These are the answers by computer scientist Lionel Elie Mamane to the ten questions about intuitionism:
(Upon hearing my answers, Herman said "you are a formalist!")
No.
Define 'define'.
Sure, I can write down "λ x . if x < 1 then 0 else 1". There, it is defined. But I cannot compute it.
If one thinks about mathematics as an abstract model for the real world, let's say that such functions don't come up in the real world, but that they are useful simplified approximations to make an analysis. On second thought, the point is moot. The real numbers have no real-world meaning: One knows nothing at an arbitrary precision there. Everything is discrete, but non-discrete spaces useful approximation. (Continuity is a trivial notion for discrete spaces.)
Even more shrewd answer:
f := λ x . x * x + 1can be defined, computed, etc. Now, consider the topology on R where all ] a; b [ are open and additionally { 1 } is open. The function is not continuous in 0: There is no neighbourhood of 0 whose image is contained in {1}, a neighbourhood of f(0).
I have just done it! Hence, it must be possible.
No.
∀ A:logic, it holds in A the way it is normally stated in A
No.
Define 'are'.
Sure, there is \aleph_{1}, \aleph_{18}, ... as formal constructions. For mathematics as a tool to model the real world, there is no infinity at all. Or maybe countable. Maybe.
No.
It is a meaningful statement (in ZFC). What do you mean by "a definite truth value"? It has both truth values, or none or ... depending how you look at it. You can choose! It has no unique truth value.
No.
Same as 4.
Mu.
Question is vague. Must I enter only one machine in the competition? Or can I enter multiple machines and only one has to win?
Mu.
At first sight, no. I don't see any reason this would hold. But maybe I can be convinced. (No, not by a truth table.)
On second thought, no. For example, "n is prime" and "3 divides n". None implies the other (but the counter-examples to the (classically seen) implication uses a different n).
(Obviously, in classical logic, the truth table argument works.)
I guess my answer is: It depends what logic you are in.
Yes.
Algorithm extraction, for example.
Yes.
Yes.
The statements true in minimal logic are more true: there are true in both constructive and classical logic! All mathematical truths are true within the mathematics they are true in.