A Short History of Modern Philosophy: From Descartes to Wittgenstein, Second Edition

by Unknown Author

  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 defini
tion 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.

  There was a particular consequence of this overthrow which Frege did not at first foresee. The old logic had taken its cue from the grammar of ordinary language. It was this that made it so difficult to represent the difference between ‘Socrates exists’ and ‘Socrates is alive’. The difference is in fact so radical that we are forced to conclude that grammatical form in ordinary language is no guide to logical behaviour. To put it in Russell’s way, the true logical form of the sentence ‘Socrates exists’ is not reflected in its grammar. How then should we represent this sentence? The natural answer is to seek for a system of symbols that would allow expression only to the true ‘logical form’ of any sentence. This intrusion of mathematical method into the foundations of logic was the first of many. Since logic itself governs much of philosophical argument, the process can be continued further; eventually it resulted in the almost entirely mathematised philosophies of atomism and positivism which I shall mention in the final chapter.

  There are more specific ways in which Frege’s adoption and extension of mathematical ideas changed the nature of philosophy. This can be seen in Frege’s theory of the nature of language. It was clear to Frege, as it had been to Leibniz, that statements of identity are different in form from statements which predicate a property of an object. The ‘is’ of identity and the ‘is’ of predication are logically distinct. If I say ‘Venus is the Morning Star’ then I make a statement of identity. The statement remains true (or, if false, false), when the names are reversed: the Morning Star is as much Venus as Venus is the Morning Star. In the sentence ‘Socrates is wise’ the terms cannot be reversed in the same way. The whole sense of the sentence depends upon my ascribing a different role to the subject term ‘Socrates’ and the predicate term ‘wise’.

  Now the distinction between subject and predicate is basic to thought. A creature who could not understand it, who spoke only of identities, would know nothing of his world; he would know only the arbitrary determinations of his own usage, whereby he is able to substitute one name for another. But he would know nothing about the things that he thereby names. It behoves us, therefore, to try to understand the relation between subject and predicate—in so far as anything so basic will yield itself to logical investigation.

  Frege’s analysis of this relation is contained in a series of articles among which the most important is ‘On Sense and Reference’. Frege there advances various theses, some of which had already proved important in describing the nature of arithmetic. Two theses of particular interest are these: first, that it is only in the context of a whole sentence that a word has a definite meaning; secondly, that the meaning of any sentence must be derivable from the meanings of its parts. These seem to be, but are not, contradictory. The first (an application of which is found in Frege’s contextual definition of number) says that the meaning of a word does not belong to it in isolation, but consists in its potentiality to contribute to a completed ‘thought’. It is because sentences can express thoughts that the words which compose them have a meaning. The second thesis states that the meaning of the whole sentence (or of any other composite linguistic entity) must be wholly determined by the various ‘potentialities’ belonging to its parts. Thus the word ‘man’ has the meaning it has because we use it to talk about men. Equally, the sentences with which we talk about men derive their meaning in part from that of ‘man’. This mutual dependence of part on whole and whole on part is characteristic of language. As linguists have begun to realise, it is what makes language learnable. If the meaning of the sentence is determined by the meaning of its parts, then, knowing only a finite vocabulary, I may yet understand indefinitely many sentences. My language-use is automatically ‘creative’, and gives me the capacity for unlimited thought.

  How then do we proceed to describe the component parts of a subject-predicate sentence? Consider the sentence ‘Socrates is wise’. Frege argues that, for the purpose of clearer representation, we can assume this to be composed
of two parts, a name and a predicate. Names may seem to be more intelligible than predicates: we understand them because they stand for objects, and if we know which objects they stand for we seem already to know what they mean. But, Frege argues, matters are more complicated than that. Consider the sentence ‘Hesperus is Phosphorus’. This uses two names, only one in fact the name of the Evening Star. Surely I could understand it without knowing it to be true? But if to understand ‘Hesperus’ is to know to which object it refers, then I ought to know that the sentence is true just as soon as I understand it. But I do not. Frege took this example as proving that there is a general distinction in language between that which we understand (the sense of a term) and that which a term refers to or ‘picks out’ (the reference of the term). The sense of a term directs us towards the reference: but it is not identical with it.

  In the case of a name the sense is something like a complex description—‘the planet which...’ or ‘the man who...’. The reference, on the other hand, is an object. This may seem intuitively acceptable— although in fact it is now widely devoted. But what about predicates? And what about the sentence taken as a whole?

  In discussing Frege’s theory of arithmetic I wrote loosely of concepts, properties and predicates, wishing to postpone the question of the interpretation of these terms. But now it is necessary to be more precise. A predicate has as its reference a particular concept: in understanding the predicate ‘is wise’ I am ‘led to’ the concept of wisdom, by its sense or meaning. What then can we say, from the philosophical point of view, about the nature of concepts? Frege was clear about one thing: concepts are public, and belong as much to the publicly recognisable aspect of language as do the words which express them. The ‘senses’ of predicates are therefore equally public. Otherwise the meaning of words could not be taught, and language would cease to be a form of communication. Senses are to be distinguished from private associations, from images and from every other merely ‘inner’ episode. They are determined by rules of usage which are available to every speaker.