For the latter purpose, as was seen, an inquirer has to know the (mathemati-

cal) identity of the function observed in a given controlled experiment. This

distinction between the two aspects of inductive inference can play a role

in our understanding of actual scienti¬c inference. For instance, the distinc-

tion between the two components of answers to experimental questions has a

rough-and-ready counterpart in a distinction that is sometimes made, even ter-

minologically, between two different kinds of activities of a scientist. In what

are typically referred to as inductive inferences, a scientist is trying to reach the

function-in-extension that is the factual component of an answer to an experi-

mental question, or at least trying to anticipate new pairs of argument values

and function values. But when a scientist is making a stab at identifying the

dependence function mathematically, what he or she is doing is often not called

inductive reasoning or inference but constructing a mathematical model. In

this way, model construction receives a natural niche among the activities of

an experimental scientist and indeed of any scientist. If one is enterprising

enough, one can even try to develop along the lines indicated here a logical

and epistemological theory of model building in science as distinguished from

inductive inference.

Even in cases where it is not natural to speak of model building, we can,

in principle, distinguish from each other two tasks involved in ¬nding a con-

clusive answer to an experimental question. One task is to ¬nd the function-

in-extension that codi¬es the looked-for mode of dependence between two

variables. We might be tempted to call it the “inductive task,” and I will in the

following yield to this temptation. We should nevertheless keep in mind that

what is usually called induction must be understood as involving both tasks.

The other task is to ¬nd the mathematical identity of the function-in-extension

that has been found inductively. I will call it the “identi¬cation task.”

In practice, these two tasks are not usually distinguished from each other,

certainly not temporally. For instance, the need for non-inferential model con-

struction results from the fact that in actual scienti¬c practice, an experimenter

cannot wait till he or she has established an entire function-in-extension codi-

fying the dependence of a variable on others before raising the question of

its mathematical identity. But this does not eliminate the identi¬cation prob-

lem; it merely pushes it to an earlier stage of inquiry. What this means is that

The a priori in Epistemology 137

there is a conceptual component in inquiry over and above the usual inductive

techniques, such as curve-¬tting. Such techniques only serve to overcome the

limitations on the number of experimental observations and on their accu-

racy, and cannot automatically facilitate an identi¬cation of the “curve” that

directly results from the experiment. Thus, any adequate future discussion of

the general problem of induction must be complemented by a discussion of the

role of conclusiveness conditions”in other words, the role of the conceptual

component of our empirical knowledge.

This injunction extends even to the history of the notion of induction. It

is, for instance, instructive to think of Aristotle™s notion of epagoge as not

being so much induction in the twentieth-century sense as a search for the

conceptual knowledge that provides the conclusiveness conditions needed to

answer wh-questions. Aristotle thought that we can, after all, easily know

which entities a concept applies to, for a rational man surely knows what he is

talking about in the sense of knowing what entities he intends to talk about.

But he need not thereby know what concept it is that picks out those entities

from others”that is to say, he need not know the de¬nition of this concept.

Thus, as a consequence, induction is for Aristotle essentially a certain kind of

search for de¬nitions or, to put the same point metaphysically, an attempt to

assemble the appropriate form in one™s soul. In Hintikka 1980, I have in fact

argued precisely for this kind of interpretation of the Aristotelian epagoge. In

this form, the search for conclusiveness conditions has played an important

role in the history of philosophy.

16. Comparison Between Different Cases

One of the insights we have reached is that there are two different things in the

kind of reasoning that is usually called inductive. On the one hand, an inquirer

is in the simple but representative case of experimental research searching

for a function-in-extension. (This is the inductive problem in the narrower

sense.) On the other hand, an inquirer is trying to identify the function in

question. This is what I suggested calling the “identi¬cation task.” These two

searches are not sequential, but can go on at the same time. There can be all

sorts of interplay between them and they can also proceed at different speeds.

This creates an interesting perspective from which to view different kinds of

scienti¬c inquiry and even different episodes in the history of science.

