Date: Jul 20, 1995 10:47 AM
Author: Chih-Han sah
Subject: Long Post on What is Math? Dimension? Symmetry?


Historically, there was not much difference between a
mathematician and a physicist. Both were viewed as 'natural philosophers'.
Each person did both. I think it is fair to say that the separation
(some would even say divorce) began near the beginning of the 20-th century.
The two disciplines not only separated, but, like many others, each further
bifurcated into *pure* and *applied*. *Flaming* sometimes became very
heated. It is only in the last 10 years or so that physics and mathematics
are getting back together. In trying to tackle *fundamental questions
about nature*, physicists have had to come to grip with some of the most
*abstract ideas* of mathematics. In the process, physicists have introduced
not only *new mathematics* but even led the way to many *new results*
in mathematics to the extent that some physicists would refer
*pure mathematics* as *applied physics* (!).

Probably, these *fundamental questions* may be summarized by the
title of the book:

H. Weyl, Space, Time and Matter, (German original, 1921, English
ed. 1950, now available in Dover Publ.)

Weyl was considered by many as one of the greatest 20-th century
mathematician/philosopher/physicist...., a universalist. He and Einstein
were both original members of the Institute for Advanced Studies at
Princeton, an independent institution from Princeton University.

The first 10 pages (Introduction) to Weyl's book are worth
reading--the first *formula* began on p. 8 and followed by a few more
that can be recognized as the equations of lines (t = at' + b, or,
y = ax + b). These were used in the definition of the algebraic
concept of a *group*. Its introduction was often attributed to
E. Galois in his fundamental work (his last letter
to his friend the night before the fatal duel had been considered
by a number of mathematicians as the most important work in human
history--measured by the ratio of ideas vs. the number of pages--needless
to say, it was decades later before the world of mathematics began to
appreciate what Galois had done), the germs were already contained
in Euclid's Elements. Weyl had authored the following books
(available in Dover Publications) among others:

The theory of groups and quantum mechanics (German original, 1928).
Symmetry (Princeton University Press, 1952).

The first of these is extremely difficult to read. Mathematicians who
were able to understand the first were viewed by many (in years past)
to have moved to a different *dimension* (!). On the other hand, the second
one was perhaps the first book with such a title and can be read by
everyone. For modern physicists, *symmetry* has now become the guiding
principle. One of the greatest admirers of Herman Weyl is the Nobel
Laureate in Physics, Chen Ning Yang (who overlapped with Weyl at Princeton's
IAS) and is himself a physicist/philosopher/mathematician.....(the
reverse alphabetical ordering is mine since Yang considers himself as
a physicist first--his father was a mathematician, there is an interesting
interview in Math. Intelligencer (1992 ?) with the title: C. N. Yang
and Contemporary Mathematics). There is also the article:

C. N. Yang, Hermann Weyl's Contribution to Physics, in
Hermann Weyl 1885-1985, Springer Verlag, 1986.
(This requires quite a bit of mathematics, but
readable if you just skip over the 'hieroglyphics'.)

C. N. Yang, Symmetry and Physics, in Oskar Klein Memorial
Lectures, vol. 1. World Scientific Publishing.
(This is very readable.)

To end this part, I should mention that the famous physicist James Jean
suggested in a meeting of physicists back in the beginning of the
century that the physicists should decide what is important mathematics.
He started by saying: The first thing we should throw out is this
concept of "group". It is totally useless! The phrase "grouppest"
can still be found on some occasions in physics arena but perhaps with
some sort of "affection" since it is one of the most basic things used
by many physicists (unfortunately, the precise definition is often lost
in the 'translation'). Perhaps it is a good thing that we are talking
about "group learning" (not the same as "learning about groups", but
then one of the main points of modern physics is the "non-commutative"
facets of "nature").

Let me now move to my re-reading of the classic:

R. Courant and H. Robbins, What is Mathematics? Oxford, 1941
(Dover 4-th ed)

Courant was generally thought of as an *applied mathematician* or a
*mathematical physicist*. He was instrumental in establishing the
Courant Institute of Mathematical Sciences at NYU. I think it is
fair to say that the above book was Courant's personal attempt with
the help of Robbins to clarify in his own mind the difficult question:

What is Mathematics?

The 5 pages at the very beginning are still very much to the point.
I would like to mention that Courant and his illustrious colleagues
were aware of the fact that

"recorded mathematics begins in the Orient, where,
about 2,000 B.C., the Babylonians collected a great
wealth of material that we would classify today under
elementary algebra".

However, they were less knowledgeable about the mathematical developments in
Eastern Asia. The problem was that they did not have access to the
original sources. Leaving all these aside, I find that the 5 pages
ended with:

Fortunately, creative minds forget dogmatic philosophical
beliefs whenever adherence to them would impede constructive
achievement. For scholar and layman alike it is not philosophy
but active experience in mathematics itself that alone can
answer the question: What is Mathematics?

A more careful reading of the book shows that it was 'heavier' in the
theoretical side than the applied side. The selected topics covered
must now be updated in view of the recent achievements in mathematics
such as the resolution of the Four Color Theorem and the Fermat's Last
Theorem. The first depends on heavy use of computer and we do not
really have what traditionalist would call a *rigorous* proof. Apparently,
the *computer assisted proof* is *robust*. The second is a theoretical
tour de force. I should mention, at the height of the FLT frenzy, there
appeared a physics preprint in the high energy physics electronic
BB where the author asserted that solutions of the Fermat's
equation had something to do with the *breaking of supersymmetry*. What
seems to be less speculative is that the entire family of Fermat's
equations (viewed over the complex numbers), after a long, long tortuous
journey, led to problems related to the *speculative* magnetic monopoles
but in hyperbolic 3-space.

