# Why is Bertrand's Postulate Called a “Postulate?”

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.

• Tradition. ${}$ – Mariano Suárez-Álvarez Sep 9 '16 at 17:49
• In modern use, axiom and postulate have the same meaning. – Mauro ALLEGRANZA Sep 9 '16 at 17:50
• 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" ... – Mauro ALLEGRANZA Sep 9 '16 at 17:51
• +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. – Ethan Bolker Sep 9 '16 at 17:54
• 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. – Mauro ALLEGRANZA Sep 9 '16 at 17:59