Ten Answers from Lionel Elie Mamane

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!")

Do you agree that it is impossible to define a total function from the reals to the reals which is not continuous?

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 + 1

can 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.
Do you agree that the intermediate value theorem does not hold the way that it is normally stated?

No.

∀ A:logic, it holds in A the way it is normally stated in A
Do you agree that there are only three infinite cardinalities?

No.

Define 'are'.

Sure, there is \aleph₁, \aleph₁₈, ... as formal constructions. For mathematics as a tool to model the real world, there is no infinity at all. Or maybe countable. Maybe.
Do you agree that the continuum hypothesis is a meaningful statement that has a definite truth value, even if we do not know what it is?

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.
Do you agree that the axiom which states the existence of an inaccessible cardinal is a meaningful statement that has a definite truth value, even if we do not know what it is?

No.

Same as 4.
Do you agree that for any mathematical question it is easy to build a machine with two lights, yes and no, where the light marked yes will be on if it is true and the light marked no will be on if it is false?

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?
Do you agree that for any two statements the first implies the second or the second implies the first?

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.
Do you agree that a constructive proof of a theorem gives more insight than a classical proof?

Yes.

Algorithm extraction, for example.
Do you agree that mathematics can be done using different kinds of reasoning, and that depending on the situation different kinds of reasoning are appropriate?

Yes.
Do you agree that all mathematical truths are true, but that some mathematical truths are more true than other mathematical truths?

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.