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ON NEW TRENDS IN THE PHILOSOPHY OF MATHEMATICS1

The aim of this paper is to present some new trends in the philosophy of mathematics. The theory of proofs and counterexamples of I. Lakatos, the con- ception of mathematics as a cultural system of R.L. Wilder as well as the con- ception of R. Hersh and the intensional mathematics of N.D. Goodman and S. Shapiro will be discussed. All those new trends and tendencies in the phi- losophy of mathematics try to overcome the limitations of the classical theories (logicism, intuitionism and formalism) by taking into account the actual prac- tices of mathematicians.

Since the beginning of the sixties a renaissance of interests in the philosophy of mathematics can be observed. It is characterized, on the one hand, by the dominance of the classical theories like logicism, intuitionism, formalism and platonism developed at the turn of the century, and, on the other, by emergence of some new conceptions. The latter are reactions to the limitations and one- sidedness of the classical views which are in fact results of certain reductionist tendencies in the philosophy of mathematics and consequently are of explicitly monistic character. Logicism claims that the whole mathematics can be reduced to logic, or, in a modern version, to set theory. Hence mathematics is nothing more than logic, or set theory. Intuitionism rejects the existence of an objective mathematical reality and says that mathematical knowledge can be founded on the activity of human mind and that this activity can be directly known. Hence the subject of math- ematics is the mental activity of mathematicians. A. Heyting, the follower of

1Originally published in: E. Orłowska (ed.), Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pp. 15–24 (1999). Heidelberg–New York: Physica-Verlag, A Springer-Verlag Company.
c Physica-Verlag Heidelberg 1999. Reprinted with kind permission of Springer-Verlag GmbH.

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. 75–84. Amsterdam/New York, NY: Rodopi, 2010. 76 Part I: Philosophy of Mathematics

L.E.J. Brouwer, the founder of intuitionism, wrote: “In fact, mathematics, from the intuitionistic point of view, is a study of certain functions of the human mind [...].”(cf. Heyting 1966, p. 10). Formalism (at least in the radical version of H.B. Curry, cf. 1951) – Hilbert, the founder of formalism, was not so radical) reduces mathematics to certain play on meaningless symbols. Mathematics is simply a study of formal systems expressed in artificial formal languages and based on certain formal rules of inference. Platonism,2 founded by Plato and taking various forms in the history of mathematics, is always alive and is in fact the philosophy of most mathemati- cians (if they do thing of philosophy at all and express their views). Platonism claims that the subject of mathematics are certain timeless and spaceless enti- ties being independent of mathematicians. The nature and existence of them was understood and explained in various ways. What is important here is the fact that a mathematician who constructs mathematics was not taken into account. Mathematics was an absolute knowledge about certain absolute objects. This attitude is characteristic for all classical theories in philosophy of math- ematics. They are giving us one-dimensional static pictures of mathematics as a science and are trying to construct indubitable and infallible foundations for mathematics. They treat mathematics as a science in which one automatically and continuously collects true proved propositions. The complexity of the phe- nomenon of mathematics is lost in this way – neither the development of math- ematics as a science nor the development of mathematical knowledge of a par- ticular mathematician are taken into account. The classical theories provide us only with one-sided reconstructions of the real mathematics. The reason for that can be seen in the fact that they were created at the turn of the century in an atmosphere of a crisis in the foundations of mathematics which was the result of the discovery of antinomies in set theory. What one needed then mostly were un- questionable and indubitable foundations on which the “normal” pursued math- ematics could be founded. One supposed here of course that mathematics should be an infallible and indubitable science and that in fact it is such a science. Logicism claims that mathematics can be reduced to logic (or: set theory); intuitionism says that mathematics can be based on the intuition of natural num- bers and the latter can be founded on the intuition of a priori time; formalism sees the resource in formal languages to which all mathematical theories should be reduced. As a result one receives idealized pictures of mathematics being really practised. All this, together with the growing interests in the history of mathematics, led to the rise of new more adequate theories in philosophy of mathematics. Its characteristic feature is the fact that the actual practices of real mathematicians

2This name was introduced by P. Bernays.