The
Chronicle of Higher Education

June
23, 2000

**By
DANIEL ROCKMORE**

"Consider a spherical cow. ..."
That's the way an old

calculus
problem I know of begins, at least here on the

Vermont/New
Hampshire border. The phrase touches on the odd

relationship
that reality has with some of mathematics, and

the
way in which mathematicians think and work.

The
description of a real cow is complicated. For starters,

the
shapes of any two real cows are different, and any

particular
cow can almost certainly not be described easily in

mathematical
terms. But rather than throw out the bovine with

the
bath water, the mathematician chooses to simplify the

situation
with abstraction, creating a problem inspired by

reality
but now mathematically tractable.

Having
simplified the problem to a spherical cow, we can now

proceed
precisely and logically, deriving truth upon truth

about
this platonic beast. The facts may or may not say

something
about real cows, but they will be forever consistent

with
a simplified -- spherical, non-four-footed, colorless,

headless
-- and unchanging model of reality. Spherical cows

allow
for universal truths; real cows don't.

Searching
for absolute truths about ideal objects -- that's

the
daily activity of many a mathematician, and it's not a

world
with which others usually have much contact. But if

you'd
like to visit this world for a few hours and see what

happens
when you try to prove theorems about life, then the

logical
thing would be to see the play Proof, written by David

Auburn
and currently at the Manhattan Theatre Club, under the

direction
of Daniel Sullivan.

Auburn
comes from an academic background -- his father was an

English
professor and is now a college dean. Auburn is not a

math
scholar, but studied calculus while at the University of

Chicago,
where, he told The New York Times, he "knew a lot of

science
and math guys." For background, Auburn said, he spoke

to
mathematicians and read a number of books.

Proof
stars Mary-Louise Parker as the 20-something Catherine,

daughter
of a famous University of Chicago mathematician. As

the
play opens, Catherine's father, Robert (Larry Bryggman),

has
just died after a long illness, probably schizophrenia,

during
which Catherine cared for him at the expense of

continuing
her own studies. She seems to have inherited her

father's
mathematical genius -- and, possibly, his disease.

The
play hinges on the correctness and provenance of a proof

of
a famous mathematical conjecture (a statement not yet known

to
be true or false) discovered among the hundred or so

notebooks
that Robert generated during his illness. The

manuscript
is found by Hal (Ben Shenkman), a former graduate

student
of Robert's, who has been going through the

professor's
papers looking for hidden gems. It turns out that

Hal
is interested in more than Robert's papers -- he has

harbored
a crush on Catherine ever since meeting her several

years
previously, while working on his dissertation.

This
trio of mathematicians is counterbalanced by Catherine's

sister,
Claire (Johanna Day), a levelheaded businesswoman who

arrives
on the scene to help take care of final arrangements.

Her
secondary purpose is to persuade her sister to return to

New
York with her, primarily so that Catherine will be well

cared
for should she turn out to have the same troubles as her

father.

Is
the proof correct? Who is the true author of the proof? Is

Catherine
doomed to madness? Will Hal and Catherine find love

with
each other, or will Catherine depart for New York? Those

are
the dramatic tensions driving the play.

As
the subdramas unfold, Proof demonstrates some beautiful and

subtle
insights about, and comparisons between, mathematical

and
real-life proof.

Proof
of a mathematical fact is the easier to confirm.

Assuming
that a person knows the language and has the

background,
anyone could, in theory, check all of the steps

and
decide on the correctness of a proof, and any two persons

would
make the same judgment. Moreover, proofs of most

interesting
theorems -- and, in particular, the theorem hinted

at
in the play -- are general enough to treat an infinity of

possibilities
at once.

The
theorem of the play is about prime numbers -- all of them,

including
those that nobody has written down yet. Since there

is
an infinity of primes, those that have not yet been

discovered
do exist. Consequently, any mathematical statement

about
an infinity of objects could not be confirmed one by

one,
since at any given point in time only a finite number of

cases
would be addressed.

For
example, one of the most important conjectures in

mathematics,
the Riemann Hypothesis -- which concerns the

distribution
of prime numbers -- has been shown to be true in

more
than one billion cases, which is more experimental

confirmation
than members of the species Homo sapiens have had

of
the rising of the sun. Thus, while the Riemann Hypothesis

remains
classified as undecided, in the words of another

Broadway
play, we'd all bet our bottom dollars on the sun's

coming
out tomorrow.

