“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 of research, 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, nor shown 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.

http://chronicle.com Section: Opinion & Arts Page: B9