@Shaun It seems that your question is simply "A recent, private conversation I had online called into question the legitimacy - the efficacy, the applicability, the rigour - of topos theory and its implications about constructive mathematics. So I was wondering: Are the different logics given by topoi legitimate?".

So it seems that there are 2 things to do: 1. Give the arguments which called into question the legitimacy, and 2. Kill the rest of the question - it just all seems irrelevant. Maybe say "I know a bit of topos theory", but giving a list of questions you've asked on Math.SE is not helpful.

Even then, the question is a bit woolly. You could rephrase it to be more objective by saying something like "is this a mainstream opinion, and does it hold any water?" It may not be accepted by the community, but it would at least be a better question :-)

Thank you, @user1729. Your comments are helpful as always. I have edited the question accordingly.

@Shaun I was really meaning kill everything up to and including the words "...from what I have read.", and everything from the words "Thoughts:...".

@Shaun, the fact that you added a meager bounty to the already downvoted, and poor question, by doing that I cannot support this question. You should have only asked in CURED, only prior to resorting to a meager bounty.

@user1729 I should have revisited this room months ago to restore you to room owner. I should have done this long ago!

^^^@Shaun I'm disappointed.

@user1729 You're far more loyal here than I am, and self-less, than anyone here is. So now you have the power to delete me as owner, if you so choose. :-)

Thank you again, @user1729. I have edited the question again.

I'm sorry you feel that way, @amWhy.

I didn't say what I felt. I said what I saw, and the I'd also suggest you stop answering Algebra PSQs.

Take good care, Shaun. :-)

@amWhy I intend to, as always. Is there anything else you'd like to add? I feel like, besides myself, you're my biggest critic.

@Shaun As rarely as we speak? Please understand that I do not expect much from users that seem to clearly not progress quickly. I support them all, whatever the level. Perhaps, among those I believe hold great promise, and shown great promise, perhaps I am more critical. That is never meant to degrade anyone; but to some degree it is commensurate with the promise I see. I don't always get it right. Like most of us here, and like you, I trust, we do the best we can. But everyone of us,

including me can use inspiration. Perhaps I need to reframe my comments so they sound more supportive, with suggestions, and less critical. And I welcome constructive feedback here. But let me assure you, I think highly of you, and certainly also of @user1729. Perhaps I should share that more often.

I've reintroduced the context, @user1729. I think it's better to have too much than not enough.