Logic and mathematics

Jan Woleński

pp. 197-210

in: Werner Depauli Schimanovich, Eckehart Köhler, Friedrich Stadler (eds), The foundational debate, Berlin, Springer, 1995

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.