by the reply.

The distinction between the two modes of identi¬cation can be used to

put the dual nature of a controlled experiment into a perspective. It may

not be far-fetched to suggest that the primary “inductive” purpose of the

physical performance of a controlled experiment is to identify the dependence

function perspectivally. Then one could describe the step from an observed

function-in-extension to the mathematical law governing it as an inference

from a perspectival identi¬cation to public identi¬cation.

The sense in which a function-in-extension does not alone constitute an ade-

quate answer to an experimental question can be seen in different ways. For a

simple historical instance, it was not dif¬cult for scientists and mathematicians

to produce good graphs of the curve formed by a perfectly ¬‚exible uniform

hanging chain. But it took a signi¬cant ingenuity on the part of early modern

mathematicians to ¬gure out what the function is that was represented by such

graphs. (See Mach 1960, pp. 85“87.) This example illustrates a more general

point. Only if a scientist knows what the function is that an experiment has

yielded can he or she predict what the function-in-extension is that another

experiment concerning the same variables is likely to produce. For knowing

what a function is is logically speaking nothing but being able to recognize it (its

manifestation) in a variety of different circumstances. For instance, unless one

knows the function that speci¬es the shape of one™s hanging chain, one does

not know what to expect of other hanging chains with a different length and

with different endpoints. In more general terms, the mathematical function

identi¬ed in a controlled experiment will contain parameters characterizing

the particular set up used in the experiment. (In the case of the hanging chain,

these would be the length of the chair and the distance between the hanging

posts.) The values of these parameters are not ordinary observable quantities,

but can serve to de¬ne one particular application of the relevant laws and the-

ories. The logical status of such applications is discussed in Hintikka 2002(a).

Socratic Epistemology

124

Generality with respect to such parameters is thus, in effect, generality with

respect to the possible scenarios presupposed by the concept of knowledge.

Thus the Lorelei problem is thus very much a fact of life”or at least of fact of

epistemology”for practicing scientists.

In general, a scientist can generalize from a single experiment only if he or

she knows the law governing the functional dependence that an experimen-

talist has found. The identi¬catory knowledge encoded in the conclusiveness

condition is not only indispensable for a genuine answer to our experimen-

tal question, it is also indispensable for the purpose of using the outcome of

an experiment as a stepping-stone to an inductive generalization. Paraphras-

ing Euler (see Youschkevich 1977, p. 68), one might say that the graph of

an unidenti¬ed function not determined (at least not yet determined) by any

known equation is, as it were, traced by a free stroke of the hand. (Of course,

in the case of an actual experiment, the hand is nature™s.) I do not know what

the same hand will draw next time unless I know the rule it is following. Even

more generally, how can one as much as to discuss a function-in-extension?

One can assign a name to it, but one can de¬ne it only in a purely nominal

sense. Such a naming does not in principle help me at all to know how the same

variables depend on each other in another experimental situation. Hence the

conclusiveness condition makes a real difference in the case of experimental

questions, too. For if I can identify the function represented by a graph in one

experimental situation, I know its graph in other possible situations. Indeed,

this is what identi¬catory knowledge means in correctly interpreted possible-

worlds semantics, for the intended “possible worlds” are precisely the different

occasions of use of the concepts in question. (See Hintikka 2002(a).)

And this difference is precisely what the failure of (18)“(21) amounts to. For

to know what an entity”in this case, the function g”is, in other words, to know

what “g” means, is to know what it refers to in different circumstances of use.

And what one™s knowledge of the mere function-in-extension fails to provide

(as was seen) is precisely knowledge of what that dependence function (even

only the function-in-extension) is in other situations in which the dependence

is manifested. As was seen earlier, such different circumstances of use might

be manifested by different values of the parameters that identify different

particular experimental contexts.

The dif¬culty of the Lorelei problem is easily underestimated. A historical

example of this is the dif¬culty that Kepler had in ¬nding the right mathe-

matical law that captures Tycho Brahe™s observations concerning the orbit of

Mars. Yet no lesser a ¬gure than Peirce had to confess that he had initially

and mistakenly thought of Kepler™s work as little more than “to draw a curve

through the places of Mars.” (Peirce 1931“35, 3.362.)

