Post #2645665
2026-04-30 15:47 UTC
Replies (1)
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@MartinEscardo@mathstodon.xyz 2026-04-30 16:27
1/ @RobJLow@mathstodon.xyz writes "MartinEscardo I suspect that there are at least some people who think that when they find the 'right' axioms, the scales will fall from everybody's eyes and those axioms will define the right mathematics." Let me try to explain my point of view in more detail. Which, by the way, is shared by many people in my mathematical community. (1) The sign "·" may mean many things. Multiplication on the reals, or in any group are examples (among others!). (2) In constructive mathematics "exists" means something different from what it means in classical mathematics. And the same goes for "or". (3) But it is perfectly possible, and it has been done, to accommodate both meanings in a single "foundation" (I so hate this terminology). (4) But now, more darlingly, just like "·" may mean different things (in different groups), "exists" and "or" may mean even more things than discussed in (2). (5) Oh, wait! Don't "exists" and "or" have fixed meanings? No, like "·" doesn't. (6) Just like we may let the meaning of "·" depend on the group we are working with, we can let the meaning of "exists" and "or" to depend on what "topos" we are working with. Never mind what toposes are, for the moment, but they are the analogues of groups in this discussion.