To count or to think, that is the question. (English) Zbl 0858.00006

Halmos, P. R.

Nieuw Arch. Wiskd., IV. Ser. 13, No. 1, 61-76 (1995).

The paper gives the text of the 1993-1994 Johann Bernoulli Lecture given at the University of Groningen on June 7, 1994. The question posed in the title is no real question for the author. Number work should be left to the computer, “what mathematicians are concerned which are designs, patterns, abstract ideas, and the logical connections among them. […] Abstract ideas are what we try to juggle – abstract ideas such as symmetry, continuity, order, chance, size, and connectedness – that’s the stuff, that is our daily bread and butter” (p. 62). The main tool for mathematical work is abstraction by building equivalence classes, accompanied by a philosophical stand which the author calls “extensionalism” and which says that a concept is its extension. Such stand can be found in definitions like “5 is the equivalence class of equinumerousness to which the set of fingers of my right hand belongs” (cf. p. 70).
The paper provides an amusing reading on a couple of important topics in the philosophy of mathematics, such as the question of usefulness of mathematics and the status of nonconstructive existence proofs. It arrives at Platonism maintaining that the typical mathematician’s attitude is not that of creation but of discovery, because abstractions are facts, “facts that we do not ‘invent’ but that are there for us to find if we can” (p. 76).

Reviewer: V.Peckhaus (Erlangen)

MSC:

00A30	Philosophy of mathematics
03A05	Philosophical and critical aspects of logic and foundations

Keywords:

extensionalism; abstraction; nonconstructive existence proofs; Platonism