One kind of interplay concerns the role of the mathematical identi¬cation of

experimental dependencies in estimating the reliability of inductive inferences

in the narrower sense. If I have to anticipate the value of an experimentally

established function-in-extension for a previously unobserved argument value,

my procedure is likely to be not only simpler but also more reliable if I have

established the mathematical law that captures the observed fragment of the

dependence function. This is especially clear when two fragments of a function-

in-extension have been established for two separate intervals of argument

Socratic Epistemology

138

values. Suppose that the corresponding mathematical functions have been

identi¬ed. If they turn out to be the same function, it enhances the two partial

inductions and encourages extending them to new argument values. If they are

different, an experimental scientist faces the further problem of reconciling

the two mathematical functions. I will return to this problem later.

Other situations call for other comments. In the schematic and highly

abstracted thought-experiment described earlier, an idealized experimenter

reaches a fully speci¬ed function-in extension and then tries to identify the

mathematical function that it is a graph of. This thought-experiment looks

more unrealistic than it need be. In some cases, only discrete values of the

argument are possible. In such cases, if measurements are reasonably accurate,

an experimenter can reach a fairly close approximation of the function-in-

extension that codi¬es the results of experimentation. Then the main problem

will be to ¬nd the mathematical function that captures the results of experi-

mentation.

One can think here of the historical situation at the early stages of the

development of quantum theory in this light. What happened at that time

does not ¬t very well into the conventional picture of inductive reasoning. In

so far as historians of science have diagnosed the thinking of early quantum

theorists, they have typically spoken of model construction. There is more to

be said here, however. Physicists had at the time a great deal of accurate infor-

mation about the spectra of different atoms. Such a situation resembles one

in which the relevant function-in-extension has been reached, at least in the

sense that the main problem facing investigators was not to make the measure-

ments of the spectral lines more accurate or more numerous. It was to ¬nd the

law that captured a reasonable number of the already known measurements.

Indeed, Bohr™s theory applied in the ¬rst place only to the hydrogen atom,

and accounted for its spectra, albeit not fully accurately. What was impressive

about Bohr™s theory was not that it predicted accurately the observed hydrogen

spectrum, but that it produced an explicit mathematical formula that yielded

a good approximation to this spectrum. In other words, success in solving the

identi¬cation counted more than accuracy in solving the inductive problem.

Bohr™s theory was presented by him in the form of a physical model, but since

this model was known to be impossible in classical physics, philosophically

speaking it was little more than a mathematical model.

In the same way, Heisenberg and Schrodinger were later trying to ¬nd

¨

an explanation for similar observed data. Their respective mathematical laws

were accepted (and were shown to be equivalent) well before there was any

¬rm conception of what the underlying physical reality is like. (Indeed, some

physicists warned their colleagues in so many words against asking too many

questions about the reality behind the mathematical laws of quantum the-

ory.) Hence it is natural to look upon their work in the ¬rst place as a search

for the mathematical law that governs the relevant phenomena. All this is

The a priori in Epistemology 139

characteristic of the identi¬cation problem as distinguished from the induc-

tive problem in the narrower sense.

More generally speaking, partial solutions to the two tasks”the inductive

one and the identi¬catory one”can help each other. If a physicist has reached

the mathematical function governing certain phenomena on the basis of partial

evidence, he or she can use it to predict further observations that can ¬ll in

the function-in-extension. This is what happens when a novel theory is used

to predict new kinds of observations or to predict the existence of previously

unknown objects.

One kind of special but important situation arises when experimentalists

have managed to establish a function-in-extension for more than one interval

of argument values so well that even the identi¬cation is possible for each of

the intervals to the limit of observational accuracy. It suf¬ces here to consider

the case in which the intervals are not overlapping. Then it may happen that

the resulting mathematical functions are different for the different intervals of

argument values. Then scientists are faced with a double task. Not only do they

have to ¬nd the function values corresponding to all the different argument

values, they have also to combine the different mathematical functions so that

they become special cases of a single function (up to the limits of observational

accuracy). This identi¬cation problem is typically solved initially by means of