>
> On Mon, 12 Apr 1999, Julio Gonzalez Cabillon wrote:
>
>>
>> In substance, my correspondent disagrees completely with the current fad
>> that encourages teaching of the (so-called) History of Mathematics either
>> to undergraduates or to normal mathematics graduates when both his/her
>> eneral historical background and or intellectual maturity are below a
>> reasonable (whatever the term means) threshold:
>>
>> "Of course, a lecturer should make the occasional historical
>> remark, and it should be well founded. But it is not possible to
>> acquire historical knowledge about the Renaissance authors without
>> reading Italian, about the Baroque authors without Latin, about
>> the 18th/19th century authors without French. And even for Peano
>> we also need latine sine flexione. Every translation is already
>> an interpretation. Second hand clothes may be acceptable; second
>> hand knowledge is not."
>>
>> Reactions?...
>>
>> Julio Gonzalez Cabillon
>
> I have kept out of this before now, but found myself feeling so
> sympathetic to the remarks of this correspondent (almost despite
> myself) that I now feel I should say why.
>
> I feel that with most students, time spent teaching mathematical
> history would almost always be better spent teaching mathematics.
> This holds even in very good universities like the ones I've taught
> in; Cambridge and Princeton, in each of which I've taught courses in
> mathematical history.
>
> The reason is that for the history to be at all worth while it
> should be more than that of elementary mathematics. It should contain
> a discussion of evolution of the subtle concepts the students are
> learning in their mathematical courses. But for most of these, a
> course specifically on mathematical history is not the most appropriate
> place, because its teacher can seldom be sure that all the students
> really are familiar with the modern versions of these concepts: there's
> no point in teaching how we got to X to students who don't really
> understand the point of X itself.
>
> So the best place to say something about this kind of X will usually
> be in the courses where X is taught. Of course this requires that the
> teacher of mathematics must also be the teacher of mathematical history,
> and doesn't work too well nowadays because too many mathematicians are
> unfamiliar with mathematical history. But the reason above isn't the
> only why the specialist courses don't work too well - another is that
> nowadays many specialist mathematical historians don't know too much
> mathematics!
>
> I think that there IS a place for courses on mathematical history -
> namely ones aimed at the best mathematical students, but only after
> they've learned a lot of mathematics, and taught by teachers who
> know both the modern mathematics and its history. In other words,
> such courses should come AFTER the students learn the mathematics
> rather than BEFORE or WHILE they do so, and with both the teacher and
> the students knowing the consequential mathematics.
>
> I believe this used to be the way it was done, but nowadays there are
> many mathematical history courses that are taught to students who don't
> know much mathematics, and which therefore consist of a few tidbits
> about number-systems and the like, which are quite interesting but
> mathematically trivial, and better displaced by teaching real mathematics.
>
> John Conway
>
I have some empathy with the above attitude. I refused to teach history
of mathematics, and more generally history of science, to freshman and
sophomore students on such grounds as these. It can also be argued that
studying history of mathematics at whatever level can actually interfere
with a student's education in mathematics. For example, it might stifle
creativity and originality when a student sees how much has been done in
some areas, and something about the ways in which it was done. When I was
on a postdotoral fellowship at Princeton for three years back in the early
1960s, I remember some faculty members arguing along those lines, e.g. John
Milnor.
However, the original question, as I understood it, was not whether or not
people should teach history of mathematics at this or that level, but
whether or not one should try to teach or maybe even learn some history of
mathematics without having read or being able to read the work one is
teaching about in the languages in which the work was written. This seems
to me to be a different matter, and utopian in the pejorative sense.
In passing, let me tell an anecdote from my experience of teaching history
of mathematics to beginning graduate students. I decided one semester to
build a history course around such matters as elliptic integrals and
elliptic functions, as worked out by a variety of 19th century
mathematicians, with some excursion into related matters involving
differential equations. To begin with, I started with a discussion of what
I assumed would be quite familiar to the students, some topics from
elementary calculus such as the definitions of inverse trigonometric
functions as integrals taken as leading to definitions of the direct
trigonometric functions, as compared with defining the trigonometic
functions geometrically, or as solutions of appropriate differential
equations. I was upended when I discovered that for most of the students,
this wasn't familiar material at all. On the whole, they knew a fair bit
about various matters of abstract algebra, some foundations of calculus as
learned in "advanced calculus" courses, something about elementary
topology, a little number theory, and the like, but they had "forgotten
their calculus", as they put it. So I had to lower my sights. I suppose
one could argue that I should have converted the history course to a kind
of second time through elementary calculus at a more advanced level.
Actually, now that I think about it, I believe that's what I did do, for 6
weeks or so.
Gordon Fisher gfisher@shentel.net