Uwagi o platonizmie matematycznym

Abstrakt

The mathematical Platonism is the most popular standpoint amongst mathematicians. The Platonism better than other conceptions explains the fact, that mathematicians discover some properties of the mathematical objects, which are independent of them and exceed the boundary of our intuition and experience. On the other hand Platonism generates some difficulties. In spite of this the Platonism is the idea worth the defence. In the paper I modify the traditional formulation of Platonism, which avoids some of the difficulties.

It seems, as if some mathematical notions have the same properties as concrete physical objects, for instance: number 5, number π, the minority relation between the natural numbers, the ring of integers. These notions determine the single mathematical objects, whereas concepts such as function, natural number, relation, ring, linear space, determine the whole classes of the objects with the same properties. Thus we can divide mathematical notions into two categories. The notions, which we can treat like the concrete objects, form one of these categories. In the second category there are notions which play the role of general notions from the colloquial language. The notions from the first class can be members of some set; they can also be combined together. The notions from the second group cannot become elements of any set.

These different ways of behaving of mathematical notions suggest that their existence is also different. The objects from the first category exist independently from mathematicians; the notions from the second group are the creations of human mind. Therefore I propose to keep Platonism regarding the objects from the first class and to accept conceptualism regarding mathematical objects from the second group.