Let me now examine the word "dimension". In its daily English
use, I think it is possible to connect it up with Space, Time and Matter.
It is related to *space* as in the "3-dimensional space" we live in.
It is related to *time* as in *time is the 4-th dimension*. It is related
to matter as in "what is the dimension of this filing cabinet?" etc.

Very loosely, the word appears to convey the qualitative idea of
"size". Mathematicians would then try to make the idea more precise.
One might view this as a *mathematical task*. One thing that all of us
like to do is to simplify and clarify. However, in the process, we are
quite likely to have to make things much more complicated and muddy.

Physicists often use the word "dimension" interchangeably
with "the number of degrees of freedom". The idea here is roughly:
how many numbers do you need to locate a point in space? For most of us,
the answer is 3. If I were to use numbers in their decimal expansion
and adopt the convension that numbers must not end in an infinite string
of zeroes, then I can record 3 positive numbers as a single positive number.
Thus, if the three numbers are: 1/2, 1/3 and 1/4, I could first write:

1/2 = 0.49999999....
1/3 = 0.33333333....
1/4 = 0.24999999....

After that, I simply sandwich them and record all three in the form:

0.432934939939939....

Watch out: the backward decoding process may lead to trouble. For
example: the positive number 0.120129129.... and the positive number
0.121120120120.... would both correspond to the three positive numbers:

1/9, 2/9, 1/10

In spite of this shortcoming, this coding process leads to the idea
that somehow the 3-dimesional space has been compressed into a 1-dimensional
space. This is of course great for computers. But it is somehow very
counter-intuitive.

In trying to clarify our intuitive *physical* understanding,
mathematicians found out that there is a variety of ways of defining
*dimension* for *reasonable spaces*. In every case, many definitions
have to be introduced and precise theorems proved. For example,
the least complicated one (still very tortuous) comes out of linear
algebra in the setting of a *vector space* over a *field*. One has
to learn the process of solving a system of linear equations, something
that Chinese mathematicians already learned around 100 BC-100 AD using
what is now called the Gauss Elimination Method. After a long shaggy
dog like tale, we finally come to the *Theorem*.

3-dimensional Euclidean space really has *dimension 3*.

By this time, the more pragmatic people have long left the room.
Rather than trying to prolong this tortuous discussion, let me
cite another book:

W. Hureciwz and H. Wallman, Dimension Theory, Princeton
University Press, 1948.

Hurewicz was a topologist at MIT. There was some claim that the
science fiction story: A subway named Moebius was a take off on
Hurewicz riding in the Boston subway system lost in thoughts.

On p. 4, one finds a definition due to Menger and Urysohn which is inductive.

a) The empty set has dimension -1.
b) The dimension of a *space* is the least integer n for
which every *point* has arbitrarily *small neighborhoods*
whose *boundaries* have dimension less than n.

Thus, the dimension is an integer.

Here are some exercises:

Exer 1. If a space A has dimension n, then any subspace
has dimension at most n.

Exer 2. The space of integers has dimension 0.

Exer 3. The numbers line has dimension 1. So is the perimeter
of a circle.

Exer 4. The plane has dimension 2. So is the surface of a solid
sphere.

Exer 5. Euclidean 3-space really has dimension 3.

As a result, a topologist and an algebraist come to the same common agreement
through two totally different routes while the returning physicist shakes
her/his head and say: why don't you guys go and solve some *real world
problems*.

The book marches on and on until p. 102. There one finds that
in 1937, Szpilrajn found a connection between the concept of *dimension*
and the concept of *measure*. Measure theory is an abstract generalization
of integral calculus. Back in 1917, F. Hausdorff (he wrote one of the
earliest texts on Set Theory--I only know the German original: Mengenlehre)
introduced the concept of dimension using an interplay of analysis and
topology. On p. 107, we find the statement:

For a *metric space X*--meaning *separable metric space*,

Hausdorff dim of X is at least equal to dim of X.

To show that equality does not have to hold, one has the bombshell:

The Hausdorff dimension of the Cantor *middle third set*
is ln 2/ln 3 = 0.63093.....

The Cantor middle third set is an example of a *nowhere dense*
*perfect* set. Here, perfect = closed + dense in itself.
Mathematicians often love to find such un-intended (or perhaps intended)
in-jokes. Here is the construction.

It begins with the closed interval [0,1]. One first removes
the open middle third (1/3,2/3) to get two remaining intervals.
From each, remove their open middle third and repeat this
process ad infinitum and what is left over is the the
Cantor Middle Third Set (Exer. Why is there anything left?
after all, a simply calculuation would have shown that the
total length of the removed intervals add up to 1. Yes, one
has to sum a geometric series.)

Some 30 years ago, *main stream* mathematicians avoid this stuff like
the plague. In recent years, one finds *fractals*, *Hausdorff dimensions*
*snowflakes*, bantered about all over the places.

Is this Mathematics? Is this Physics? Is this Marine Biology?
Is this Archeology? Is this Art?

(For the last three, I refer back to pictures in H. Weyl, Symmetry.)

I don't know the answer, but I tend to agree with Courant
and Robbins's conclusion.

Han Sah, sah@math.sunysb.edu