__Mathematics as an Aesthetic Discipline__

** **

*J.D. Phillips*

*Department of Mathematical Sciences*

*Saint Mary's College*

*Moraga, California 94575*

** **

** **

**0. INTRODUCTION **

This brief paper offers a defense of the study of
mathematics. It is intended for those people who are convinced either
that mathematics is not worth studying or that mathematics is "just not
for them."

This paper is especially intended for those humanists and the
literati who thrive in the world of art, music and literature, but who think
that mathematics is a mechanical, cold, unimaginative discipline, suitable only
for unartistic, uncreative "computer-types." This paper will
suggest that these humanists have confused mathematics with the discipline that
went by that name in their schooling. In short, this paper will suggest that
the literati who think that the study of mathematics needs defending are
completely unaware of the meaning of the word "mathematics."
And thus, they are really demanding a defense of something else; namely, the
memorization of formulae and equations and the mechanical manipulation of
numbers that was forced upon them in school. They will find no such
defense here.

The reader is alerted to a caveat: this paper is not intended for
those who find the entire academic enterprise in need of defending. Those
who demand a defense of the study of music, poetry, philosophy, biology,
chemistry, and mathematics, are advised to look elsewhere. They will not
find it here.

**1. THE COMMON DEFENSE**

** **

Usually, the study of mathematics is defended almost exclusively
along the lines of its effectiveness as an instrument. Legions of
so-called "mathematics" teachers attempt to sell mathematics to their
students as nothing more than a manipulative and a practical tool.

Of course mathematics is useful and practical as a utensil, but
only to the professional scientist and engineer. Almost everyone else
will use no more "mathematics" in their everyday life than the
simplest of grammar school arithmetic: balancing a check book, counting
change. One needs little more. The notion that anyone other
than a scientist will *ever* use even the most elementary trigonometry or
algebra is laughable. Imagine the absurdity of being in a car or on a
plane when suddenly the need arises to solve a quadratic equation or to graph a
trigonometric function. But this is precisely* *the scenario that the
traditional defense has coerced us into accepting as realistic. Clearly
this is absurd. And so is our complicity.

Of course, students realize this. They become apathetic or
openly hostile towards this "mathematics." And who can blame
them? Why should anyone care about mathematics if its only value is its
practicality, a practicality relegated either to the simplest of childish
arithmetic or to the arcanely out of reach, complex world of the professional
scientistÍs mathematics? If this is mathematics, then something is wrong
with the student who likes* *mathematics!

** **

** **"*My early teachers chanted the notion
of practical value like a litany. It was repeated at each level, in each
course, from grade one*

* through high school. They meant to justify
mathematics on the basis of its utility in the conduct of one's daily life.*

* *

* There is nothing wrong with this except they
went too far and claimed too much. Mathematics is useful in this sense.
But, with*

*this narrow connotation of 'value,' a
little goes a long way. Counting change, measuring carpet, or balancing
one's checkbook *

*requires only the slimmest knowledge of
mathematics. From early on, I wondered why such pedestrian activity
required so much schooling.*

* *

* The
true value of mathematics lies outside commonplace activity*." Jerry P. King
[Ki].

** **

** **

**2. THE NEW DEFENSE**

** **

The new defense of the study of mathematics does not rely on the
utility of mathematics. The cornerstone of this new defense is the beauty
of mathematics, a notion singularly alien to the general public.

**2.1.
****AESTHETICS**

We study mathematics for the same reasons we study poetry or music
or painting or literature: for aesthetic reasons. Simply put, we study
mathematics because it is one of the loveliest disciplines known to man.

** **"*A mathematician, like a painter or a poet, is a
maker of patterns. ... The mathematician's patterns, like the painter's
or the*

*poet's, must be beautiful; the ideas, like
the colours or the words, must fit together in a harmonious way. Beauty
is the*

*first test: there is no permanent place in
the world for ugly mathematics*." G. H. Hardy [Ha].

* "...this
character of beauty and elegance [in mathematics is], capable of developing in
us a sort of aesthetic emotion." *Henri Poincare [Po].

"*There
is, first of all, the motivating force for mathematics which is beauty."* Jerry King
[Ki].

** **

The fashioners of this sublime beauty, artists indeed, must
possess a rare creativity and an imagination of the highest order.

"*The moving power of mathematical invention is
not reasoning but imagination*." Augustus de Morgan [HA].

