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I think that when a person put on a true conjecture in mathematics (we have verified it to be true) ,proof must came in his mind in the form of what we called intuition. I mean if his saying is universally correct how can it be a fluke? Can it be true that he just forgot the whole process of arriving to his conjecture or just couldn't fill all the gaps in form of mathematical axioms. Shouldn't we call that a proof in philosphical terms?

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  • Whether a conjecture is "true" depends on the background assumptions. The parallel postulate is true in Euclidean geometry but not in others, continuum hypothesis is true in the constructible universe but not in ZFC, etc. Since "intuition" is by its nature blurry on what the assumptions are it does not even settle what is or is not true. And it certainly need not be based on a proof, even a vague "subconscious" sketch of it. It often comes from empirical induction, analogies, wrong or insufficient reasons, or mere wishful thinking for the sake of aesthetics. Can it, yes, but usually isn't.
    – Conifold
    Nov 2, 2019 at 8:05
  • @conifold: They do indeed, but because they are in the background we don't take them into account until we reach a foundational problem like non-Euclidean geometry. They are part of the background intuition, so to speak. Can you tell me why analogy is not part of intuition? Nov 2, 2019 at 12:51
  • "If his saying is universally correct how can it be a fluke?" Why shouldn't it be? A conjecture can be motivated by nothing more than seeing an apparent pattern and guessing it continues. Pretty much by definition, sometimes that guess will be right and sometimes it will be wrong. Nov 2, 2019 at 17:47
  • @MoziburUllah I couldn't understand your comment (you said "They do indeed"), can you elaborate more on on this please..
    – ANUJ GUPTA
    Nov 19, 2019 at 18:34
  • @ANUJGUPTA: Why ask me now, this was weeks ago. If you wanted elaboration you should have asked me then. I can't be bothered now. Nov 20, 2019 at 1:31

3 Answers 3

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Conjectures are often stated without any knowledge that there might be a proof, and with no intuition about a proof. Conjectures are often wrong.

There are many, many conjectures that didn't require any intuition at all. Just a bit of statistics, or heuristics. "It's unlikely to be wrong" is often a good reason to state a conjecture.

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Poincare on his musings on the mathematical process wrote how when he was trying out a proof on Fuschian functions that he sat at his desk every morning for two weeks trying out innumerable combinations until one morning it came to him as he was stepping off a bus. He felt certain that it was correct and when he, later that morning sat at his desk and wrote out the proof, he verified his intuition was correct.

The matter of mathematics, unlike that of many other arts, is invisible. If you are a potter, your material is clay. If you are a painter, your medium is paint. But what of mathematics? Where do Fuschian functions hang about? Or the covariant derivative? Nowehere, except in the mind, even when we write them down.

Thus, when Poincare was trying out many conjectures what he was doing was pickig up the Fuschian functions in his mind and turning it over, feeling its shape, discovering its properties. In this way, Fushcian functions become more concrete in his mind. Until he solves the problem he has set himself.

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Until someone either proves or disproves it, a mathematical conjecture is often named after the person that first proposed it. E.g. Goldbach's conjecture - Wikipedia is still unproven.

"Shouldn't we call that a proof in philosphical terms?"

No. Just because a mathematician, even a very famous one, made a conjecture, that doesn't prove it is true.

For instance, Euler's sum of powers conjecture - Wikipedia was proven false nearly 200 years after he made the conjecture.

We can hardly say that Euler's conjecture "proved to be true" for nearly 200 years until someone proved that it wasn't.

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