# 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. ${}$ Sep 9, 2016 at 17:49
• In modern use, axiom and postulate have the same meaning. Sep 9, 2016 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" ... Sep 9, 2016 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. Sep 9, 2016 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. Sep 9, 2016 at 17:59