"*There
is an astonishing imagination even in the science of mathematics ...We repeat,
there is far more imagination in the*

* head of Archimedes than in that of Homer*."
Voltaire [HA].

"*The
essence of mathematics is its freedom*." Georg Cantor [HA].

* "The science of pure mathematics, in its modern
developments, may claim to be the most original creation of the *

*human spirit." *A.N. Whitehead [Wh].

One of the most compelling aesthetic features of mathematics is
its refined austerity. Its unadorned gracefulness is unique among the
arts. In fact, part of the very essence of mathematics is its
precision. People are referring to this quality when they suggest that
mathematics teaches "clear thinking." Mathematics' precision
does not lie in any claims of universal truth. But rather this precision,
and hence power, lie in the acknowledgement of exactly the points at which
mathematics consciously and deliberately abandons claims of universal
truth. Mathematics is the only discipline that I am aware of that does
this. And this precision and austerity allow for an elegant economy, an
economy that comes from the elimination of the cluttering mire of imprecision.

* "Strange as it may sound, the power
of mathematics rests on its evasion of all unnecessary thought and on its
wonderful*

* saving of mental operations.*" Ernest
Mach [Be].

* *

*"Mathematics is precise or it is
nothing*." Jerry P. King [Ki].

*"Mathematical knowledge adds vigour to the mind,
frees it from prejudice, credulity, and superstition." *John Arbuthnot [Mo].

*"One cannot escape the feeling that these
mathematical formulae have an independent existence and an intelligence of*

*their own, that they are wiser than we
are, wiser even than their discoverers, that we get more out of them than we *

*originally put into them."* Heinrich Hertz
[HA].

*"Calculus is the most powerful weapon of thought yet
devised by the wit of man."* W.B. Smith [Mo].

** **

The mathematician, however, is not merely an ascetic, cold and
austere. He or she is an expressive artist involved in the richly human
struggle to create and to discover.

** **"*...a mathematician experiences in his
work the same expression as an artist; his pleasure is as great and of the same
nature.*"

Henri Poincare [Be].

* "I have heard myself accused of
being an opponent, an enemy of mathematics, which no one can value more highly
than I,*

* for it accomplishes the very thing whose
achievement has been denied me." *Goethe [Be].

* "A mathematician who is not also
something of a poet will never be a complete mathematician." * Karl
Weierstrass [Mo].

*"Other qualities of a far more subtle sort, chief
among which in both cases is imagination, go to the making of a good artist or*

*a good mathematician." *Maxime Bocher [Mo].

Sadly, most people, including the otherwise sensitive and
culturally sophisticated, are completely unaware of the intrinsic aesthetic
features of mathematics.

** **

* "The useful combinations are precisely the
most beautiful, I mean those best able to charm. This charm is the special*

*sensibility that all mathematicians know,
but of which the profane are so ignorant as often to be tempted to smile."
*

Henri Poincare [Po].

"*Nothing lives further from the
intellectual experience of members of the educated public than the notion that*

*mathematics can have aesthetic
value." *Jerry P. King [Ki].

** **

The common defense is not, however, supplanted by the new defense,
but rather it is subsumed by it. This subsumption takes the unexpected
form of an *appreciation* for the utility of mathematics. By this I mean
that to most students of mathematics, the utility of mathematics should be
presented in something like the same fashion as music is presented to students
of music history, namely as a marvel to be appreciated, not an instrument to be
operated. Those students interested in actually creating music (i.e. in
becoming musicians or composers) are advised to study performance or
composition. Similarly, those students interested in actually harnessing
the utilitarian powers of mathematics (i.e. in becoming engineers and scientists
and mathematicians) are advised to study engineering and applied
mathematics. But for the vast majority of mathematics students, a simple,
honest *appreciation* of the remarkable utility of mathematics should be seen
as the ultimate "real world" goal. In short, the sense of agency
developed in most students regarding the utility of mathematics should be of an
appreciative nature, not an instrumental nature. And since
"appreciation" is an aesthetic term, not a scientific term, for most
students, the traditional defense of the study of mathematics as a tool is
subsumed by the aesthetic perspective of the new defense.

** **

** ***"There is no branch of mathematics,
however abstract, which may not someday be applied to phenomena of the real
world." *

Nicolai Lobachevsky [HA].

