[Gordon Fisher wrote:]
>
>> >Did Grattan-Guiness
>> >mean we just stick to the records of an epoch when we are describing
>> >mathematical work of that epoch, and not be guided by a current state of
>> >mathematics, as we see it? If the latter, we are faced with vexing
>> >historiographical questions about how we can sufficiently eliminate
>> >relevant parts of ourselves and our cultures from our historical
>> >interpretations,
>
>It is my experience that my undergraduate students have very
>little knowledge of the mathematics of our own modern culture.
>I find that in order to make any sense out of the history,
>I first have to teach them some of the more recent stuff.
>For instance, in order to explain what infinitesimals are
>and what Dedekind's definition of the reals meant and
>why angles can't be trisected and several
>other historical topics, I first have to give a modern definition
>of a "field." I suppose things could be taught in chronological
>order, but I think it would take much much longer that way.
>The modern, abstract definition gives us a framework in which
>we can talk about all these older topics, skipping over the
>old confusions and misunderstandings -- or perhaps discussing
>some of the confusions and misunderstandings *after* we've
>considered the correct explanations.
>
>I guess this makes me a "Whig" historian. Can someone recommend
>one or two elementary things I should read, to begin to reduce
>my Whigginess?
>
>
>
Teaching history of mathematics to undergraduates and doing research in
history of mathematics are, I think, two quite separate endeavors, not
always happily married. In fact, an attempt at marriage might lead to
divorce.
I remember deciding to teach a graduate course in history of mathematics in
which for some reason I wanted to sketch a history of elliptic integrals
and elliptic functions in the 19th century. I was jolted aback when I
started with what I thought all the students would know, or at least recall
with little effort, as a prelude to Abel's treatment using the notion of
inverses, namely a development of trigonometric functions as inverses of
appropriate algebraic functions which the students would have seen in
elementary calculus.
I found that most of the students had no memory to speak of concerning this
topic (they'd studied it when they were sophomores or so), nor could many
of them grasp with any ease the idea of "doing it the other way around",
i.e. starting with the inverse trigonometric functions (as integrals) and
proceeding to the trigonometric functions. So I spent some time doing
elementary calculus in a graduate history of mathematics course, and
naturally never was able to treat elliptic integrals and functions the way
I'd hoped for.
As to using axioms for a field as a background for treating aspects of the
history of mathematics which were developed before or along with
present-day attitudes, it seems like a reasonable way to connect what
undergraduate students probably have studied in an algebra course or two
with what happened over the course of a few thousand years. One no doubt
should emphasize, though, that explicit formulation of a set of field
axioms as we now customarily present them early in the career of a
mathematics student did not emerge until relatively late in the game,
although some implicit use preceded the explicit formulation.
By the way (he asked innocently), what do you think infinitesimals are?
Gordon Fisher gfisher@shentel.net