Modern Logic: From Frege to Gödel: Brouwer and Intuitionism
MODERN LOGIC: FROM FREGE TO GÖDEL: BROUWER AND INTUITIONISM
The intuitionist conception of mathematics was developed by the Dutch mathematician Luitzen Egbertus Jan Brouwer (1881–1966). According to Brouwer mathematics is not a system of formulas and rules but a fundamental form of human activity, an activity that has its basis in our ability to abstract a conception of "twoness" from successive phases of human experience and to see how this operation may be indefinitely repeated to generate the infinitely proceeding sequence of the natural numbers. In the system of mathematics based on this primordial intuition, language serves merely as an aid to memory and communication and cannot of itself create a new mathematical system; our words and formulas have significance only insofar as they are backed by an essentially languageless activity of the mind. In particular, the wording of a theorem is meaningful only if it indicates the mental construction of some mathematical entity or shows the impossibility of the entity in question. Brouwer's conception of proof as essentially mental is useful as a corrective to a narrow formalist account that would construe proof as proof in a given formal system, although his psychologism is philosophically questionable—Wittgenstein's work has rendered more than doubtful the thesis that language is only an incidental accompaniment to thought, required solely for purposes of memory and communication. What is important in intuitionism is not so much its psychologistic features as its emphasis on constructibility and the form of mathematics which its criterion of meaningfulness determines.
Implicit in classical mathematics is the notion that to know the meaning of a statement it is sufficient to know the conditions under which the statement is true or false, even though these conditions may be such that we could never be in a position to determine whether or not they held. The possibility of a gap between what can be meaningfully stated and what can be recognized either as true or as false is not admitted by the intuitionists. On their theory we can know the meaning of a statement only when we can recognize a proof of it; indeed, to understand a statement simply is to know what constitutes a proof or verification of that statement.
This emphasis on verification leads to an explanation of the logical constants and of a number of mathematical concepts that results in the rejection or reinterpretation of large parts of classical mathematics. Thus, whereas in classical mathematics the truth-table definition is adequate to giving the meaning of the constant "∨" ("or"), for the intuitionist we can explain the meaning of a statement of the form "A ∨ B " only by indicating under what conditions we should be warranted in asserting such a statement. These conditions are that we should be warranted in asserting A or that we should be warranted in asserting B, and it is clear that neither condition may hold, even when A is the negation of B.
Assume, for instance, that A is an existentially quantified statement, ∃xP (x ), with the quantifier ranging over the natural numbers. To suppose that this holds is to suppose that we can actually construct a number with the required property. On the other hand, what is it to suppose that ∃xP (x ) is false? It cannot mean that a case-by-case examination of the numbers will provide a refutation of the statement, since a case-by-case investigation of an infinite totality is not a real possibility—it is a picture to which the classical mathematician is wedded by a mistaken analogy with finite totalities. But if ∼∃xP (x ) is to have a meaning which we can grasp, it can mean only that there is a contradiction in the idea of a number's having the property P. Given this explanation of the sense of the proposition and its negation, we are obliged to abandon Aristotelian logic as no longer trustworthy in this context, for asserting the disjunction ∃xP (x ) ∨ ∼∃xP (x ) is tantamount to asserting that we either are in a position to construct a suitable number or can show the impossibility of such a construction. We are not entitled to assert a priori that at least one of these possibilities must obtain, but to do so would simply be to commit ourselves to the unfounded belief that all mathematical problems are solvable.
This insistence on the identification of existence with constructibility can be traced back to Leopold Kronecker (1823–1891), and a precise formulation of principles of intuitionist logic was carried out in 1930 by a pupil of Brouwer's, Arend Heyting (1898–1980). Several branches of mathematics have been redeveloped from the intuitionist standpoint, but the reconstructions are often complicated, and in some cases, particularly where set-theoretic notions are involved, there has been a question of outright rejection, rather than reconstruction, of classical mathematics. Thus, impredicative definitions, hierarchies of transfinite numbers, and nonconstructive postulates such as the axiom of choice (and hence the well-ordering theorem), while important classically, are rejected in toto by the intuitionists, a rejection which has led many mathematicians to discount the claims of intuitionism without giving sufficient attention to the arguments, admittedly often obscurely expressed, on which they are based.
Bede Rundle (1967)