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.