Subject: Re: [HM] Mathematics and Time
From: Ralph A. Raimi (rarm@math.rochester.edu)
Date: Thu Mar 09 2000 - 12:08:52 EST
Clark Kimberling <ck6@evansville.edu> began this discussion about
mathematics and time by noting:
> One difference between mathematics and other sciences is the dependence of
> the latter on time...
> These musings lead to a historical question: how far back in time can we
> date recorded recognitions of the time-independence of mathematics?
He took as an example a definition of a sequence by induction, for
which English provides a convenient set of words invoking before and
after, now and later; indeed "successor", which is at the root of the
Peano system, invokes, at least etymologically, the notion of time.
Kimberling went on to point out that time is of course not essential here,
the language of quantifiers making such statements complete when
written in any order; he asked his historical question with that
background in mind.
The comments that have reached HM since his question have
concerned just the other question: In what way *does* or *did* our
intuition of time get mixed up with our mathematical ideas. I don't
believe one has to drag in Hamilton or Kant to answer this; it is obvious,
since all our conversation and writing is linear, one part of it following
the other in time as we speak or read. There is hardly another literate
way, and it might be that even two and three dimensional works of art are
read and appreciated with earlier and later perceptions. No college
freshman will write: "Then Socrates is mortal; Assume Socrates is a man;
Assume all men are mortal."
However, to put the Socrates argument in that order changes
nothing. After all, the German habit of putting the verb at the end of a
sentence always impresses English ears as suspenseful, though saving the
direct object for the end might impress Germans that way. Both German and
English manage to say the same things, and recognition of this fact is
exactly the thing Kimberling was asking about in mathematics: When did it
become explicit?
It has been part of the training of a mathematician in the 20th century
to recognize something of this sort, beginning with the 19th Century
developments in "the arithmetization of analysis", Weierstrasse and all
that, though clearly this recognition is far from universal even today, or
at least quite recently, among people whose understanding of those
developments was less than adequate.
For example, J.H. Minnick, a professor of mathematics education at
the University of Pennsylvania and later Dean of the school of education
there, wrote a book called *Teaching Mathematics in The Secondary Schools*
(Prentice-Hall, 1939), in which the following occurs on page 253:
If k is the limit of the variable x, then three things must be true:
1. k must be a constant quantity.
2. It must be possible to make the difference between x and k less than
any chosen quantity,however small, that is, it must be possible to make x
come as near to k as may be desired; and
3. When the difference between x and k has become less than any desired
quantity, it must be impossible by the same process to make it larger.
Observe the "has become less than", and "by the same process".
This sort of thing was being taught to future high school teachers even in
the days I went to college. What Minnick had in mind there, and which
gave rise to the oversight in (3), where he implies limits must be defined
only for monotone functions, was the proof that a line parallel to the
base of a triangle divides the sides proportionately, in the case of an
incommensurable division of the sides. His method is to use a line
parallel to the base, but very close to the base and interior to the
triangle, situated so that the partitions formed by the upper vertis, the
original line parallel to the base, and the new "false bottom" *are*
commensurable. He then permits or causes the false bottom to creep down
to the actual base. Unfortunately, Minnick did not understand Euclid (or
Dedekind), and never did give a definition for the word "limit". Or for
"quantity", for that matter, or for incommensurable ratio. But the idea of
a continuum of time in the "process" made such definitions superfluous in
his mind.
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.
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.
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.
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.
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.
Ralph A. Raimi Tel. 716 275 4429 or (home) 716 244 9368
Dept. of Mathematics FAX 716 244 6631
University of Rochester Webpage http://www.math.rochester.edu/u/rarm
Rochester, NY 14627 (Webpage contains links to papers)
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