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
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
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
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.