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I have two questions:

(1) Is a postulate the same thing as an axiom? This answer seems to suggest the answer is yes. I've always thought the two were the same but the question below suggests a difference.

(2) If the answer to (1) is yes, then why is Bertrand's Postulate so named? It is a theorem, one can prove it. Was it taken as an axiom at one point or is this just a sort of "abuse of terminology?"

I guess, if said naming is indeed an abuse of terminology, my opinion is that this abuse is not totally benign. When I first heard of Bertrand's Postulate, it was mentioned in a proof in the following manner: "We see that Statement X follows immediately from Bertrand's Postulate." Not familiar with the theorem, I assumed this was another name for some famous axiom, which was confusing to me since I knew of no axiom which readily implied the result. Upon looking up Bertrand's Postulate, my misconception was cleared up but I was left wondering why they called it a postulate.

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    $\begingroup$ Tradition. ${}$ $\endgroup$ – Mariano Suárez-Álvarez Sep 9 '16 at 17:49
  • $\begingroup$ In modern use, axiom and postulate have the same meaning. $\endgroup$ – Mauro ALLEGRANZA Sep 9 '16 at 17:50
  • $\begingroup$ It was a conjecture, named by someone (who ?) postulate; it has been proved, and thus now is a theorem. There is no reason to think that (human) mathematicians are "logical" ... $\endgroup$ – Mauro ALLEGRANZA Sep 9 '16 at 17:51
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    $\begingroup$ +1 for the question. I hope it gets a good etymological answer. Bertrand seems to have conjectured it, not assumed it as an axiom. My guess is that calling it a postulate is an abuse and that we're stuck with the times folks read it your way and then have to take the time to clear things up. $\endgroup$ – Ethan Bolker Sep 9 '16 at 17:54
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    $\begingroup$ See Bertrand's paper, page 129 : "In order to prove [...] I'll assume as a fact, for every number $n > 6$, the existence at least of a prime number ... This statement is true for every number less than six millions, and thus it seems to be true in general." It is clearly stated as a conjecture. $\endgroup$ – Mauro ALLEGRANZA Sep 9 '16 at 17:59
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It is a theorem since a long time. Bertrand conjectured it (in 1845) but did not prove it.

Postulate means something that is assumed to be true. As an axiom or tentatively as a conjecture. As detailed in a comment by Mauro Allegranza this matches exactly what happened. That is, Betrand assumed this to be true (tentatively as a conjecture) in order to be able to progress with an argument.

It seems he did not use the word himself though. Yet, Chebyshev already used this term few years later (1852) writing in the paper where he proved it "ce qui est le postulatum connu de M. Bertrand" meaning "which is the known postulatum of Mr. Betrand" (my translation).

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"Postulate" is used here as a synonym of "conjecture". Bertrand conjectured this in 1845, Chebyshev proved it in 1852.

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  • $\begingroup$ Clearly "postulate" is meant here as a synonym for "conjecture". I think (hope) the OP is asking whether that usage has a history/ $\endgroup$ – Ethan Bolker Sep 9 '16 at 17:56

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