Version: v1, Published online: 1998
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5. The setbacks; a brief evaluation
The first of the setbacks for the logicist view was the discovery of a contradiction in Frege’s logic (see Paradoxes of set and property §2), which arises from the assumption, already noted, that every concept has an extension – a very strong form of what has come, in the theory of sets, to be called a ‘comprehension principle’. Frege himself was eventually led to abandon his logicist view of arithmetic, and to suggest instead that arithmetic is not only – as Kant thought – synthetic a priori, but is actually to be grounded in the ‘geometric knowledge-source’ ([1924–5] 1979: 276–80). Others, however – notably Russell ( 1937: introduction), Ramsey (1925) and Carnap (1931) – felt no need to abandon the logicist thesis, but reconciled themselves to a modified view of logic, designed to circumvent the contradiction and still allow the essential modes of reasoning. Most celebrated among logicians as such a modified logic is Russell’s ‘theory of types’; but the axiomatic theory of sets, which has proved most serviceable as a medium for mathematics, could perfectly well also claim to be a form of ‘logic’ (see the characterization quoted from Zermelo, the pioneer of axiomatic set theory, in §4 above).
The second setback is more profound. Frege’s own invention of a way of rendering logic purely formal made it possible to apply modes of mathematical reasoning to the formal linguistic structures of logical systems themselves. The undertaking of such application was a fundamental contribution of Hilbert. This process has further revolutionized our way of looking at logic itself. Among the most basic results of this examination are two theorems of Gödel (1930, 1931), one of which tells us that ‘elementary logic’ – or first-order logic – can be formulated so that every proposition that deserves to be considered logically true can be derived by ‘purely formal’, or ‘mechanical’, means (for example, these propositions can be generated by a computer program); whereas the other tells us that no such thing can be achieved for a system – whether a higher-order logic (of which the theory of types is an example), or a first-order axiomatized theory (of which set theory is an example) – that is strong enough to generate the theorems of elementary arithmetic.
With Gödel’s theorems there should be mentioned a discovery that preceded them by several years: the Löwenheim–Skolem theorem (see Löwenheim 1915 and, for clearer and more adequate proofs, Skolem 1920, 1923). This theorem implies that any theory axiomatized in first-order logic that has an infinite model (domain that satisfies the axioms) has a countably infinite model (one whose domain can be placed in one-to-one correspondence with the natural numbers). In a very remarkable paper, Skolem (1923) invoked this theorem (and other considerations) to cast grave doubt on the view that set theory could be ‘a satisfactory ultimate foundation for mathematics’. Indeed (restricting attention here to the implications of the Löwenheim– Skolem theorem), it is, for example, a fundamental proposition of classical analysis that the set of all the real numbers is uncountably infinite. The existence of a model for the allegedly foundational theory containing only countably many elements seems, then, prima facie to show that the foundational attempt has failed: that, in particular, the assertion within the theory that a certain set is uncountable has not captured what ought to be the genuine meaning of the term ‘uncountable’. The issue is both deep and subtle; here, it must suffice to remark that both Gödel’s theorems and the Löwenheim– Skolem theorem lead to the notion of nonstandard models – the theory of which has since become an important part of logic (see Löwenheim–Skolem theorems and nonstandard models).
These results may be summed up by saying that there is not, and in principle cannot be, a ‘completely formalized’ logic within which all mathematically statable questions are correctly answerable. Whether one chooses to call this a refutation of logicism, or an unexpected result about the ‘uncompletable’ nature of logic itself, the result is unquestionably a deep and interesting one simultaneously concerning mathematics and logic; and, for its discovery, a great debt is owed to those who advanced the thesis of logicism.
Stein, Howard. The setbacks; a brief evaluation. Logicism, 1998, doi:10.4324/9780415249126-Y061-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/logicism/v-1/sections/the-setbacks-a-brief-evaluation.
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