The Chronicle of Higher Education

June 23, 2000

Uncertainty Is Certain in Mathematics and Life



 "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

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.