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Foundations as truths which organize mathematics. (English) Zbl 1271.18001
This article is a detailed and analytical answer to Feferman’s claims about categorical foundations of mathematics. Summarising, Feferman objects that categorical foundations are not a foundation for mathematics since (1) it is based on earlier set theories, and (2) membership is not really derived or constructed but just reduced by means of a trick to an arrow-oriented representation.
The author argues against these objections: in the first place, the development of category theory, and of its foundations, started from set-theoretic ideas, but it was the categorical view that emerged as the guiding principle to developments; in the second place, categorical set theory was developed not as a foundation to derive all the mathematical truths, even if this is possible in principle, but, on the contrary, as a way to organise the mathematical body of knowledge.
In fact, the author favours a view of foundations which develops as a recognised body of truths adequate to organise definitions and proofs, at least in their presentation. He claims that finding concise principles to organise mathematical knowledge has been a huge achievement by mathematician and logicians.
In the spirit of the journal it is published in, the article is conversational rather than rigidly technical, and quite easy to read even for the non-expert, although previous knowledge of Feferman’s objections is required to have a correct picture of the discussed questions.

MSC:
18A15 Foundations, relations to logic and deductive systems
00A30 Philosophy of mathematics
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