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C

OGNITIVE

A

CCESSIBILITY OF

M

ATHEMATICAL

O

BJECTS

JERZYPOGONOWSKI1

Department of Logic and Cognitive Science Adam Mickiewicz University

Jaroslav Hašek wrote that it is difficult to describe non-existing animals but it is much more harder to show them to the audience. We are not going to discuss the old dilemma: is mathematics created or discovered? Rather, we will focus our attention on the access which we have to mathematical objects themselves. More- over, this access will be characterized inside mathematics and not based on, say, philosophical considerations about perception.

In turn, John von Neumann expressed the opinion that in mathematics we are not aiming at understanding it but we rather get accustomed to it. Is this dictum a play with words only? We think of mathematics as a science of patterns which is also an art of solving problems, according to prescribed rules. The meanings of ma- thematical concepts are determined by the underlying theory. Understanding these concepts is obtained in the context of transmission of mathematical knowledge, with the help of intuitive explanations.

Mathematical objects can be standard, exceptional or pathological. Whether they are considered as well-behaving depends on the goals they are supposed to serve. Domestication of mathematical objects is a result of accumulation of know- ledge about them and widening the scope of their applications.

The objects of each mathematical domain may be classified with respect to their accessibility for the cognitive subject. We have different cognitive access to several sorts of numbers: integers, rational, algebraic, constructible, computable, irrational, transcendental, normal, etc. numbers. There are easy sets (finite, Borel, constructible) and difficult ones (Bernstein, Cantor, Vitali, large cardinals, inde- composable continua). Functions are classified in the Baire’s hierarchy. One can distinguish degrees of computability and incomputability. Some examples will be discussed in details in our talk.

We are going to emphasize the fact that degrees of accessibility of mathema- tical objects can be characterized in mathematics itself, thus without support of metaphysical assumptions. However, the accessibility in question is relativized hi- storically and depends on the expressive power of mathematical discourse, as we will try to show.

1The work on this paper has been sponsored by the National Scientific Center research grant 2015/17/B/HS1/02232 Extremal axioms: logical, mathematical and cognitive aspects.

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