria of such knowledge? Admittedly, these criteria depend on the pur-

poses of the mathematicians in question and the state of the art in the entire

mathematical community. But this relativity does not imply vagueness. Once

the standards are ¬xed, however tacitly, the conceptual situation is what has

been diagnosed in this chapter. It is simply a fallacy to conclude from the

The a priori in Epistemology 129

fact that a concept is not sharp that it is not applicable, important, or even

expressible in a logical situation

In this respect, the situation with respect to simple wh-questions and experi-

mental questions is similar. For instance, people™s actual choice of the opera-

tional criteria of knowing who someone is vary widely. (For their messy choices,

see, e.g., Boer and Lycan 1986.) But even though the choice between different

¨

criteria may be underdetermined, it does not imply that each of the possible

standards is not objective. The notion of knowing who does play a useful role

in our language and in our thinking. Once the criteria of knowing the iden-

tity of entities of different logical type have been ¬xed, the logical relations

are as I have described them. Likewise, it can make a concrete difference to

mathematicians™ and physicists™ working life whether they know which func-

tion mathematically speaking a certain function-in-extension “really” is, even

if the criteria of such “real” knowledge are different on different occasions.

This analogy between the identi¬cation of persons and the identi¬cation of

mathematical functions may not be immediately convincing, however. There

is more to be said here, in any case.

One unmistakable fact is that the identi¬cation of functions obtained em-

pirically is a job that is constantly being performed by scientists and mathemati-

cians. And whatever the precise criteria are or may be of succeeding in such

an enterprise, it makes a concrete difference when one does succeed. Kepler

spent years trying to ¬nd the function represented by Tycho Brahe™s observa-

tions concerning the orbit of Mars, and when he reached the right one, there

was no hesitation in his mind. Even experimental scientists frequently report

their results in the form of a function represented by a mathematical formula,

as we saw earlier. Obviously there are, in scienti¬c practice, some objective

criteria for when an empirically established dependence relation is found to

be instantiating some speci¬c mathematical function. I am not overlooking

the problems occasioned by the limits of observational accuracy, presence of

errors, and so on, only arguing that they present a problem different from the

identi¬cation of functions.

This does not mean that mathematicians, logicians, and philosophers would

not have a hard time if they were asked to specify the precise conditions

in which one can be said to know a mathematical function. In this day and