A Short History of Modern Philosophy: From Descartes to Wittgenstein, Second Edition (37 page)

The ground was prepared for Frege’s logic by certain discoveries in the foundations of mathematics, and in the techniques of formalisation. But the new logic arose also from Frege’s sense of the deep connection between logic and metaphysics, and of the philosophical errors that had been per-petuated in the name of logic. In particular Frege believed that the Kantian theory of mathematics—that all mathematical truth is synthetic
a priori
—was mistaken, and could be shown to be mistaken by the adoption of a logic free from the Aristotelian preconceptions that had mesmerised Kant. Frege offered to demonstrate that arithmetical truth is not synthetic but analytic, in the sense of following from laws of logic so basic that they cannot be denied without self-contradiction. Frege was a kind of ‘Platonist’; he believed in a realm of mathematical truth independent of the human capacity to gain knowledge of it. Nevertheless, as a result of his ideas, the science of mathematics was soon to be construed, not as the exploration of a realm of timeless entities, nor as a prime example of synthetic
a priori
knowledge, but as the projection into logical space of our own propensities towards coherent argument. What appears as an independent realm of mathematical entities or mathematical truth, is simply a shadowy representation of our own intellectual powers. The number one is no more an entity than is the average man, and the laws of mathematics no more truths about an independent world than the assertion that ‘all bachelors are unmarried’.

On this account (which Frege made possible but only partly accepted), if we have
a priori
knowledge of mathematical truth it is because we ourselves have
constructed
that truth. (This explanation of
a priori
knowledge is an old one, and was given by the mediaeval nominalists, who lacked the means to determine whether it could be applied to mathematics.) Clearly such an interpretation of mathematics has enormous philosophical consequences. Not only Platonism, but also the entire rationalist tradition, had relied in one way or another on mathematics as giving an immediately intelligible example of the ‘truths of reason’, and so demonstrating the superiority of reason over empirical investigation, in point of certainty, completeness and ultimate veracity. Since Kant had identified metaphysics with the realm of synthetic
a priori
knowledge, and given mathematics as the most persuasive example of this knowledge, the demonstration that mathematics is analytic would open the way to a wholly new and characteristically modern rejection of metaphysical argument.

Empiricists had attempted to reject the Kantian theory of mathematical truth, and these attempts were renewed by J.S.Mill, in his
System of Logic.
This work, as the most systematic nineteenth-century exposition of the tenets of British empiricism, deserves lengthier treatment than I can here accord to it. Not only did Mill present a sustained and, in many ways, convincing theory of the distinction between logic and science (between the logic of deduction and the logic of induction), thus laying the foundations for the modern philosophy of science; he also addressed himself to many of the patterns of thought that had given rise to prevailing metaphysical illusions. The fact that his own illusions escaped him in the course of this examination is more a cause for satisfaction than surprise, for it was the absurdity of Mill’s theory of mathematics that made clear to Frege the strange fact that mathematics can be completely known to someone who wholly misunderstands it.

For Mill our ideas of numbers are abstractions from experience. The number three is made familiar to us in the perception of threesomes, four in the perception of foursomes and so on. Moreover, mathematical truths themselves, such as 2 + 3 = 5, can be seen as reflecting very basic laws of nature, which have been observed to govern the aggregates to which they refer. Frege argued, in his
Foundations of Arithmetic
(1884), that neither this, nor any other empiricist account of the nature of numbers, could be accepted. Not only does Mill give us no clue as to how we understand the number zero; he also fixes the limit of our mathematical knowledge at the limit of our experience. But ‘who is actually prepared to assert the fact which, according to Mill, is contained in the definition of an eighteen-figure number has ever been observed, and who is prepared to deny that the symbol for such a number has, none the less, a sense?’ In asserting that the laws of arithmetic are inductive generalisations, Mill confuses the application of mathematics with mathematics itself. Mathematics is intelligible independently of its applications. Finally, Frege points out, ‘induction must base itself in the theory of probability, since it can never render a proposition more than probable. But how probability theory could possibly be developed without presupposing arithmetical laws is beyond comprehension.’

