Version: v1, Published online: 1998
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During the first four decades of the twentieth century, each of the post-Kantian programmes outlined above came under attack. Frege’s logicism was challenged by Russell’s paradox. Russell’s logicism encountered difficulties concerning its use of certain existence axioms (namely his axioms of reducibility and infinity) which did not appear to be laws of logic. Both were challenged by Gödel’s incompleteness theorems, as was Hilbert’s formalist programme. Finally, the intuitionists were criticized both philosophically, where their idealism was called into question, and mathematically, where their ability to support a significant body of mathematics remained in doubt. Various modifications have been proposed.
Modifications of logicism. On the mathematical side, a chastened successor to logicism can perhaps be seen in the model-theoretic work of Abraham Robinson and his followers. They are interested in determining the mathematical content latent in purely ‘logical’ features of various mathematical structures or the extent to which genuinely mathematical problems concerning these structures can be solved by purely logical (that is, model-theoretic) means. They have been particularly successful in their treatment of various algebraic structures (see Macintyre 1977; Robinson 1979; Hodges 1993).
Philosophically, too, there have been attempts to renew logicism. It re-emerged in the 1930s and 1940s as the favoured philosophy of mathematics of the logical empiricists (see Carnap 1931; Hahn 1933). They did not, however, develop a logicism of their own in the way that Dedekind, Frege and Russell did, but, rather, simply appropriated the technical work of Russell and Whitehead (modulo the usual reservations concerning the axioms of infinity and reducibility) and attempted to embed it in an overall empiricist epistemology.
This empiricist turn was a novel development in the history of logicism and represented a serious departure from both the original logicism of Leibniz (§10) and the more recent logicism of Frege (and Dedekind). It was less at odds with Russell’s logicism which saw mathematics and the empirical sciences as both making use of an essentially inductive method (the so-called ‘regressive’ method – see Russell 1906, 1907).
Like all empiricists, the logical empiricists struggled with the Kantian problem of how to account for the apparent necessity of mathematics while at the same time being able to explain its cognitive richness. Their strategy was to empty mathematics of all non-analytic content while, at the same time, arguing that analytic truth and inference can be substantial and non-self-evident.
Their ideas came under heavy attack by W.V. Quine, who challenged their pivotal distinction between analytic and synthetic truths (1951, 1954). He argued that the basic unit of knowledge – the basic item of our thought that is tested against experience – is science as a whole and that this depends upon empirical evidence for its justification. Mathematics and logic are used to relate empirical evidence to the rest of science and, so, are inextricably interwoven into the whole fabric of science. They are thus part of the total body of science that is tested against experience and there is no clean way of dividing between truths of meaning (analytic truths) and truths of fact (synthetic truths).
Within a relatively brief period of time, Quine’s argument became a major influence in the philosophy of mathematics and the logicism of the logical empiricists was largely abandoned. Newer conceptions of logicism have, however, continued to appear from time to time. For example, Putnam (1967) addressed the difficult (for a logicist) question of existence claims, arguing that such statements are to be seen as asserting the possible (as opposed to the actual) existence of structures. They are therefore, at bottom, logical claims, and can be established by logical (or metalogical) means. Hodes (1984) takes a somewhat different approach, arguing that arithmetic claims can be translated into a second-order logic in which the second-order variables range over functions and concepts (as opposed to objects). In this way, commitment to sets and other specifically mathematical objects can be eliminated and, this done, arithmetic can be considered a part of logic.
Field (1980, 1984) also presents a kind of logicist view – namely, that mathematical knowledge is (at least largely) logical knowledge. Mathematical knowledge is defined as that knowledge which separates a person who knows a lot of mathematics from a person who knows only a little, and it is then argued that what separates these two kinds of knowers is mainly logical knowledge; that is, knowledge of what follows from what.
Modifications of Hilbert’s programme. Hilbert’s programme too has its latter-day adherents. For the most part, these have adopted one of two basic stances: that of extending the methods available for proving the consistency of classical ideal mathematics; or that of diminishing the scope and strength of classical ideal mathematics so that its consistency (or the consistency of important parts of it) can more nearly be proved by the kinds of elementary means that Hilbert originally envisaged.
Some in the first group (for example, Gentzen, Ackermann and Gödel) have argued that there are types of evidence that exceed finitary evidence in strength but which have the same basic epistemic virtues as it. Others (for example, Kreisel 1958; Feferman 1988; Sieg 1988) argue for a change in our conception of what a consistency proof ought to do. They maintain that its essential obligation is to realize an epistemic gain, and that finitary methods are not the only epistemically gainful methods for proving consistency.
Those in the second group – the so-called ‘reverse mathematics’ school of Friedman, Simpson and others – try to isolate the mathematical ‘cores’ of the various areas of classical mathematics and prove the consistency of these ‘reduced’ theories by finitary or related means. So far, significant success has been achieved along these lines. (See Hilbert’s programme and formalism §4.)
Modifications of intuitionism. Regarding intuitionism, Heyting’s work in the 1930s to formalize intuitionism and to identify its logic has led to a vigorous programme of logical and mathematical investigation (see Heyting 1956; Troelstra 1973, 1977; Beeson 1985 for descriptions of some of this work). In addition to Heyting and his students, Errett Bishop and his followers have extended a constructivist approach to areas of classical mathematics to which such an approach had previously not been extended (see Bridges 1987 for a survey).
On the more philosophical side, the most important development is the construction by Michael Dummett and his anti-realist followers of a defence of intuitionism based upon – in their view – the best answer to the question ‘What is the logic of mathematics?’. Their answer is based upon what they take to be a proper theory of meaning – a theory which, following certain ideas set out by Wittgenstein in his Philosophical Investigations, equates the meaning of an expression with its canonical use in the practice to which it belongs. They then identify the canonical use of an expression in mathematics with the role it plays in the central activity of proof, and from this they infer an intuitionist treatment of the logical operators (Dummett 1973, 1977).
Detlefsen, Michael. Modifications. Mathematics, foundations of, 1998, doi:10.4324/9780415249126-Y089-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/overview/mathematics-foundations-of/v-1/sections/modifications.
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