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Logical positivism

DOI
10.4324/9780415249126-Q061-1
DOI: 10.4324/9780415249126-Q061-1
Version: v1,  Published online: 1998
Retrieved March 29, 2024, from https://www.rep.routledge.com/articles/thematic/logical-positivism/v-1

3. Logic and the foundations of mathematics

Whereas the positivists appealed to Poincaré’s concept of convention (as realized, so they thought, in relativistic physics) to give a new answer to Kant’s question concerning the possibility of pure natural science, they appealed to modern developments in logic and the foundations of mathematics to give a new answer to Kant’s question concerning the possibility of pure mathematics. There were in fact two distinguishable sets of developments here. The formal point of view, typified by David Hilbert’s logically rigorous axiomatization of geometry, freed geometry from any reference at all to intuitively spatial forms and instead portrayed its subject matter as consisting of any things whatever that satisfy the relevant axioms (see Geometry, philosophical issues in §2; Hilbert’s programme and formalism). Geometry is rigorously and a priori true, not because it reflects the structure of an intuitively given space, but rather because it ‘implicitly defines’ its subject matter via purely logical – but otherwise entirely undetermined – formulas. Mathematical truth, on this view, is identified with logical consistency. The ‘logicism’ of Gottlob Frege and Bertrand Russell, by contrast, aimed to construct particular mathematical disciplines (especially arithmetic) within an all-embracing system of logic. On this view mathematical disciplines (like arithmetic) indeed have a definite subject matter about which they express truths: namely, the subject matter of logic itself (propositions, classes, and so on). As thus purely logical, however, such pure mathematical disciplines express merely analytic truths and are not synthetic a priori (see Logicism).

Hilbert’s formal point of view was pursued especially by Schlick, who in a sense made the notion of implicit definition, together with the associated distinction between undetermined form and determinate (given) content, the centrepiece of his philosophy. The logicist point of view, by contrast, was pursued especially by Carnap, who studied with Frege and then was decisively influenced by Russell. Indeed, Carnap was inspired by Russell’s conception of ‘logic as the essence of philosophy’ to reconceive philosophy itself on the model of the logicist construction of arithmetic. He began, in Der logische Aufbau der Welt (1928), by developing a ‘rational reconstruction’ of empirical knowledge – an epistemology – within the logical framework of Russell and Whitehead’s Principia Mathematica (1910–13). By defining or ‘constituting’ all concepts of empirical science within this logic from a basis of subjective ‘elementary experiences’, Carnap’s reconstruction was to show, among other things, that the dichotomy between empirical truth and analytic/definitional truth is indeed exhaustive.

Yet the logic of Principia Mathematica was afflicted with serious technical difficulties: the need for special existential axioms such as the axioms of infinity and choice. Partly in response to such difficulties, Ludwig Wittgenstein asserted in his Tractatus Logico-Philosophicus (1922) that logic has no subject matter after all: the propositions of logic are entirely tautological or empty of content (see Wittgenstein, L.J.J. §§3–7). Carnap eagerly embraced this idea, but he also attempted to adapt it to the new, post-Principia technical situation – which involved the articulation of the ‘intuitionist’ or ‘constructivist’ point of view by L.E.J. Brouwer and the development of meta-mathematics by Hilbert and Kurt Gödel (see Intuitionism §§2–3; Logic in the early 20th century §§6–9; Mathematics, foundations of). In Logische Syntax der Sprache (1934) Carnap formulated his mature theory of formal languages and put forward his famous ‘Principle of Tolerance’ – according to which logic has no business at all looking for true or ‘correct’ principles. The task of logic is rather to investigate the structure of any and all formal languages – ‘the boundless ocean of unlimited possibilities’ – so as to map out and explore their infinitely diverse logical structures. Indeed, the construction and logical investigation of such formal languages became, for Carnap, the new task of philosophy. The concept of analyticity thereby took on an even more important role. For this concept characterizes logical as opposed to empirical investigation and thus now expresses the distinctive character of philosophy itself.

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Citing this article:
Friedman, Michael. Logic and the foundations of mathematics. Logical positivism, 1998, doi:10.4324/9780415249126-Q061-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/logical-positivism/v-1/sections/logic-and-the-foundations-of-mathematics.
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