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In the context of classes, it is very often that discussion on the history of mathematics arises, whether it'd be on who should a lemma be attributed to or a certain event that occurred during the discovery of the proof (the elementary proof of the prime number theorem is one such example).

My question is:

What does a math historian do? Is he simply a mathematician who dabbles in search for the history behind his research or does he commit his time fully investigating past mathematical facts? Also, is it closer in nature to mathematics or is it closer in nature to history (i.e. is the context behind the discovery of the proof emphasized or is the insight that led to the proof emphasized)?

EDIT: Due to the nature of some of the answers, I am now curious to as to whether math historians are mathematicians or historians (ie do they work in math departments or history departments). Does anyone have an answer to this?

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History of mathematics is a highly specialized branch of scholarship that requires full commitment. It is not a collection of anecdotes, fun though these can be. And yes, intellectual context is the main thing. –  André Nicolas Jun 4 '12 at 19:44
@AndréNicolas That sentence came out not as I intended. I meant that I don't often see math history programs being offered at the PhD level at math departments. So I was wondering if most math historians are actually hired as research mathematicians in other specialized fields but do math history on top of their research as well. From the answer though is seems that they are offered by history departments instead rendering that part moot. –  Eugene Jun 4 '12 at 20:09
The ones I have known are in history departments, occasionally in Classics, or have joint appointments. –  André Nicolas Jun 4 '12 at 20:56
You might take a look at Judy Grabiner’s home page; it’ll give you an idea of the what one well-known historian of mathematics does and what her academic background is. –  Brian M. Scott Jun 4 '12 at 21:29
@Gigili Ok then. I'm removing my comments to avoid cluttering up this space. –  Eugene Jun 5 '12 at 5:25
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2 Answers

First of all, as far as I know, serious historians of mathematics are or were in their majority mathematicians: the reason is that mathematics is very difficult and you can't analyze it in depth without a very serious technical background.
There might be exceptions for very ancient mathematics, but even there I wouldn't trust a historian studying Diophantus who wouldn't have some knowledge of arithmetic/algebraic geometry and number theory.

What often happens is that aging mathematicians start writing about the history of the subject they have devoted their life to.
Prestigious examples are for example:
Weil on number theory,
Dieudonné and his wonderful histories of algebraic geometry and algebraic topology,
Marcel Berger on differential geometry,
Dickson and his monumental history of the theory of numbers.

Younger mathematicians may also be interested :
Bourbaki has very nice historical surveys at the end of some of his chapters, written at the time by necessarily young members (there was an age limit for participants)
Schappacher is an excellent research mathematician who already as a young researcher wrote about the history of number theory,
Krömer has written a great thesis on the genesis of category theory (including the incredible beginnings of sheaf theory in a prisoner of war camp ) ,
and to finish on a personal note, here is the fairly recent thesis on the birth of group cohomology by Nicolas Babois, whom I taught at the undergraduate level (but I had no rôle in his thesis).

In conclusion, my point of view is that a historian of mathematics is essentially a mathematician, and historical science in the usual sense is of secondary importance.
This is certainly controversial.
My convictions on this subject essentially derive from Dieudonné's and Houzel's points of view. (Houzel is an other example of a mathematician with high technical skills attracted very early by the history of mathematics)

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To add to your list, Thomas Jech as brief historical notes in his books; and Akihiro Kanamori writes wonderful historical overviews in most of his books. I am not sure whether or not to classify either as young or aging mathematicians (right now both would be the latter, but Jech has historical notes in books he wrote 40 years ago). Perhaps a math historian could aid with an answer to this question... :-) –  Asaf Karagila Jun 4 '12 at 21:07
@AsafKaragila I was actually hoping a math historian would have an answer to my question. –  Eugene Jun 5 '12 at 4:48
I will agree with one thing in your answer, Georges: "This is certainly controversial." Dickson's "History", for example, is a marvelous piece of work, but it isn't history, any more than a list of all the battles of World War II would be a history of that war. It has no narrative, no flow, no synthesis, no elucidation of pattern.... Judith Grabiner, mentioned elsewhere, writes history, not compendia of facts. There's room enough for both Dickson and Grabiner, but Dickson was a mathematician with an interest in history, Grabiner's a historian of mathematics. –  Gerry Myerson Jun 5 '12 at 5:13
Weil wrote the Bourbaki's historical notes. Other members might have also written some of the notes, but I'm not sure. –  Makoto Kato Jun 5 '12 at 5:39
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It depends on the historian. Most historians of mathematics probably have deep associations with mathematics in some way, but being a mathematician it is not required any more than a sports historian should be a professional athelete. But surely the more mathematics one knows the more one tends to appreciate its history.

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