Categorical Foundation of Mathematics?

jodmos.horon jodmos.horon at
Tue May 24 21:25:14 EDT 2022


Harvey Friedman wrote:
> No one has challenged me on here to show how to present the usual f.o.m. with philosophical coherence. And no one has tried to present categorical foundations in a philosophically coherent way on here.

I do not see the need, honestly. For neither.

Set theory is doing fine at the problems it solved. Category theory (broadly construed) offers a way to (at a minimum) reencode what set theory has done (and this is not the main reason I am interested in category theoretical approaches).

Doesn't change the fact that category has provided criticism of the usual framework not only of set theory but of logic itself. It indeed made a case that syntax comes up from something else, for instance when quantification has been shown to be an adjunction.

I do not believe it is acceptable to brush away criticism of theory A by theory B on the grounds, whether true or not, that theory A has foundations or justification that theory B doesn't.

> This suggests hardened positions that are not possible to budge.

Well, I do not even see, speaking for myself, what positions I should be considering budging.

But yes, I am dissatisfied with set theory. It makes too many assumptions. If sets are based on points rather than previous philosophical understandings of parts and wholes, it's because Peano found it convenient (and rightly so). Nothing more.

I do not see special foundational philosophical status to the idea of founding set theory on that hypothesis of sets based on points. There should be a way to dispose off of this hypothesis.

It's in fact a fairly recent introduction (late 19th century) in the discourse of logic. And I fail to be moved by it in any intellectual capacity.

I guess I'm a kroneckerian of a kind.

Jodmos Horon.

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