his research did not follow the main trend. However, his views and conceptions were interesting. The following three chapters are devoted to three centres of logic and mathematics in the interwar Poland, namely: Warsaw (Warsaw School of Mathematical Logic), Cracow and Lvov (Lvov School of Mathematics) and to the presentation of philosophical views on logic and mathematics developed and proclaimed there. In particular, views of Tarski, Andrzej Mostowski (Warsaw), Jan Sleszyński, Stanisław Zaremba, Zygmunt Zawirski, Witold Wilkosz and Leon Chwistek (Cracow; in fact Chwistek was active both in Cracow and Lvov) as well as Hugo Steinhaus, Stefan Banach and Eustachy Żyliński (Lvov) are discussed. The volume is closed by a chapter devoted to the description and analysis of philosophical views on logic and mathematics of members of the so-called Cracow Circle. This term is used to describe a group of scholars who tried to apply the methods of modern formal/mathematical logic to philosophical and theological problems, in particular they attempted to modernize the contemporary Thomism (the trend which was then prevailing) by the logical tools. The group consisted of the Dominican Father Józef (Innocenty) M. Bocheński, Rev. Jan Salamucha, Jan Franciszek Drewnowski as well as the logician Bolesław Sobociński who collaborated with them.
Papers included into this volume (with one exception) were published earlier in journals and collective volumes as separate and independent items. Putting ←00 | 7→them now together in one volume implies that there appear some unavoidable repetitions. I hope that this circumstance will not be an obstacle for the reader.
I would like to thank all who helped me in the work on this book. First of all, I thank the co-authors who agreed to include into the volume our joint chapters, in particular Professors Thomas Bedürftig and Jan Woleński. I thank also the publishers of particular papers for the permission to reprint them in the present volume. I thank the Faculty of Mathematics and Computer Science of Adam Mickiewicz University in Poznań for the financial support as well as Ms Magdalena Stachowiak for her help in converting some files and Doctor Paweł Mleczko for his advices concerning TEX. Last but not least I thank Mr. Łukasz Gałecki from Peter Lang Verlag for his helpful assistance.
Roman Murawski
Poznań, in June 2019
Contents
On the Philosophical Meaning of Reverse Mathematics
On the Distinction Proof–Truth in Mathematics
Some Historical, Philosophical and Methodological Remarks on Proof in Mathematics
The Status of Church’s Thesis (co-author: Jan Woleński)
Between Theology and Mathematics. Nicholas of Cusa’s Philosophy of Mathematics
Mathematical Foundations and Logic in Reborn Poland
Tarski and his Polish Predecessors on Truth (co-author: Jan Woleński)
Benedykt Bornstein’s Philosophy of Logic and Mathematics
Philosophy of Logic and Mathematics in the Warsaw School of Mathematical Logic
The philosophy of Mathematics and Logic in Cracow between the Wars
Philosophy of Logic and Mathematics in the Lvov School of Mathematics
Cracow Circle and Its Philosophy of Logic and Mathematics
←10 | 11→
On the Philosophical Meaning of Reverse Mathematics
The aim of this chapter is to discuss the meaning of some recent results in the foundations of mathematics – more exactly of the so-called reverse mathematics – for the philosophy of mathematics. In particular, we shall be interested in implications of those results for Hilbert’s program.
Hilbert’s program
One of the reactions on the crisis in the foundations of mathematics on the turn of the 19th century was Hilbert’s program. Hilbert’s aim was to save the integrity of classical mathematics (dealingwith actual infinity) by showing that it is secure.1 He saw also the supra-mathematical significance of this issue. In 1926 he wrote: “The definite clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences, but for the honor of human understanding itself ”. Being first of all a mathematician, he “had little patience with philosophy, his own philosophy of mathematics being perhaps best described as naïve optimism – a faith in the mathematician’s ability to solve any problem he might set for himself ” (cf. Smoryński 1988).
Hilbert’s program of clarification and justification of mathematics was Kantian in character (cf. Detlefsen 1993). Following Kant, he claimed that the mathematician’s infinity does not correspond to anything in the physical world, that it is “an idea of pure reason” – as Kant used to say. On the other hand, Hilbert wrote in (1926):
Kant taught – and it is an integral part of his doctrine – that mathematics treats a subject matter which is given independently of logic. Mathematics, therefore can never be grounded solely on logic. Consequently, Frege’s and Dedekind’s attempts to so ground it were doomed to failure.
As a further precondition for using logical deduction and carrying out logical operations, something must be given in conception, viz., certain extralogical concrete objects which are intuited as directly experienced prior to all thinking. For logical deduction to be certain, we must be able to see every aspect of these objects, and their properties, differences, sequences, and contiguities must be given, together with the objects themselves, as ←11 | 12→something which cannot be reduced to something else and which requires no reduction. This is the basic philosophy which I find necessary not just for mathematics, but for all scientific thinking, understanding and communicating. The subject matter of mathematics is, in accordance with this theory, the concrete symbols themselves whose structure is immediately clear and recognizable.2
According to this, Hilbert distinguished between the unproblematic, finitistic part of mathematics
