existing framework by locating empirical input within it.

The analogy between the identi¬cation of individuals and the identi¬cation

of functions also re¬‚ects some light back on what is necessary for knowing

who someone, say “b,” is. What is needed is not to be able to refer to “b” in

some special way”for instance, by a proper name or some other supposedly

rigid designator. (See here Kripke 1972.) What is needed is to have so much

information that one can rule out all those possible scenarios in which “b”

refers to a different individual. Such information is needed even in the case

where “b” is a proper name, and it is not gained by a Kripkean dubbing

ceremony. In an analogous case, I can, so to speak, admire a beautiful curve on

a graph paper, representing a function-in-extension, and solemnly pronounce:

“I name you a lemniscata.” Needless to say, such a ceremony has not in the

least helped me to come to know what that function is, even though in the case

of a mathematical object, we do not even have to worry about keeping track

of its future career by means of causal links as, according to Kripke, we have

to do with physical objects. In mathematical practice, curves are identi¬ed by

their equations, not by their proper names. By the same token, a Kripkean

dubbing act does not help me, in the least in the case of physical objects, to

know who or what the dubbed individual is, in outright contradiction of the

idea that a proper name so introduced is a vehicle of de re knowledge about

that very individual.

Kripkean dubbing is at bottom nothing more than a dramatization of the

old idea of ostensive de¬nition. Now, the role of an ostensive de¬nition can

be pinpointed completely accurately. Such a “de¬nition,” like other forms

of identi¬cation, locates the de¬nendum in a certain space. In the case of

ostension, this space is one™s visual space. Identi¬cation by means of such a

space is a form of perspectival identi¬cation, not a public one. Visual space

does not lie at the bottom of our normal identi¬cation of public entities, be

they persons, physical objects, places, events, or whatnot. Kripke is simply

mistaking perceptual identi¬cation for public identi¬cation.

Kripke thinks that once an entity has been ostensively identi¬ed and given

a name, its continued identity is guaranteed by causal chains. This might seem

to work for physical objects, but in the case of higher-order entities such as

functions, there is not even a plausible candidate to play the role of causal

continuity.

On the other hand, the identi¬catory character of the mathematical knowl-

edge involved in the presuppositions con¬rms the earlier insight into their

status as conceptual truths. There is perhaps not the same temptation to label

our knowledge of the identity of functions linguistic or semantical, let alone

lexical, as there is in the case of our knowledge of the identity of individu-

als. However, once the precise nature of the analogy between the two cases

of identi¬cation is realized, it can be seen in what sense the identi¬cation of

functions, too, is a matter of conceptual knowledge.

Socratic Epistemology

134

We are dealing here with extremely important questions. The possibility

of making a distinction between conceptual and factual information in an

experimental context casts a long shadow on all attempts to consider, for

instance, logical truths as the most general factual truths. The two kinds of

knowledge involved in answering an experimental question differ in kind, not

just in degree of generality.

One more clari¬cation is in order. I am not assuming a separate class of enti-

ties called functions-in-intension. The ways of identifying different functions-

in-extension cannot be rei¬ed into a class of intensions. This is shown, among

other things, by the fact that there may be more than one way of identifying

a function in a full enough sense to justify one in saying that one knows what

function it is. For instance, the system of spectral lines of a certain kind of atom

can be considered as a function-in-extension in a sense relevant here. As far as

the account given here is concerned, a scientist can prove that he or she knows

what that function is either by deriving it from the Schrodinger equation or

¨

from matrix mechanics. It is a hugely oversimpli¬ed view of the actual criteria

of knowing who someone is to think that they can be summed up in know-

ing some particular facts about him or her, including knowing some mythical

“essence” of his or hers. Likewise, knowing what a function is does not in

general reduce to knowing some one thing about that function, for instance

knowing its intension.