*"The mathematician, carried along on his flood
of symbols, dealing apparently with purely formal truths, may still*

*reach results of endless importance for
our description of the physical universe."* Karl Pearson
[Be].

*"Algebra is the intellectual instrument which has been
created for rendering clear the quantitative aspects of*

* the world." *A.N. Whitehead [HA].

*"Mathematics is the queen of the sciences."* Carl Fredrich
Gauss [Be].

* "It is mathematics that offers the exact
mathematical sciences a certain measure of security which, without mathematics,*

*they could not obtain." * Albert
Einstein [Be].

*"A book on the new physics, if not purely descriptive
of experimental work, must be essentially mathematical."*

P.A.M. Dirac [Di].

*"The great book of nature can be read only by those
who know the language in which it was written. And this *

*language is mathematics."* Galileo [Be].

** **

** **

**2.2. GREAT THINGS**

The study of great things, including the study of great ideas,
needs no defense. And many of the greatest of human thoughts have taken
the form of mathematics.

** **"*. . . not the mere fact of living is
to be desired but the art of living in the contemplation of great things*."

Bertrand Russell [Ru].

*"This therefore is Mathematics, she reminds you of the
invisible forms of the soul; she gives life to her own*

*discoveries; she awakens the mind and
purifies the intellect; she brings light to our intrinsic ideas; she abolishes*

*oblivion and ignorance which are ours by
birth."* Proculus Diadochus [HA].

* "Mathematics
is the only good metaphysics." *Lord Kelvin [Be].

* "To create a healthy philosophy you should renounce
metaphysics but be a good mathematician." *Bertrand Russell
[Be].

* "Number rules the universe,"* Pythagoras
[Be].

*"God ever geometrizes." *Plato [Be].

* "The Great Architect of the Universe now begins to
appear as a pure mathematician." *J.H. Jeans [Je].

Mathematics is created by human beings. It was not carved on
tablets and handed down by a god. The most brilliant members of our
species have exerted, and continue to exert, the noblest effort to give us this
mathematics.

When school children study analytic geometry, they should be made
aware that this seemingly trivial and esoteric subject exists to us only
because of the heroic efforts of a succession of brilliant minds, culminating
in the work of Descartes. Its depth, originality and profundity are lost
on students. It has been carefully polished and refined so exquisitely,
presented so elegantly and simply, that students myopically receive it as a
trifle.

** **"*Though the idea behind it all is childishly
simple, yet the method of analytic geometry is so powerful that very ordinary*

* boys of seventeen can use it to prove
results which would have baffled the greatest of the Greek geometers - Euclid, *

*Archimedes, and Apollonius. The man,
Descartes, who finally crystallized this great method had a particularly full *

*and interesting life."* E.T. Bell
[Be].

* "(Analytic geometry), far more than any of his
metaphysical speculations immortalized the name of Descartes, and *

*constitutes the greatest single step ever
made in the progress of the exact sciences." *John Stuart Mill
[Be].

** **

When calculus students give a sleepy, disinterested yawn during
the discussion of the fundamental theorem of calculus, they should be told that
the most outstanding human minds struggled for over two millennia to find this
seductively simple formula. Until Newton and Liebnitz finally uncovered
it for us, no human eyes had ever gazed upon it, although the greatest
intellects had searched for it.

Today, we present this masterpiece to teenage students in a ten
minute lecture. And students receive it in the same spirit that it's
presented : as just another boring, god-given, inhuman formula to
memorize. Clearly this is unacceptable. Students must learn that
mathematics is the most human of endeavors. Flesh and blood
representatives of their own species engaged in a centuries long creative
struggle to uncover and to erect this magnificent edifice. And the
struggle goes on today. On the very campuses where mathematics is
presented and received as an inhuman discipline, cold and dead, new mathematics
is created. As sure as the tides.

Students deserve the truth: Mathematics is vibrant and
dynamic, an incredibly rich and human discipline, a liberal art and a humanity
in the purest sense.

** **

"*...the mathematics of a mathematician is
profoundly personal.*" Seymore A. Papert [Pa].

"*Although mathematics itself is 2,500
years old, more has been created in the last fifty years than in all the
previous ages*

* combined. ..." *Jerry King [Ki].

* *

"*In mathematics alone each generation builds a new
story to the old structure*." Hermann Hankel [Kl].

