sity of Chicago Press, Chicago.

Robinson, Richard, 1971, “Begging the Question 1971,” Analysis, vol. 31, no. 4,

pp. 113“117.

Robinson, Richard, 1953, Plato™s Earlier Dialectic, Clarendon Press, Oxford.

Ryle, Gilbert, 1971, Collected Papers I-II, Hutchinson, London.

Sentas, Gerasimos Xenophon, 1979, Socrates, Routledge and Kegan Paul, London.

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5

The Place of the a priori in Epistemology

1. The Unreasonable Effectiveness of the a priori in Epistemology

Aristotle said that philosophy begins with the experience of wonder. But dif-

ferent phenomena are experienced as wondrous by different thinkers and to

a different degree. The wonder that is the theme of this chapter seems to have

struck some non-philosophers more keenly than most professional philoso-

phers. Indeed, the most vivid formulation of the problem is probably in the title

of Eugene Wigner™s 1960 paper, “The Unreasonable Effectiveness of Mathe-

matics in the Natural Sciences.” Historically speaking, Wigner™s amazement is

nevertheless little more than another form of the same reaction to the success

of mathematics in science as early modern scientists™ sense that the “book

of the universe is written in mathematical symbols.” For a philosopher, this

question is in any case a special case of the general problem of the role of our

a priori knowledge, mostly codi¬ed in the truths of logic and mathematics, in

the structure of our empirical knowledge.

This overall philosophical problem assumes different forms in different

contexts. Wigner™s puzzle has been taken up in the same form by relatively

few philosophers, most extensively by Mark Steiner in his 1998 book, The

Applicability of Mathematics as a Philosophical Problem. This problem can be

given technical turns”for instance, by asking whether scienti¬c laws can be

expected to be computable (recursive), and even more speci¬cally by asking

which known laws are in fact computable. (See, e.g., Pour-El and Richards

1989.)

More commonly, philosophers have adopted a narrower and more skepti-

cal stance. They have raised the question as to whether a priori theories, in

the ¬rst place mathematical theories, are really indispensable in science. The

extensive discussion of this question has been prompted largely by what is usu-

ally referred to as Quine™s argument for the indispensability of mathematics in

science. (See e.g., Quine 1981(b); Colyvan 2001.) In the light of hindsight”or

perhaps in the light of a more general perspective”this has turned out to be

107

Socratic Epistemology

108

an inauspicious beginning in more than one respect. Quine™s indispensability

claim has deep roots in the idiosyncrasies of his overall philosophy. Quine

had to resort to the kind of argument he used because of the poverty of his

epistemological theory. In a nutshell, this theory can be explained in terms of

Quine™s famous “world wide web” metaphor. The structure of our total sys-

tem of knowledge is like a huge net that is held in place at its boundaries by

observations (sense-registrations). Its different nodes are connected with each

other by relations of logical implication. But what is the place of mathematics

in Quine™s picture? There does not seem to be much left for mathematics to do

in such a scheme. On the face of things, the Quinean picture is not so far from

Ernst Mach™s idea of science as an economic description of our experiences.

And Mach, too, leaves next to nothing for serious logic and mathematics to do.

For him, all deductive reasoning, explicitly including mathematical reasoning,

is tautological. Similarly, Quine™s analogy does not seem to leave any place for

mathematical knowledge.

What lies behind this predicament is Quine™s rejection of the analytic“

synthetic distinction. According to him, the network itself is our creation. We

can in principle change the inferential links between its nodes. Mathematics is

simply the best possible way we have of organizing those links”and thereby

organizing the entire texture of our empirical knowledge”in the best possible

way. The indispensability of some mathematics or other is thus simply part

and parcel of this overall epistemology and philosophy of science. We need

logic and mathematics in order to connect the nodes of the net with each

other. In other, more commonplace words we need mathematics in science for

the purpose of deductive systematization. This is not a conclusion of Quine™s

theory; it is its presupposition. It is in this sense that philosophers have been

able to speak of a “failure to explicate what is meant by ˜indispensable™ in

Quine™s argument” (Colyvan 2001, p. 76.) It can only be evaluated as a part of

Quine™s overall epistemology.