In
statements about life, proofs of similarly absolute

certainty
are difficult, if not impossible, to derive. People

are
neither abstractions nor instances of general theories.

But
as mathematicians, Catherine and Hal can't seem to keep

themselves
from foisting this misplaced paradigm of certainty

upon
their own lives.

Hal
conjectures the authorship of the proof, and then does his

best
to settle the conjecture. We watch him bring all of his

logical
tools to bear on the question, and the process mimics

that
which any mathematician might go through in attacking a

conjecture.
Evidence is accumulated for and against the

conjecture.
Different approaches are tried. He attempts to

distance
himself from his feelings for Catherine and his

particular
knowledge of the principals involved, thereby

"abstracting"
the setting. Hal is looking for an airtight

argument,
which by the end of the play certainly seems

convincing
-- yet still could be wrong.

In
life, if not in math, the axiomatic method does not provide

a
good tool for predicting the future. Personality traits or

genes
are not axioms pointing to some inescapable conclusion

--
at best, they're mental ticklers, worriers, and warnings.

Nevertheless,
we see Catherine trying to prove or disprove to

herself
that she is doomed to repeat her father's demise. In

her
eyes, a string of implications points frighteningly to a

necessary
madness.

As
they grapple with such issues, Hal gives as good a

description
as I've ever heard of the beauty that can be found

in
a wonderful mathematical argument. Like a great romance

novel,
a beautiful proof can be full of twists and turns,

dashing
heroes, and surprise appearances of characters whose

import
is only slowly revealed -- all sewn together with a

driving
narrative line that compels the reader ever onward

toward
a satisfying and inevitable conclusion.

Hal
also touches upon the frustration ofresearch, and the

stereotypes
of the young genius and the math geek (although

there
seems to be overcompensation in his discussion of the

latter).
The portrayal of Robert's legendary creativity and

illness
recalls the true story of the Nobel laureate John

Nash,
who, before battling mental illness, established

important
principles of game theory (rivalry among competitors

with
conflicting interests).

Despite
Hal's protestations, the suggestion that Catherine has

inherited
her father's blend of genius and madness --

especially
in juxtaposition with Hal's normalcy and

concomitant
fears of intellectual mediocrity -- lends a bit

too
much credence, for this viewer's taste, to the equation:

Intelligentsia
equals dementia.

That's
a small quibble with a wonderful drama that elegantly

describes
the world of mathematics, and suggests how

ill-suited
the mathematical notion of truth is for life. It's

impossible
to divine the future, and it's no easier to derive

it.
We're only as certain as our next best guess. Genius can

turn
to madness, love to hate, and joy to sadness. You make

that
best guess based on what you know, and you move on.

So
says Claire, and the viewpoint is echoed in her

professional
choice to apply mathematics to everyday life.

She's
willing to bet on the sunrise, and ultimately, for all

of
Hal's and Catherine's careful, axiomatic reasoning, each of

them,
too, is required to make a few leaps of faith to get on

with
things. And so the boundary has been drawn; life this

side,
please, math the other.

Or
has it? The play ... no, actually it's math itself ... has

further
twists of plot. For within mathematical research,

there
is analogous acknowledgment of the limitations of formal

reasoning.
They are delineated by Godel's Theorem.

Briefly
put, Godel proved that in any finite collection of

logically
consistent axioms, there must be statements that can

neither
be derived from the axioms, norshown to be false by

reasoning
from the axioms. Even if those undecidable

statements
are appended to the list of axioms, as long as this

enlarged
system remains consistent, there will still be other

such
undecidable statements. Even in mathematics, logic has

its
limits.

So
neither in math nor in life does knowledge foretell all. In

both
realms, as long as you have a sensible notion of truth,

it
will include a sensible notion of mystery. And, confronted

by
mystery, all you can do is dive into it and see if things

work
out. Emerge successfully with new truths, and new

mysteries,
too, cling to you.

As
Proof draws to its conclusion, the implications of the

characters'
decisions are far from determined. Like Catherine,

Hal,
and Claire, we find reasons to take stock, to be awed, to

be
uneasy and hopeful. Those are the effects that a good

proof,
or a good play, can have, and one needn't be a genius

or
a madman to appreciate them.

Daniel
Rockmore is a professor of mathematics and computer

science
at Dartmouth College.