Subject: Re: [HM] Mathematics and Time
From: Gordon Fisher (gfisher@shentel.net)
Date: Thu Mar 09 2000 - 13:49:59 EST
"Ralph A. Raimi" wrote:
[large deletion]
>
>
> All this has nothing to do with time as the *subject* of study, as
> with Galileo. Time has been the oldest study of mathematics I can think
> of, with calendars long preceding geometric models for the heavens. But
> while it is inevitable that whatever mathematics is used for calendars
> gets confused with the thing measured, which is time, and not only time
> as unrolled in the course of conversation, it is not inevitable that a
> systematized geometry will get that confused. Euclid states the
> Pythagorean theorem and then its converse. People tend to think of the
> right angle as the *cause* of the sum of the squares property because the
> converse is not much emphasized in schoolbooks, but Euclid knew better,
> and I don't think time as such is visible in Euclid's system.
>
I agree that time as such is not visible in Euclid's system. I note that Plato,
as a predecessor of Euclid, spent some effort on discussing and recommending
that mathematics "proper", or as we now say, "pure" mathematics, is or should be
conducted from a timeless point of view, and that this is one of Plato's prime
steps toward a general theory of what we now call Platonic Ideas, claimed to be
"eternal verities", in the domain of which Truth (with a capital T) resides.
Aristotle, on the other hand, disagreed with Plato to some fair extent, in this
connection.
>
> Nor is it visible (to me) in arithmetic, of whatever era.
>
> But the answer to the question of when the *world* realized --
> explicitly -- that all mathematics was time-independent is one I'd like to
> hear about. Of course it is implicit in the ancient geometry, and when it
> comes to the arithmetic of positive integers it probably clearly so even
> for the man in the street, though unworthy of explicit mention. Time may
> well enter his conversation when talking about adding a column of figures,
> just as the entries might seem to be the *cause* of the total, but even
> the man in the street will admit time as a metaphor only. That the column
> is inextricably connected to the total and has been since the beginning of
> the universe, is quite plain. And when epsilons and quantifiers entered
> analysis, it was not for nothing that the result was called
> "arithmetization", since it rendered static, like arithmetic, what
> previously was being confused with motion and intuitions of continuity.
>
I would say that it isn't that "time" that is implicit in Euclid's work, but
rather "an absence of time". Also, while mathematicians of the 17th century,
including Newton and Leibniz, usually, perhaps always, worked with time as a
basic constituent of mathematics, I would argue that they were in fact engaged
in what we now call "applied mathematics".
I wouldn't say that Newton and Leibniz were "confused" in regarding limits as
essentially involved with time, but rather that they can be considered as
applying what *later* became formulated timelessly (Weierstrass via Cauchy, and
all that), to a physical world in which time is an ingredient. Aside from
limites, when Newton used his version of Euclid's work in the *Principia* , for
example, I take him to have been *applying* Euclid to what we now think of as
physics, although Newton would no doubt have said something to the effect that
he was using Euclid in doing "natural philosophy", as the full title of the
*Principia* indicates -- "The Mathematical Principles of Natural Philosophy".
>
> So I believe the time-independence of mathematics was, apart from
> clearly metaphorical language rooted in the fact that we speak and write
> linearly, so obvious in ancient times that it did not deserve mention.
But time is mentioned, quite a lot, in the works of Plato and Aristotle, and in
some of the works of the pre-Socratic Greek philosophers. Think, for example,
of Zeno's paradoxes, or more generally the contrast between the things we have
that Heraclitus said, and the things Parmenides said. Numerous historians of
philosophy, and classic scholars, have characterized as one of the major topics
of ancient Greek philosophy, contrasts between what is in time and what is
timeless. To paraphrase Karl Marx, a specter was haunting Greek natural
philosophy -- the specter of time and decay, versus timelessness and eternity.
Think, for example, of the works of Aristotle such as what can be translated as
"On Coming-to-Be and Passing-Away" (de generatione et corruptione), as
contrasted, for example, with his work called "Metaphysics".
>
> Things got muddier when mathematics began to be applied to dynamics, and
> the confused (1600-1900) talk about variables and limits, echoes of which
> are serious troubles in school mathematics to this day, has not helped.
> So I would say that the linking of mathematics *in principle* with
> considerations of time was a period with a beginning *and* an end, roughly
> speaking the period when the notion of function was being discovered and
> elaborated, beginning perhaps with the medieval Scholastics' discussion of
> change (uniform, difform, etc.) and ending with Weierstrasse and Dedekind.
> I'm speaking of the European tradition, I should say, since I do not know
> even this much much about other places.
>
I would say that Aristotle began application of time to dynamics in, for
example, his attempted "law" of falling bodies, which, as it turned out, only
works fairly well for objects falling in a viscous medium, such as honey or
molasses (cf. what we nowadays sometimes call Stoke's theorem in fluid
dynamics).
>
> And since I am not a historian at all, I look forward to comments
> from those who know more. It seems easy to me to see how time got *into*
> mathematics; I'd like to know how it got out, and especially how it got
> *known* that it was gone. Minnick didn't realize it even sixty years ago,
> and he was a teacher of teachers of mathematics, some of whom are still
> with us.
>
So I suggest that time got out of mathematics at least as long ago as ancient
Greece, and that it was very likely to have been known that it had been taken
out, e.g. by Euclid.
The question of variables and limits and all that, is another matter, as far as
I'm concerned. The explicit introduction of these, and explicit comparisons
between concepts of these which involve time and concepts which don't involve
time belongs seem to belong principally to a later era than that of ancient
Greece. As you note, Galileo can perhaps be said to have been concerned with
such matters explicitly, and so was Descartes and others of that era. I note in
passing, though, that both of these, however, have been shown to be working in
this regard on the basis of various thinkers of the European Middle ages. Cf.
the work of Annaliese Meyer, Marshall Clagett and his students, Alexandre Koyre,
and others.
Those of us brought up, mathematically speaking, in the 20th century, have
usually been educated to think of such concerns as becoming a central theme of
mathematics beginning in the 19th century, perhaps with the work of Bolzano, and
Cauchy, accelerated up through Weierstrass and others who worked to further the
so-called program for the "arithmetization of analysis". I think there can be
no doubt that Kant and Hegel and others of similar persuasions had influence on
the development and attention paid to this program, sometimes fairly directly
(recall, for example, that Hilbert paid homage to Kant on a number of occasions)
and sometimes indirectly, in ways that occur in cultures and societies.
Gordon Fisher gfisher@shentel.net
Prof Emeritus, Math & Comp Sci
James Madison Univ home: 344 Franklin St
Harrisonburg VA 22801
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