"*(Arithmetic) is one of the oldest branches, perhaps
the very oldest branch, of human knowledge; and yet some of its most *

*abstruse secrets lie close to its tritest
truths.*" H.J.S. Smith [Be].

Educated men and women, from the dilettante to the
cognoscente, must be at least modestly literate in all fields of
intellectual inquiry. Imagine the literate who is not acquainted with the
theories of evolution, relativity or quantum mechanics. Imagine the
sophisticate who is unfamiliar with the works of Shakespeare, Picasso or
Mahler. But most so-called educated people *know nothing *of mathematics.

"*One's
intellectual and aesthetic life cannot be complete unless it includes an
appreciation of the power and the beauty of *

*mathematics. Simply put, aesthetic
and intellectual fulfillment requires that you know about mathematics*." Jerry
P. King [Ki].

* What is there about mathematics that compels so many men
and women to work at it with the fervor of dedicated artists*

*and yet keeps it simultaneously outside
the experience of the rest of intellectual society? * Jerry P. King
[Ki].

"*Outside of the closed circle of professional
mathematicians, almost nothing is known of the true nature of mathematics *

*or of mathematics research.*" Jerry P.
King [Ki].

Most people, at least most 20th century Americans, are interested
in the lives of public figures. Even the lives of some intellectuals are
of interest to the average citizen: Einstein is a pop icon. __Amadeus__,
a movie about Mozart, was a popular success. There was a recent movie about the
physicist Stephen Hawking shown in American popular movie houses. The
rank and file recognize references to artists and thinkers as diverse as
Heisenberg, Schroedinger, Beethoven, Picasso, Stravinsky, Monet, Plato,
Aristotle, Freud, Jung, Camus and Sartre. But almost no one knows even
the names of the most important mathematicians. Who but the mathematician
has heard of Gauss, Galois, Cantor? They are thinkers of the first
rank. But unlike their counterparts in every other discipline, their
names are completely unfamiliar. Clearly, if the masses were aware of the
humanness of the mathematics enterprise, natural human curiosity would demand
that mathematicians be included in the class of thinkers worth knowing.

* "Those
who have never known a professional mathematician may be rather surprised on
meeting some, for mathematicians *

*as a class are probably less familiar to
the general reader than any other group of brain workers. The
mathematician is a *

*much rarer character in fiction than his
cousin the scientist." *E.T. Bell [Be].

The human essence includes an amazingly robust sense of
wonder. If students realize that they have been banned access to a
tremendously rich body of knowledge (mathematics),* *this natural wonder,
if properly cultivated, will transform the "banned" into the
"tempting." And students will demand to know of it.
Bertrand Russell perfectly captured this refined sense of wonder in his
autobiography. It is a fitting epigram for this paper.

** **"*There was a footpath leading across
fields to New Southgate, and I used to go there alone to watch the sunset and
contemplate *

*suicide. I did not, however, commit
suicide, because I wished to know more of mathematics*." Bertrand
Russell [Ru]**.**

** **

** **

**References**

** **

[Be] E.T, Bell, Men of Mathematics,
Simon and Schuster, New York, 1937.

[Di] P.A.M. Dirac, Quantum
Mechanics, 1930.

[HA] A. P. Hillman, G. L. Alexanderson, A
First Undergraduate Course in Abstract Algebra,

Wadsworth, Belmont, CA, 1973.

[Ki] J.P. King, The Art of
Mathematics, Plenum, New York, 1992.

[Ha] G.H. Hardy, A Mathematician's
Apology, Cambridge University Press, New York, 1940.

[Je] J.H. Jeans, The
Mysterious Universe, Cambridge University Press,1931.

[Kl] M. Kline, Mathematical
Thought from Ancient to Modern Times, Oxford University

Press, New York, 1972.

[Mo] R. E. Moritz, Memorabilia
Mathematica, The Mathematical Association of America,

Washington, D. C., 1914.

[Pa] S.A. Papert, The
Mathematical Unconscious, in *On Aesthetics and Science*, ed.

JWeschsler, Birkhauser, Boston, 1988.

[Po] H. Poincare, The Foundations of
Science, Science Press, New York,1929.

[Ru] B. Russell, The Autobiography
of Bertrand Russell, Bantom, New York, 1951.

[Wh] A.N. Whitehead, Science and the
Modern World, 1925.