I am giving an example of something I can do “for any x” but not “for all x”. In the first case, the x is given in a fully constructed, reified form, and I can look at its internals to build a bespoke response. In the second case, I would have to give a general procedure that can work with all x while interacting with the x only by means of its external interface.
Ah okay, I think I understand, if I’m remembering my type theory correctly. I think this is downstream of “standard type theory” i.e. type theory created by Löf not accepting the excluded middle? Which does also mean rejecting choice, for sure.
EDIT: But fwiw, I think the excluded middle is much less controversial than Choice (it should technically be strictly less controversial). I think that may be a less interesting post, but I’m sure philosophers have already written that. Though I think a post defending rejecting the excluded middle from a type theory perspective actually could be quite good, because lots of people don’t seem to understand the arguments from the other side here, and think they’re just being ridiculous.
I am giving an example of something I can do “for any x” but not “for all x”. In the first case, the x is given in a fully constructed, reified form, and I can look at its internals to build a bespoke response. In the second case, I would have to give a general procedure that can work with all x while interacting with the x only by means of its external interface.
Ah okay, I think I understand, if I’m remembering my type theory correctly. I think this is downstream of “standard type theory” i.e. type theory created by Löf not accepting the excluded middle? Which does also mean rejecting choice, for sure.
EDIT: But fwiw, I think the excluded middle is much less controversial than Choice (it should technically be strictly less controversial). I think that may be a less interesting post, but I’m sure philosophers have already written that. Though I think a post defending rejecting the excluded middle from a type theory perspective actually could be quite good, because lots of people don’t seem to understand the arguments from the other side here, and think they’re just being ridiculous.