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