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?
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.
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.
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.