Frege was not the first philosopher to believe that the truths of arithmetic are analytic. Leibniz had attempted to prove the same. However, since Leibniz believed that all subject-predicate propositions are, at least from God’s point of view, analytic, this can hardly be called a distinctive theory of arithmetic. Moreover Frege was the first to develop a logic in which this theory could be stated and proved. The details of the theory lie beyond the scope of the present work, but one or two important steps in the argument need to be grasped as a prelude to understanding Frege’s philosophy as a whole.

If we ask the question ‘What are numbers?’ we find ourselves, Frege argues, at a loss for an answer. Are they objects? Are they properties? Are they abstractions? None of these suggestions seems satisfactory. When I say, ‘Socrates is one’, I do not attribute a property to Socrates, as I attribute a property in calling him wise. If Socrates is wise and Thales is wise then I conclude that Socrates and Thales are wise: they each possess the property singly, and so continue to possess it when described as a pair. But from ‘Socrates is one’ and ‘Thales is one’ we cannot conclude that ‘Socrates and Thales are one’.

If, on the other hand, numbers are objects, how do we identify them? We ought to be able to indicate which objects they are. This is where we fall into a philosophical vertigo—we seem unable to give a definition, ostensive or descriptive, of any actual number. Numbers are like objects in this: that they are the subject of identities. When we say that the number of planets is nine we are asserting that two names, ‘the number of the planets’ and ‘nine’, refer to one thing. But numbers are unlike objects in that reference to them is entirely dependent upon the identification of a concept to which they are attached. If I point to an army in the field and ask the question ‘How many?’, then the only sensible answer is: ‘How many of
what?
I may say 12,000, 50 or 2 depending on whether I am counting men, companies, or divisions. In other words, the answer is indeterminate until I have specified a concept according to which counting is to be carried out. Is a number then a property of a concept, a second-order property, as it were? This was the suggestion from which Frege began, and he took his inspiration from an area of logic the discovery of which was largely his—the logic of existence (or quantification, as it is now called).

Kant had argued, against the ontological argument, that existence is not a true predicate (or property), but he had failed to develop a logic that would accommodate this fact. Leibniz, who made certain advances in formal logic, recognised the differences between existential propositions (propositions of the form
‘x
exists’) and subject-predicate propositions, but again was unable to represent these differences in a systematic way. This deficiency in the traditional logic was far-reaching. It was what had erected the artificial barrier (as Frege considered it) between arithmetic (the logic of quantity) and logic (the logic of quality).

We know, independently of theory, that there is a coherent logic governing terms like ‘exists’. We know that the statement ‘Something exists which is not red’ entails the falsehood of the generalisation ‘Everything is red’. The traditional Aristotelian logic had no way of representing this relation. It can be represented, Frege argued, only when we realise that ‘exists’ and ‘all’ have a special logical character. They denote not properties of objects but, as it were, second-order properties of properties. To say that a red thing exists is to say of redness that it has an instance. And to say that all things are not red is to say that redness has no instances.

It proved possible on this basis to give a formal logic of existence and universality, and to vindicate Kant’s insight that existence is not a predicate and leads to fallacies when treated as one. New analytic truths now have to be recognised, which are not of subject-predicate form, and the laws of logic must be extended to cover them. It seems natural to suggest that this logic of existence and universal quantification should provide the basis for a general ‘logic of quantity’.

