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PHILOSOPHY OF MATHEMATICS IN THE 20TH CENTURY. MAIN TRENDS AND DOCTRINES1

The aim of the paper is to present the main trends and tendencies in the phi- losophy of mathematics in the 20th century. To make the analysis more clear we distinguish three periods in the development of the philosophy of mathematics in this century: (1) the first thirty years when three classical doctrines: logicism, intuitionism and formalism were formulated, (2) the period from 1931 till the end of the fifties – period of stagnation, and (3) from the beginning of the sixties till today when new tendencies putting stress on the knowing subject and the research practice of mathematicians arose.

1. The First 30 Years – the Rise of Classical Conceptions

Philosophy of mathematics and the foundations of mathematics entered the 20th century in the atmosphere of a crisis.2 It was connected mainly with the problem of the status of abstract objects. In set theory created by Georg Cantor in the last quarter of the 19th century antinomies were discovered. Some of them were al- ready known to Cantor – e.g., the antinomy of the set of all sets or the antinomy of the set of all ordinals (the latter was independently discovered by C. Burali- Forti). He was able to eliminate them by distinguishing between classes and sets. New difficulties appeared when B. Russell discovered the antinomy of irreflex- ive classes (called today Russell’s antinomy) in the system of logic of G. Frege

1Originally published in: J. Malinowski and A. Pietruszczak (eds.), Essays in Logic and On- tology, pp. 331–347 (2006). Amsterdam–New York: Editions Rodopi.
c Editions Rodopi B.V., Amsterdam – New York, NY 2006. Reprinted with kind permission of the Editions Rodopi. 2It is usually called the second crisis and distinguished from the first one which took place in the antiquity and was connected with the discovery of incommensurables by Pythagoreans.

In: Roman Murawski, Essays in the Philosophy and History of Logic and Mathematics (Pozna´n Studies in the Philosophy of the Sciences and the Humanities, vol. 98), pp. 59–73. Amsterdam/New York, NY: Rodopi, 2010. 60 Part I: Philosophy of Mathematics

(to which the latter wanted to reduce the entire mathematics). Besides those anti- nomies one discovered also so called semantical ones (G.D. Berry, K. Grelling) connected with semantical notions such as reference, sense and truth. In this situation there was a need to find a new solid and safe foundations for mathematics. Attempts to provide such a foundations and to overcome the crisis led, among others, to the rise of new directions and conceptions in the philosophy of mathematics, namely of logicism, intuitionism and formalism. One used here of course achievements of earlier mathematicians, especially of those living in the 19th century. In particular important were the very rise of set theory (G. Cantor), the idea of the arithmetization of analysis (A. Cauchy, K. Weierstrass, R. Dedekind), axiomatization of the arithmetic of natural num- bers (G. Peano), non-Euclidean geometries (N.I. Lobachevsky, J. Bolayi and C.F. Gausss) and the complete axiomatization of systems of geometry (M. Pasch, D. Hilbert) and, last but not least, the rise and the development of mathematical logic (G. Boole, A. De Morgan, G. Frege, B. Russell). New doctrines in the philosophy of mathematics looking for solid, sure and safe foundations of mathematics saw them first of all in mathematical logic and set theory. The most important proposals are called logicism, intuitionism and formalism.

1.1. Logicism Logicism was founded by Gottlob Frege (1848–1925) and developed by Bertrand Russell (1872–1970) and Alfred North Whitehead (1861–1947). It grew out from the tendency to arithmetize the analysis – this tendency was characteris- tic for foundational studies in the 19th century. The aim of it was to show that the theory of real numbers underlying the analysis can be developed on the basis of the arithmetic of natural numbers. The realization of this task led to a new one, namely to find foundations for the arithmetic of natural numbers, i.e., to found it on a simpler, more elementary theory. This task was undertaken just by G. Frege who tried to reduce arithmetic to logic (cf. Frege 1884, 1893, 1903). Unfortu- nately the system of logic used by him turned out to be inconsistent what was discovered by B. Russell in 1901 (Russell discovered that one can construct in this system the antinomy of irreflexive classes called today Russell’s antinomy). In this situation Russell together with A.N. Whitehead undertook anew the task of reducing arithmetic to a simpler theory. They introduced so called ramified theory of types and showed that the whole mathematics can be reduced to it. This was the beginning of a mature form of logicism. Its main thesis says that (the whole) mathematics is reducible to logic, i.e., mathematics is only a part of logic and logic is an epistemic ground of all mathematics. This thesis can be formulated as the conjunction of the following three theses: (1) all mathemat- ical concepts (in particular all primitive notions of mathematical theories) can