Ονομοκρατία, Εννοιοκρατία και Πλατωνισμός στα Μαθηματικά
This paper has been motivated by an article of A. Fotinis under the title A Critical Evaluation of Universals in Nominalism, published in this Yearbook (3/1973, 382-404). My aim is a critical evaluation of universals, particularly in mathematics where universals play a significant role. In the philosophy of mathematics, where instead of universals we speak about classes and sets, we find the well-known alternatives of nominalism, conceptualism, and Platonism. In this paper I understand nominalism and Platonism in the modern meaning of these terms on account of this restriction, since their respective classical versions are out of considerations. The influence of Platonism in the philosophy of mathematics, even in mathematics itself, remains to our own day, in spite of the appearance of the antinomies that has diminished its former prestige. On the other hand, nominalistic language has proved unable to translate all classical mathematics. The disagreement, however, of constructive conceptualism with Platonism has not proved so big, and the gap between the two alternatives can be bridged over. For many philosophers and mathematicians there exists a certain eclectic affinity between these two alternatives, which makes it possible to regard the first as a variation of the second. I also consider, as seems necessary, the correlation between the three above mentioned philosophical alternatives, on the one hand, and the principal mathematical schools, namely formalism, intuitionism and logicism, on the other.