A mathematician (it was as mentioned part of a mathematics degree, and the purpose was to chart the development of mathematics), which is why I would certainly not be able to respond knowledgeably to the later post about why the contribution of Europeans to mathematics was so significant and the factors which lead to this. EDIT: However, I would definitely like you to clarify the bolded.
The most obvious way to explain the difference are historians of math are professional scholars of history that are applying historical methods, training, and perspectives to the study of the historical development of math. They are interested in math the same way historians can be interested in any phenomenon. Mathematicians who dable in the history of mathematics are amateurs who have an interest in the source material. They are generally interested in Math in a way that is more antiquarian in nature than historical.
It's of course fine that mathematicians are interested in this, the problems tend to come from the fact that often such highly skilled individuals as mathematicians will ignore the fact that they are indeed out of their area of expertise and within that of the historian. In my experience, this is actually much more of a problem in the sciences, especially physics, but you'll still get people that very stubbornly don't actually think about what it means to deal with history.
As I've brought up a few times here, and many times on GAF, it reminds me of Steve Shapin's joke on the subject. A surgeon asks his historian friend for some tips on how to writing history as he's planning on working on the history of medicine in his retirement. The historian replies, "sure, but you'll need to give me some tips on surgery as I was thinking about picking that up when I retire."
It was studied under a mathematician, but his background originates from the social sciences (particularly history and philosophy which were the subjects of his undergraduate; so certainly has more exposure than most to the social sciences even if it's by no means his area of expertise) which is why he teaches the course. The course was designed to chart the development of mathematics and to teach mathematics by considering contemporary methods rather than designed to be a historical course or focus on/analyse the primary sources from which the information was derived (even though the historical context was indeed important and clarified, given that it's necessary to understand why mathematics developed as it did).
Not looking at primary sources too much, though they should always be there, isn't too much of an issue in a broad survey course when the material is pretty introductory and simple enough to be held in rough consensus by historians. The issue is when you take a field like math or the natural sciences, whose innerdisciplinary understanding is significantly apocryphal, and then teach without them. It's like teaching math without proofs, with the added isssue that as a result much of the content will be wrong or questionable at best.
While I cannot gain access to a contemporary source, the reference given in respect to the discoverer of incommensurability and the effect it had on the Pythagoreans and who it may be attributed to was first Thomas Heath's "A History of Greek Mathematics" on page 154 (which details the reasoning to believe why it is attributed to the Pythagoreans) and Kurt Von Fritz's "The Discovery of Incommensurability by Hippasus of Metapontum" (which, as the name suggests, makes the case why it was Hippasus who first proved this even if this is a much less clear manner).
This is the very issue though. You basically don't have a reason to believe this. The story about the guy being murdered for asking the Pythagoreans about the square root of 2 doesn't have a basis in contemporary primary sources. It seems to be a myth that built up over time, and was fun, interesting, and made math seem important to people's daily lives so it was built into the popular understanding of the subject. The problem is to do that one must ignore history and historical methods.
You can't deflect from a call for sources from the 6th century BC by referencing a book from 1921.
If you're questioning it being done by Pythagoras (the individual) rather than Pythagoras (the cult) then indeed, to my knowledge there is no evidence to suggest it was Pythagoras of Samos who personally proved this (but I don't think this is a common misunderstanding as I've never heard it claimed, even if their is a very substantial misunderstanding about what he accomplished mathematically; most of his 'contributions' are that of the cult rather than as the individual or in some cases falsely associated).
It might have been done by him or the cult. We don't really know. The issue is asserting it being either as positive knowledge based on essentially popular stories and without reference to contemoporarry sources. There's a lot of bullshit, around both the man and the cult, that can't be substantiated by contemporary sources talking about either. This is including the assertion that members of the cult attributed their work to him.
It's not a misunderstanding about what he accomplished mathematically, it's positive knowledge that, at least at the moment, we can't construct what he did.
It is not a very controversial statement that the Ancient Greeks were aware of incommensurability and how to prove their existence. We know this directly from Aristotle and Euclid's writing. We just don't know what the first example found was or the initial proof (even if Aristotle's comments are very suggestive).
Right, like I said the OP is clearly doing some weird stuff and I'm not supporting him. It's fine to say the Greeks came up with this stuff, we have Euclid, not to mention the word irrational and its connotation, so we know they did. The issue is passing off apocrypha as knowledge.
The story you're referring to is in reference to commentary by Euclid in The Elements (where he details a proof of Pythagoras' Theorem, for which you are indeed accurate) in which he attributes the proof to the Pythagoreans (despite little evidence to suggest this theorem originates from the Pythagoreans). It is perhaps this to which you are referring to the claim that many mathematicians attribute this to Pythagoras despite no evidence existing that the Pythagoreans discovered or were the first to prove the theorem in spite of the name.
No, I'm referring to the fact that if we poll mathematicians most will say Pythagoras "proved", and that's a whole extra element here, the theorem. This isn't one mistake made by one guy.
I'm not talking about any specific reference, there are a number of sources dating from the 2nd century B.C. onward that make increasingly bizarre claims about Pythagoras and math. Many of these have been accepted into the popular understanding, which mathematicians generally subscribe to, of the history of math.
Pythagoras isn't the only "mathematician" that gets this treatment, though due to his antiquity and mythologizing he probably gets it more than anyone else. Most mathematicians reading of Gauss is rather interesting as well. The root problem is what I wrote about at the beginning of this post.
I'd call it philosophy or religion at this point.
Physics is always philosophy, but I agree with your sentiment. String theory is pretty far into the metaphysics side of things.