14. Identi¬cation of Functions and the History of Mathematics

The identi¬cation of functions can also be looked upon in a historical perspec-

tive. When the needs of scientists thrust upon mathematicians functions they

had not studied before, some historians of mathematics have described what

happened as a gradual widening of mathematicians™ idea of function. I ¬nd

such a claim profoundly misleading. (See Hintikka 2000.) The general idea

of a function as any conceivable correlation of argument values and function

values is older than many of such alleged extensions of the concept. It certainly

occurs in Euler. (See here Euler, quoted by Youschkevich 1977, e.g., pp. 62,

68.) Moreover, it is hard to imagine that Leibniz was not in effect relying on

it when he spoke of all possible worlds, each of them governed by a different

“law.” If Leibniz did not identify these different laws with so many different

functions, it is only because the term “function” was not yet a household word

even in mathematical households. The so-called widening of the concept of

function should be thought of as a widening of the sphere of functions mathe-

maticians know in the sense of being familiar with. Indeed, the main engine of

the extension of the class of functions considered by mathematicians was the

need to master functions obtained as solutions of partial differential equations

of physics. (See Youschkevich 1977, pp. 62, 66.)

The observations made here throw some light on other historical mat-

ters. A suggestion has been made by several scholars to the effect that the

The a priori in Epistemology 135

methodological “secret” of the early modern science was an extension of the

idea of geometrical analysis to science. (See Hintikka and Remes 1974, last

chapter; Becker 1959; and Cassirer 1944.) Indeed, Descartes acknowledges

the Greek method as one of the starting points of his own thinking. (Regulae,

Adam, and Tannery, vol. X, p. 374“378.) More fully expressed, the gist of

the Greek method of analysis was systematic study of the interrelations of

different geometrical objects in a con¬guration. “Analytic” geometry came

about when Descartes undertook to represent these dependence relations

algebraically. The revolution in natural sciences began when scientists began

to look at physical con¬gurations in the same way in terms of different kinds

of dependence relations between their ingredients. This is what Oskar Becker

called the idea of “analytic experiment.” What has been seen in this chapter

is that there is an intrinsic connection between the two enterprises. We cannot

fully understand given physical dependence relations unless we understand

the mathematical nature of the functional dependencies they instantiate.

15. Experiments and the Problems of Induction

It can also be seen here that the epistemic logic of questions and answers

puts the problem of induction into a new light”or is it old light? For one

thing, it shows the ambiguity of what is meant by the problem of induction.

Induction is often thought of (in a situation such as a simple controlled experi-

ment) as a step from a ¬nite number of observations (observed points on the

x-y coordinate system) to the function f that codi¬es the dependence of y on x,

y = f(x). Expressed in this way, an inductive step (“inductive inference”) is by

de¬nition underdetermined, and hence no absolute a priori justi¬cation can

be given to any particular choice of f. But what is equally often overlooked

is that this underdeterminancy has two different sources. This is because the

experimenter is trying to accomplish two different things. He or she is trying to

¬nd a function-in-extension, but usually also”and often in the ¬rst place”to

identify this function-in-extension. This makes a concrete difference. If what

the experimenter is merely looking for is the function f in extension, then the

impossibility of reading off f from actual observations is due to the limita-

tions of an experiment to a ¬nite number of observations and to the limited

accuracy of these observations. Such kinds of limitations can be reduced by

better experimental techniques and by additional observations. The function-

in-extension f can be approximated more and more closely through such

improvements.

But if what the experimenter is looking for is”as is normal in scienti¬c

practice”a de¬nite identi¬cation of the mathematical function in question,

then he or she faces another kind of problem”namely, the step from a

function-in-extension to its mathematical identity. Insofar as this step is part of

an “inductive inference,” then scienti¬c induction has a mathematical (more

generally speaking, conceptual) component. This mathematical component is

Socratic Epistemology

136

different from the factual tasks involved in induction. One can, for instance,

get closer to the function-in-extension f by improving one™s experimental tech-

niques, but the step from a function-in-extension to its mathematical incarna-

tion is not automatically aided and abetted by such techniques. It requires

enhanced mathematical rather than experimental sophistication.

Thus, two entirely different kinds of generalization are involved in induc-

tion. There is, on the one hand, the step from observed values of the function

y = f(x) to other observable values in the same experimental situation. On the

other hand, there is the generalization required to extend the mode of depen-