Abstract
There are two possible strategies for investigating questions on logic and mathematics. First, one can adopt the pattern recommended by the phenomenologists, which consists in looking for the actual essences of logic and mathematics in order to relate both fields. The second approach, adopted in this paper, starts with a historical review of the foundational standpoints. I will then try to extract on this base some insights on how logic and mathematics are mutually related. In particular, I am interested in the concept(s) of logic suggested by the development of the foundational debate. Yet one more preliminary remark is here in order. I will take into account the classical foundational positions: logicism, intuitionism, and formalism, as well as their later modifications and changes.1 Of course, this, so to speak, Foundational Trinity, does not exhaust the complete map of the foundations and philosophy of mathematics. For instance, I entirely neglect views wich might be exemplified by Lakatos, Dieudonné or Hersh, who, roughly speaking, radically deny that the study of logic and its relation to mathematics provides any interesting problem. I do not feel competent myself to judge this matter from the point of view of working mathematicians. On the other hand, philosophers are traditionally interested in the nature of logic and its links with other fields. Consequently, I regard my problem rather as a way of understanding what logic is, than whether it is important for mathematics or not.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
On the other hand, since I am basically interested in the place of logic in the founda- tional schemes, my account of logicism, intuitionism, and formalism will be rather selective.
Gottlob Frege. Die Grundlagen derArithmetik. Breslau: Koebner 1884, p.99e; page reference to Eng. tr. by J. L. Austin, Oxford: Basil Blackwell 1950.
Bertrand Russell, Introduction to Mathematical Philosophy,London: Allen&Unwin 1919, p. 194/195.
Rudolf Camap, “Die logizistische Grundlegung der Mathematik”, in: Erkenntnis II, H. 2/3, 1930, pp. 91–105; Eng. tr. in: Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics Selected Readings, Cambridge: Cambridge University Press 1983, pp. 41–51.
Jean van Heijenoort, “Logic as Calculus and Logic as Language“, in: Jean van Hei- jenoort, Selected Essays,Napoli, Bibliopolis 1985, pp. 11–16; originally in Boston Studies in the Philosophy of Science III (1967), pp. 440–446.
See Henryk Mehlberg, “The Present Situation in the Philosophy of Mathematics“, in Yehoshua Bar-Hillel et all., Logic and Language Studies Dedicated to Professor Rudolf Carnap on the Occasion of His Seventieth Birthday, Dordrecht: D. Reidel 1962, pp. 69–103.
Luitzen E. J. Brouwer, “Historical Backgrounds, Principles and Methods of Intuition- ism“, in Luitzen E. J. Brouwer, Collected Works,vol. 1, Philosophy and Foundations of Mathematics, Amsterdam: North- Holland/Elsevier 1975, p. 509/510; the quoted text is italicized in the original.
For a detailed treatment of this topic see Gregory H. Moore, “The Emergence of the First-Order Logic“, in: William Aspray and Philip Kitcher (eds.), History and Philosophy of Modern Mathematics (Minnesota Studies in the Philosophy of Science, vol. XI ), Minneapolis: University of Minnesota Press 1985, pp. 95–135.
Constructivist“ and ”formalist“ in the frameworks of classical logic.
See Franzisco Rodriguez-Consuegra, “Russell, Gödel, and Logicism, in: Johannes Czermak (ed.), Philosophy of Mathematics,Proceedings of the 15th International Wittgenstein Symposium, Part I, Wien: Hölder/Pichler/Tempsky, pp. 233–242.
For a defence of logicism by use of infinitary logic see J. Wolfgang Degen, “Two Formal Vindications of Logicism”, in: Johannes Czermak (ed.), ibid.,pp.243–250.
A. Mostowski, Thirty Years of Foundational Studies Lectures on the Development of Mathematical Logic and the Study of the Foundations of Mathematics in 1930–1964, Helsinki: Societas Philosophica Fennica 1964, p. 9.
Jon Barwise, “Model-Theoretic Logics, Background and Aims”, in: Jon Barwise and Solomon Feferman (eds.), Model-Theoretic Logics,Berlin: Springer Verlag 1985, p. 4/5.
Jon Barwise, ibid.,p. 6.
ibid.,p. 23.
On reverse mathematics see Solomon Feferman, “What Rests on What? The Proof-Theoretic Analysis of Mathematics”, in: Johannes Czermak, ibid., pp. 147–171, Roman Murawski, “On the Philosophical Meaning of Reverse Mathematics”, in: Johannes Czermak, ibid., pp. 173–184, Roman Murawski, “Hilbert’s Program: Incompleteness Theorem versus Partial Realizations”, in: Jan Wolenski (ed.),Philosophical Logic in Poland, Dordrecht: Kluwer Academic Publishers 1994, pp. 103–127.
Here and in the next sections, I follow, with some changes concerning the scope of logic, my paper “In Defence of the First-Order Thesis”, in: Peter Kolsr and Vladimir Svoboda (eds.), Logica ‘93, Proceedings of the 7th Conference, Praha: Filosofia 1994, pp. 1–11.
See Stewart Shapiro, Foundations without Foundationalism A Case for Second-Order Logic, Oxford: Clarendon Press 1991, pp. 35–40.
Gilbert Ryle, Dilemmas,Cambridge: Cambridge University Press 1954, p. 115.
Of course, Cn axiomatized in this manner generates classical first-order logic. However, it is possible to define consequence operations related to alternative systems. Taking classical logic as the base for my approach to logic, I do not decide which logic, the classical or a non-classical one, is “correct”.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Wolénski, J. (1995). Logic and Mathematics. In: Depauli-Schimanovich, W., Köhler, E., Stadler, F. (eds) The Foundational Debate. Vienna Circle Institute Yearbook [1995], vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3327-4_15
Download citation
DOI: https://doi.org/10.1007/978-94-017-3327-4_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4617-8
Online ISBN: 978-94-017-3327-4
eBook Packages: Springer Book Archive