But what now of numbers? We speak of them as objects (which are the subjects of identity), and yet we do not allow them to be determinate independently of a concept to which they are attached. To resolve this seeming paradox, Frege proposed a general ‘criterion of identity’ for numbers. This criterion had to be provided contextually, he argued, since numerical expressions can be used to say true things only when attached to a concept which determines what is being counted. In other words, it is only in a given context that a number-term denotes anything specific. Suppose one could specify what makes an arithmetical statement of the form
‘a =
b’ true without invoking the concept of number. One will then have explained the use of the arithmetical concept of identity. One will also have provided what was later to be called an ‘implicit’ definition of number. An analogy might make this clear. Suppose you wish to know what is meant by the direction of a line. I can give a general definition of ‘same direction’ which does not invoke the idea of direction. (Lines have the same direction if and only if they are parallel.) I have then, in effect, defined direction. The direction of a line
ab
is given by the concept: lines which have the same direction as
ab.

In like manner, Frege derives his famous definition of number in terms of the concept ‘equinumerosity’, a concept which had been introduced into the discussion of the foundations of mathematics by Georg Cantor (1845-1918). The word ‘equinumerosity’ can be defined in purely logical terms, and denotes a property of a concept. Two concepts are equinumerous if the items falling under one of them can be placed in one-to-one correspondence with the items falling under the other. Frege shows that this idea of one-to-one correspondence can be explained without invoking that of number. He then defines the number of a concept
F
as the extension of the concept ‘equinumerous to
F
’. I have used the term ‘extension’ here, as Frege does—the usage goes back to the ‘Port-Royal’ logic discussed in chapter 4. The extension of a term or concept is the class of things to which the term applies. Hence the definition of number incorporates the generalisation of the idea, already invoked in the logic of existence, of the ‘instance’ of a concept. The definitions of the individual numbers can be derived from the general definition, Frege thought, by the use of the basic laws of logic. It suffices to define the first of the natural numbers—zero—and the relation of succession whereby the remaining numbers are determined.

Zero is the number which belongs to the concept ‘not identical with itself’. Frege chose this definition because, he argued, it follows from the laws of logic alone that the concept ‘not identical with itself has no extension. At every point in the argument Frege wished to proceed in that way, introducing no conceptions which could not be explained in logical terms. Following this method he was able to derive the definitions and laws of arithmetic so as to show, he thought, that all mathematical proofs were complex applications of logic, and all arithmetical statements were, if true, true by virtue of the meaning of the terms used to express them.

Frege’s achievement was astonishing. But it was marred by Russell’s discovery of a paradox, and the resolution of this paradox seemed to require a departure from purely logical ideas in a direction of the kinds of metaphysical assumption that Frege had wished to eliminate from the foundations of mathematics. Moreover, Kurt Godel in a famous theorem (1931) demonstrated that there are arithmetical truths which are unprovable in any logical system which can be proved to be self-consistent. Hence logic cannot, in principle, embrace the content of mathematics. In the light of these results it might seem that we should reject Frege’s ‘hypothesis’ (as he put it) of the analyticity of arithmetic, and reinstate some version of Kant’s theory, that mathematics is synthetic
a priori
and
sui generis.
However, Frege came so
near
to reducing arithmetic to logic, and Godel’s result is so puzzling, that the issue of the status of mathematical truth has in consequence become one of the most important modern philosophical problems. It seems impossible to abandon the direction in which Frege pointed us, and yet also impossible to proceed further along it. It is no mean achievement to have created an irresolvable philosophical problem from something which every child can understand.

Frege’s researches into the foundations of mathematics were to have profound philosophical consequences, not the least of which was the recognition that mathematical conceptions could be and should be used to give form to otherwise nebulous problems in the philosophy of logic and language. In the
Begriffsschrift
(1879) Frege set forward the first truly comprehensive system of formal logic. His purpose was to give clear philosophical background to the arguments of his earlier work on the foundations of arithmetic, and also to represent logic in a manner that freed it from the confusions imported into it by its use of ordinary language terms. He thereby invented the modern science of formal logic; and in the course of doing so he overthrew the theories of Aristotelian and post-Aristotelian logic that had impeded advance in the subject for two thousand years.