Julio Gonzalez Cabillon, has told me, very reasonably, to stick
to history since "There are many interesting issues in mathematics but
this is not the place to discuss all of them; our topic is the *history
of mathematics*."
He suggests that I take my discussion private unless I have issues
focused on the interests of serious students of the history of mathematics.
I'm hereby doing that. I'd be grateful if anyone with an interest
in the subject matter I've spoken of would e-mail me, as some have already
done. I'll correspond with those who have shown an interest.
I'll read *historia-matematica* with interest, as I have been doing
for some months, and will respond to postings involving me here. If I feel
the need to respond at much length, I'll put the long part of the response
on my web page, and cite its address, rather than filling space here.
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I'd like a closure here that DOES relate to the history of
mathematics, and also to my interest in this *historia-matematica* group.
I've said the following, and then asked "Is this thinkable?"
There is an error - an "obvious" but wrong assumption made in
Newton's time and not changed since. It is at the level of arithmetical
procedure. There is an unrecognized but important constraint on permitted
scale of algebraic simplification on certain crossterm quantities.
The error is based on an unexamined belief that the arithmetic
of the real numbers can be projected, without modification or
augmentation, onto physical circumstances. The error can
generate false infinities, and false infinitesimals. The error
is as old as our models of coupled physical circumstances, which
is to say, at least as old as Newton.
I've gotten significant help on the question "Is this THINKABLE?".
That help is much appreciated. I expect that the help given will be
practically useful.
To address the "Is this THINKABLE" question, Samuel S. Kutler and
Gordon Fisher asked very good, clarifying questions. They asked what I
was speaking of as mathematics. Their questions weren't historical, but
mathematical-technical. So were my answers.
I'd argue that when my point about modeling becomes "thinkable" -
when it is thought to be well defined, then it becomes of historical
interest. It is not that the matter is of only historical interest.
But it does seem central to discussions of "the history of troubles
between mathematics and physical representation."
If I were to suggest a particular place where the issue might be
of special historical importance, I'd choose the work and many troubles
of James Clerk Maxwell. Many physicists would name Maxwell as one of
the three greatest physicists in history (others often nominated are
Newton and Einstein). Maxwell is also interesting because an enormous
fraction of his career was spent struggling with modeling of coupled
physical circumstances. To a remarkable extent, it was professionals'
reactions to the FAILURES of Clerk Maxwell that caused mathematics and
physics to separate as disciplines. To a remarkable extent, the
doctrines and cultural differences that make physics and mathematic so
different today, and make both so different from their 19thcentury
counterparts, key off of Maxwell's failures to "mathematically derive"
his physical laws.
I believe that Maxwell's troubles, and some troubles before and
since, relate to a misunderstanding of the rules of physical representation.
These rules, themselves, are not "historical". I believe that the
implications of these rules are historical.
Thank you,
M. Robert Showalter