different intervals of argument values rather than by means of new observa-

tions. This illustrates, once again, the conceptual character of the identi¬cation

problem.

A paradigmatic example is offered by Planck™s reasoning in ¬nding his

famous radiation law. Planck had available for him two different laws of black

body radiation”one of them (Wien™s radiation law) holding for low frequen-

cies and the other one (the Rayleigh“Jeans law) holding for high frequencies.

No help could be expected from additional observations. Planck™s initial line

of thought was simply to manipulate the explicit mathematical formulas that

express the two laws. Only later did he ask what possible physical signi¬cance

the uni¬ed law might have. In doing so, he was led to realize that a natural

quasi-argument for the integration involved the assumption of the quantiza-

tion of energy. Thomas Kuhn (1978) has called attention to the remarkable fact

that initially neither Planck nor any other physicist paid much attention to the

idea of quantization. In the light of the structure of experimental inquiry, this

is perhaps not surprising. The quantum hypothesis was completely irrelevant

to the inductive task of ¬nding a function-in-extension for the radiation law.

And the other task”the mathematical identi¬cation of the radiation law”was

easily thought of as being merely a matter of conceptual inquiry, with no direct

connections with empirical facts. In such a perspective, the quantum hypothesis

must have appeared to be only a heuristic idea that served to amalgamate the

two different partial radiation laws into a single one. Admittedly, the target

Socratic Epistemology

140

of the identi¬cation was an experimentally obtained function-in-extension.

But the physical explanation and even the physical import of this function-

in-extension might very well have seemed to Planck as hypothetical as Bohr™s

atom in its literal form as a miniature planetary system was later for its inventor.

The kind of reasoning involved in interpolating and extrapolating partial

generalizations (in the sense of generalizations established only for different

restricted classes of argument values) played an important role in earlier phi-

losophy of science. It had a name: It was called “induction.” For instance,

Newton to the best of my knowledge consistently uses the terms induction (or

its Latin cognate “inductio”) in this way. Later, this sense of induction was

forgotten, and was replaced by its Humean sense of inference from particu-

lars to general truths or to unknown particulars. Moreover, I have argued that

the older “Newtonian” sense is but a quantitative counterpart to Aristotle™s

epagoge. (See Hintikka 1993.)

Thus the perspective on experimental inquiry that has been reached throws

an interesting light on different episodes in the actual history of science, and

helps us to analyze them.

17. Approximating a Function-In-Extension Mathematically

The distinction between induction in the sense of a search of a function-in-

extension and in the sense of a search of the mathematical function capturing

this extension is not entirely new. Jevons (1877) discusses it using such terms

as “empirical mathematical law” and “rational formula or function.” He calls

the controlled variable and the observed variable in an experiment “variable”

and “variant.” The ¬rst of these terms does not really refer to a function-in-

extension obtained from an experiment, but to mathematical approximations

to it. (The closest term for a function-in-extension in Jevons is probably the

result of what he calls a “collective experiment.” (See Jevons 1877, p. 447.))

Jevons™ discussion of experiments brings out one aspect of the interrelation

of the two different inductive tasks. In actual scienti¬c practice, a scientist does

not wait for the full function-in-extension before trying to ¬nd the correspond-

ing mathematical function, nor treat the partial observations of the function-in-

extension only non-mathematically, in contrast to the mathematical function

sought for in the identi¬catory half of the inductive task. Instead, a scientist

sets up mathematical approximations to the function-in-extension. When new

observations come in, such a mathematical approximation is adjusted so as to

take them into account, too. It is apparently sometimes thought”or at least

hoped”that these mathematical approximations converge in the limit to the

right mathematical law, Jevons™ “rational formula.”

Some such idea underlies the current theories of automated discovery. (See,

e.g., Langley et al. 1987, Zytkow 1997, or Arikawa and Furukawa 2000.) These

theories are of interest, but they do not embody a foolproof discovery method

in the sense of a way of approximating asymptotically the true mathematical

The a priori in Epistemology 141

law. As we might say, approximating mathematically a function-in-extension

does not necessarily yield in the limit the correct mathematical function. Hence

the entire research project of automated discovery is subject to important

conceptual limitations. Sub specie logicae, it ¬‚irts with a category mistake,

insofar as it involves an attempt to reach new mathematical knowledge by

utilizing ways of reaching new factual knowledge.

