Strukturalismus a jeho problémy
Structuralism and its problems

Summary/Abstract: In Shapiro’s opinion, it should be taken seriously what mathematicians say (faithfulness constraint) and at the same time do not attempt to revise the results they reach (minimalism constraint). When arithmetic is viewed from this perspective, one reaches the conclusion that a number is a relational object and therefore structuralism is justified. However, difficulties arise once the limits of common arithmetic are crossed. Then it turns out that numbers are also operated in ways that contradict structuralism. That can lead one either to doubt structuralism as a whole, or to reject Shapiro’s constraints. We prefer the latter alternative, whereby we reject Shapiro’s constraints only partially. They remain valid in the context of well-established mathematical practice, let us say arithmetic; it poses difficulties in spheres that as yet lack clear contours, for instance when mathematicians say 2real=2nat. Although they understand statements of such type, we think that the way they express them is misleading and often confused. And we think that in such circumstances philosophers have the right to take part in creating more exact means of expression. Obviously, this proposal weakens both Shapiro’s constraints.

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