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"Do you believe ...?" It is not a matetr of belief. Usual mathematical practice assume something as starting point: Axioms, already proven Theorems, and deduce (by way of rules of inference) new statement: new theorem.This logical move has the form "if Axioms, then Theorems" and amounts to proving that Theorems are logical consequence of Axioms.

See e.g. Ethan Bloch, Proofs and Fundamentals A First Course in Abstract Mathematics (Springer, 2011): "Theorem 2.1.1 (Pythagorean Theorem). Let ABC be a right triangle, with sides of length a, b and c, where c is the length of the hypotenuse. Then a2+b2 = c2. [...] we want to consider its logical form. Although the words “if..., then” do not appear in the statement of the theorem, the statement is nonetheless a conditional statement. 1/2

@MauroALLEGRANZA 1. No, obviously not the same question. 2. "Pythagorean Theorem" Sure, it is obvious that mathematical reasoning is based on the conditional form. The question is whether mathematicians believe that the material implication is the correct formal model of that particular type of reasoning. 3. "the theorem has the form P ⊃ Q." I know this is what is said in the context of mathematical logic. This is said explicitly in many logic textbooks. I want to know if a majority of mathematicians outside mathematical logic believe it.

So, in the end, your question amounts to the "eternal question" : "why the conditional in classical logic is TRUE when the antecedent is FALSE?"

@MauroALLEGRANZA No, my question is explicitly about what mathematicians believe.

Probably the most frequently used rule of inference is Modus Ponens: it is based on the conditional and it needs only the case when the antecedent is TRUE. But also everywhere used is contraposition: we prove P ⊃ Q through a proof of not-Q ⊃ not-P. Using it with MP again, this amounts to the fact that the conditional not-Q ⊃ not-P is TRUE when not-Q is TRUE, i.e. when Q is FALSE.

What mathematicians believe is written into mathematicians books...

There are well-know "alternative" conditionals: strict implication, relevance logic. And there are some minor attempt to use them in mathematics; see e.g. Revantha Ramanayake, Relevant Arithmetic (2008)

@MauroALLEGRANZA They do exist, but working with them, in any practical sense, is a pain. I think they might work for the foundational end of the spectrum, but I don't see anyone abandoning material conditionals for proofs like OP mentioned any time soon.

@Speakpigeon I think this question is very subjective. Do I believe MC is necessary? No. Is it pragmatic? Sure.

You are also forgetting, you don't only use MC for proofs of the form: (if a, then b) & (a); therefore, (b). Every subsequent step given an assumption is conditional: Suppose P, deduce Q; therefore, Q.

@MauroALLEGRANZA "What mathematicians believe is written into mathematicians books" Are you aware at all that this sounds something really weird to assert? Are logic textbook was like the Bible or the Little Red Book?! You think mathematicians don't have a brain to decide what they want to believe?

@BertrandWittgenstein'sGhost 1. "this question is very subjective" Your answer may be subjective, the question is objective. It is displayed on your computer screen. 2. "MC is necessary? No. Is it pragmatic" Your answer is not only subjective it does not address my question. I know that the material implication is not necessary and I know why mathematicians use it. The question is whether they think it is the same thing as a conditional. Why is it that my questions are always wrongly interpreted?

@Speakpigeon "You think mathematicians don't have a brain to decide what they want to believe?" I think Mauro's point is that mathematicians are the ones writing those books. Mathematicians tend to write down the things they believe. If basically every math book out there agrees on a particular topic, then it's safe to assume that that's the majority view on that topic among mathematicians.

@Speakpigeon I know that you know what I said and meant. Let's drop the semantic hoop−jumps. The question is subjective because every answer to it is subjective (which, btw, is one of the listed option one can choose to flag the question for closure, but that's besides the point). I don't know a single logician that has claimed Material conditional is THE logic for conditional. I will tell you this though it's the most pragmatic approximation.

That's of course my subjective opinion. xD.

@BertrandWittgenstein'sGhost "I don't know a single logician that has claimed Material conditional is THE logic for conditional" Read the question again: Is the material implication the correct model of conditional reasoning in mathematics? 2. What is this obsession about closing questions that are perfectly legitimate? And this is just about mathematics. No wonder there are goulags.