One cannot meaningfully speak of approximating a number of measure-

ment results by a mathematical function unless this function belongs to some

prescribed class of functions. Successive approximations can converge to the

right function only if the true mathematical law belongs to this class or is a limit

of functions belonging to it. The observed behavior of a function-in-extension

can yield clues as to what a suitable class of functions may be. For instance, if

the function-in-extension is obviously periodic, the relevant class might con-

sist of trigonometric functions rather than, say, polynomials. But, in general,

there is no absolutely certain method of selecting, a priori, a suitable class of

functions.

This illustrates the difference between the two different experimental tasks.

It also throws an interesting critical light on the entire project of automated

discovery in science.

18. The Inseparability of Factual and Conceptual Knowledge

Another interesting feature of the conceptual situation revealed by the insights

we have reached is a kind of inseparability of factual and conceptual knowl-

edge. For instance, it was seen that both are needed to answer a perfectly

ordinary wh-question. Any physicist who raises an experimental question in

effect hopes for an answer that codi¬es both factual and conceptual knowledge.

We have to treat this inseparability carefully, however. Quine is often said

to have pointed out the inseparability of factual and linguistic (and other con-

ceptual) knowledge. If so, he is vindicated by the inseparability that has been

identi¬ed in this chapter. But if so, he does not reach an adequate diagnosis of

what is unsolved in this inseparability. To put the matter in the simplest possi-

ble terms, Quine has denied that there is any hard-and-fast distinction between

the two kinds of knowledge to begin with. If this rejection is taken literally,

it becomes inappropriate to speak of either inseparability or separability, for

there is, according to such a view, no two things that could be separated from

each other in the ¬rst place. The inseparability that has been diagnosed here is

entanglement of the two kinds of knowledge in our epistemic practices, such

as raising and answering questions. What is even more directly obvious, the

two kinds of knowledge are distinguished in some cases by their logical form.

The kind of conceptual knowledge that has been dealt with in this chapter

is of the form “knowing who someone is” and its analogues for categories

other than persons. In contrast, it is not clear that “knowing that” involves any

conceptual knowledge. Thus, ironically, Quine™s denial of any hard-and-fast

Socratic Epistemology

142

difference between factual and conceptual truths has to be rejected for the

very purpose of doing justice to inseparability that has been claimed to be one

of Quine™s insights.

This is not the end of the ironies of the situation. What was just pointed out

is that factual and conceptual knowledge are, in the paradigm cases, different

grammatical and logical constructions in which the notion of knowledge occurs.

From this difference in construction it does not follow that a different concept

of knowledge itself is unsolved in the two cases. On the contrary, drawing the

very distinction between the two constructions with “knows” presupposes that

this concept is the same in both. The ultimate distinction is between the refer-

ence system and the identi¬cation system mentioned earlier. In the formalism

of epistemic logic, there is only one symbol “K” for knowledge, not a separate

symbol for analytic and synthetic (conceptual and factual) knowledge. Hence

Quine is, in a sense, verbally right in maintaining the inseparability of the two

kinds of knowledge. But being right in this sense does not make Quine or

anyone any wiser substantially. We can reach understanding and clarity only

by analyzing the precise nature of this inseparability.

What has been done here is to clarify the distinction between factual and

conceptual knowledge and then to argue that they are both nevertheless inter-

tangled in an interesting way in scienti¬c (and everyday) practice.

In all these different ways, the initially simple-looking logical insights

expounded here have been seen to have important general consequences.

The indispensability of mathematics in experimental science is an especially

instructive case in point. In brief, we have seen how a great deal of philosophy

of science can be distilled into a drop of logic.

References

Adam, Charles, and Tannery, Paul, 1897“1910, Oeuvres de Ren´ Descartes, published

e

under the auspices of the Ministry of Public Instruction, L. Cerf, Paris.