It follows that Quine cannot offer any explanation as to why this or that

particular mathematical theory is useful or perhaps indispensable in science.

Quine offer no solutions to Wigner™s problem.

2. The Nature of Deductive Systematization

Even though Wigner™s and Quine™s puzzles offer a handy introduction to the

problem of this chapter, the way of solving the problem is not to pursue their

lines of thought further. The reason is that the solution of the problem of

the role of the a priori in epistemology has to be looked for elsewhere. In

order to clarify the problem situation, a brief discussion of Quine™s approach

is nevertheless in order. This discussion will focus mostly on ideas that have not

¬gured prominently in the earlier literature on the indispensability problem.

First, Quine™s web analogy does not bear critical scrutiny. It is part and par-

cel of the analogy that the consequence relations holding the net together are

The a priori in Epistemology 109

syntactical and recursive or at least recursively enumerable (axiomatizable).

Otherwise, human reasoners do not have any general method of deciding

which node is connected with which other one. The resulting overall view is so

unrealistic that it is to my mind surprising that it is still being taken seriously.

For one thing, it is totally alien to all serious logical semantics. After the work

of Godel and Tarski, it simply is impossible to maintain that the meaning of

¨

propositions is constituted via their inferential relations to other propositions.

Meaning is a matter of semantics, and the overwhelming impact of the work

that began with Tarski and Godel is to make it clear that the semantical rela-

¨

tionships cannot be reduced to syntactical ones. Furthermore, any associative

relationship must be grounded in syntactical relations as providing the clues

that prompt those inferences or associations, and a fortiori fails to serve as a

ground of meaning. Instead, its “lateral” relations to other propositions, the

semantical properties of a proposition, are determined by its truth conditions

or its other “vertical” relations to reality. Small wonder, therefore, that Quine™s

ideas in language theory have for a long time been totally alienated from all

real work in logical or linguistic semantics.

We can thus see that Quine™s entire “world wide web” picture is predicated

on his syntactical and inferential viewpoint. This viewpoint is inadequate, how-

ever, and once its inadequacy is realized, the motivation for Quinean holism

evaporates. The truth-conditions of a given sentence do not refer to its deduc-

tive relations to other sentences. One can do model theory without ever men-

tioning rules of inference, but in the last analysis, one cannot understand, let

alone justify, one™s inference rules except model-theoretically. The same holds,

of course, for scienti¬c theories employing more powerful mathematics than

elementary arithmetic. In brief, the Quinean argument for the indispensability

of mathematics in science is predicated on an antiquated pre-Godelian dogma.

¨

It may be that the function of logical words in guiding logical inferences is of

great practical interest to philosophers and scientists who are applying logic.

But philosophically, this function is secondary in relation to the role of logical

words in determining the meaning, including the truth-conditions, of the sen-

tences in which they occur. To think that this determination takes place via

inferential relationships is to fail to appreciate the insights reached in logical

theory in the last eighty or so years.

This point becomes evident when we scratch the surface of any particular

problem about the relationship of inference rules and meaning. For instance,

intuitionists do not mean something non-classical by their statements because

they are using different inference rules. They have to use different inference

rules because their statements are not only calculated to assert the truth of a

certain mathematical statement S. They assert that the truth-making functions

(witness functions, Skolem functions) of S are known. (See Hintikka 1996(a)

and 2002(b).)

In any case, the last nail in the cof¬n of the idea that inferential relationships

can serve to systematize a theory is contributed by independence-friendly (IF)

Socratic Epistemology

110

¬rst-order logic. This logic has a much better claim to be our true unrestricted

basic logic than the so-called ordinary ¬rst-order logic (predicate calculus),