On ontology and realism in mathematics (Q2919947)

The author of this paper proposes that a position in the philosophy of mathematics should not be classified in terms of existence of objects; instead, one should characterize his position in terms of the statements one accepts as having a definite truth value. According to the author, the general orientation towards the category of the object can be misleading, and the philosophy of mathematics should follow a different path. The author remarks that there are other related distinctions in the literature, attributed to Shapiro and Kreisel.NEWLINENEWLINEIn the light of his general proposal, the author surveys the area of foundations of mathematics: he analyses briefly (i) several positions, from ultrafinitism to set-theoretic realism, (ii) Quine's criterion of ontological commitment, (iii) the ``ontology/ideology'' distinction, (iv) the possibility of ``concrete'' reductions of mathematical theories, with special attention to finite sets, natural numbers, and syntactical objects, and (v) epistemological issues related to realism, with special attention to the incompleteness phenomenon.NEWLINENEWLINEThe paper is clearly written and contains some background information about basic technical points in logic. However, it is hard to imagine that it could be read profitably by someone who has never seen the arithmetic hierarchy.