A partial analogue of the difference between knowing the function-in-

extension and knowing its mathematical law is receiving a coded message and

decoding it. This analogue is apt precisely in that it illustrates the complexities

of the Lorelei problem. Indeed, some early modern scientists compared the

The a priori in Epistemology 125

task of a scientist to deciphering the “Book of Nature.” This simile perhaps

has”or can be interpreted as having a more speci¬c meaning than we ¬rst

realize.

10. The Conceptual Component of Experimental Answers is

Mathematical

The same analogy serves to extend the distinction between conceptual and

factual components of an answer from simple wh-questions to experimen-

tal questions. In the case of simple wh-questions, the indispensable collateral

knowledge concerned the identity of individuals, and can be thought of as

semantical. But now the entities whose identity is at issue are functions, not

individuals, as is seen from (18)“(21). Accordingly, the requisite identi¬catory

knowledge is in this case mathematical, not semantical in a narrowly linguis-

tic sense. In any case, whatever one takes the relation of these two kinds of

knowledge to be, the identi¬catory knowledge needed here is not physical

knowledge. That is, ¬nding out what mathematical function is represented by

a given experimentally established curve is not a problem to be solved by an

experimental physicist. It is a task for mathematicians, or mathematical physi-

cists, not for experimental scientists, even though knowledge of the relevant

physical laws may be needed to solve it. Solving the problem of the iden-

tity of a curve produced by a controlled experiment can indirectly increase

our factual knowledge, even though in a different sense it does not. By itself,

identifying a function (coming to know it) contributes only our mathemati-

cal knowledge, in other words, to conceptual (a priori) knowledge. Such an

identi¬cation typically means ¬nding a place for the function to be identi¬ed

within the framework of some theory of similar functions. Developing a the-

ory that enables a mathematician to do so is no mean feat in most cases. Yet

such is this mathematical knowledge that has been found to be indispensable

in experimental science. Indeed, this need of being able to tell physicists what

the functions are that their experiments have produced has been an important

incentive in the actual development of mathematics. (See Hintikka 2000.)

In brief, the requisite knowledge of the identity of experimentally discov-

ered functions is not factual, but mathematical in nature. Lorelei problems are

solved in departments of mathematics, not in physics laboratories. Yet such

mathematical knowledge is needed for the purpose of conclusively answering

empirical, especially experimental, questions. What an experimental physicist

is trying establish is not just a smooth curve on a graph paper indicating his or

her data, but establishing what function it is that governs the dependence of

different variables on each other. For one small example, when a co-discoverer

of Wien™s radiation law reported his ¬ndings, he did not just communicate a

function-in-extension to other physicists, he gave them a mathematical law:

“Furthermore, I am pleased to inform you that I have found the function I

of Kirchoff™s law. From my observations, I have determined the emissitivity

Socratic Epistemology

126

function I = (c1 / ) e(exp(c2 / T)).” (Paschens to Kayser, June 4, 1896; quoted

in Hermann 1971, p. 6.)

The mathematical character of the knowledge of conclusiveness conditions

of experimental questions is also of interest from the vantage point what used

to be called historisch-kritisch. Michael Friedman has aptly argued that the

most important ingredient of the philosophy of logical positivists was their

conception of the a priori element as being mathematical and logical. What

has been found vindicates this conception in the instructive test case of ex-

perimental knowledge in a literal sense. The a priori knowledge needed in

experimental science is the knowledge codi¬ed by conclusiveness conditions,

and this knowledge has been seen to consist, in this case, of knowledge of

mathematical functions.

11. The Indispensability of Mathematics

Thus, even though identifying the function which an experiment yields is a

mathematical problem, it is an integral part of conclusively answering a physi-

cal (experimental) question. Hence, mathematics is indispensable in science.

Without its help, we literally could not answer conclusively perfectly empirical

scienti¬c questions. And hence Quine was right: Mathematics is indispensable

in science, just as he claimed. We have not only found a place for a priori

knowledge in the world of facts, we have shown the indispensability of such

knowledge.

It is nevertheless of interest to see precisely what that place of the math-