These are the answers by philospher Andrei Rodin to the ten questions about intuitionism:
No.
Except if definability is properly restricted. I assume that such restrictions can make perfect sense but I don't assume that they are always necessary.
No.
With similar reservations concerning assumed logical setting.
No.
With a similar reservation. A interpret the above questions as posed about any possible setting, and I assume that there are different acceptable settings. Hence reservations (which don't mean uncertainty about given answers).
No.
Again if the question concerns any possible setting. In some settings this might appear to be true.
No.
Not in any setting.
No.
No.
No.
But quite often so.
Yes.
And I also think that a "kind of reasoning" in a sense relevant to mathematics could be itself a mathematical matter. Which doesn't exclude the possibility of universal (kind of) mathematical reasoning (to leave aside universal reasoning in a wider sense).
Mu.
But I think this wording could be given good sense, in particular when one does mathematics in a topos. If one doesn't like it for some philosophical reason, one could perhaps avoid such wordings even working in a topos. I rather like. An example is found in the end of: Conceptual Mathematics by Lawvere and Schanuel. In any event I don't believe that only bi-valuated (or only bi-valuatable) mathematical statements are meaningful.