/sci/ - Science & Math

>>5840614
You don't really give an argument why not start with both at the same time. I see how it's practical for doing "fuzzy" generalizations (although I don't know why using pairs where the second entry just contains the additional information wouldn't work).

The point is that both, functions and set membership are natural - and so dropping set membership and relying on a subobject classifier even for standard sets... feels stupid.

Maybe that's just the pragmatic physicist speaking.

(logic and topology: the abstract topology concept coincides somehow with... non-full heytin models of second order arithmetic or something, right?)

Mhm, e.g. by discovering a mathematical tool or fact, and then coming across evidence that someone did it before, maybe in a more general case.
(sidenote: I also feel this is also the way of remembering things better, i.e. when you learn something as the solution for a problem you worked on before. If you get to learn a theory without having asked the question it answers, then you're less in awe.)

I discovered how equality of terms (interpreted as functions, say <span class="math">\sum_{k=0}^n 7^n=\frac{7^{n+1}-1}{7-1}[/spoiler]) actually doesn't see that there is a difference in compexity of evaluation (reduction, really).
And now I discover that people actually DO make a big deal out of the lacks of extensionality.

What do some foundations without only the rough extensional function identity (i.e. frameworks which destinguish sharper than set theory) know about complexity of functions?