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Proof

Proof


What do the statements 2 + 2 = 4 and "the sky is blue" have in common? One might say that they are both true. The statement about the sky can be confirmed by going outside and observing the color of the sky. How, then, can one confirm the truth of the statement 2 + 2 = 4? A statement in mathematics is considered true, or valid, on the basis of whether or not the statement can be proved within its mathematical system.

What does it mean to prove something mathematically? A mathematical proof is a convincing argument that is made up of logical steps, each of which is a valid deduction from a beginning statement that is known to be true. The reasons used to validate each step can be definitions or assumptions or statements that have been previously proved.

As an example of a proof, look at finding the sum of the first n whole numbers.

The sum of the first three whole numbers is 1 + 2 + 3, so n = 3, and the sum is 6.

The sum for n = 4 is 1 + 2 + 3 + 4, and this sum is 10.

Is there a pattern here? Can the sum be found (without doing all the addition) for n = 8?

In 1787, a teacher gave this problem to a 10-year-old boy, Carl Friedrich Gauss. Gauss pointed out the following pattern:

What is the sum of each of the number pairs indicated by the arrows? The sum is four pairs that sum to 9, and there are eight numbers in all.

What would the mean of these numbers be? The mean is (4 × 9) divided by 8. The mean of the whole set is which is also the mean of the first and last numbers in the set. The sum of eight numbers, whose mean is , is 8 , or 4 × 9, which is 36. Maybe the sum of the first n whole numbers can be found by finding the mean of the first and last whole numbers, and then multiplying that mean by n.

Try the following pattern to see if it works for other values of n. What happens if an odd number is chosen for n ?

Is the sum the same as the mean of 9 and 1 ( or 5) multiplied by n (which is 9 in this case)? Is the sum 5 × 9, or 45?

Is the sum of the first n whole numbers always equal to the mean of the first and last multiplied by n ? This seems to be true, but in mathematics even a huge number of examples is not enough to prove the truth of the statement. Therefore, a proof is needed.

The first number in the sum is always 1. The last number in the sum is always n. The mean of n and 1 is according to the definition of mean. In algebra, n multiplied by this mean looks like . A proof must show that the sum of the first n whole numbers is always .

First, consider n as an even number. In that case, there is an even number of pairs, and each pair has a sum of n + 1, regardless of the size of n. The truth of this pattern does not depend on the size of n, as long as n is an even number so pairs can be made, each of which adds to (n + 1).

There are pairs, each of which adds to (n + 1), so the total is or . This means that the pattern is proved true as long as n is an even number.

Next, consider the case where n is an odd number:

The circled number in the middle will always be the mean of the first and last numbers. In algebra, the middle number will be . So the middle number adds one more mean to the (n 1) means that were made by the paired numbers.

So, again, the total sum is n multiplied by the mean of the first and last numbers, or . These two cases, for n, an even number, and n, an odd number, together make up the proof.

There are several forms of mathematical proofs. The one just given is a direct proof. Indirect reasoning, or proof by contradiction, can also be used. A third kind of proof is called mathematical induction.

Although many examples do not prove a statement, one counterexample is enough to disprove a statement. For example, is it true that y + y = y × y ? Try substituting values of 0 and then 2 for y. Although the statement is true for 0 and 2, it is not true in general. One counter-example is y = 1, since 1 + 1 is not equal to 1 × 1 because 2 is not equal to 1.

Here is a well-known proof that 0 = 1. Try to find the flaw, or mistake, in this proof.

  1. Assume that x = 0. Assumption
  2. x (x 1) = 0 Multiplying each side by (x 1)
  3. (x 1) = 0 Dividing each side by x
  4. x = 1 Adding 1 to each side
  5. 0 = 1 Substitute x = 0, the original assumption

All the steps except one are valid. In Step 3, the proof divided each side by x. The reason for this is that, if a = b, then if c is not equal to 0. But the original assumption said that x was equal to 0, so Step 3 involved dividing by 0, which is undefined. Allowing division by 0 can lead to proving all sorts of untruths!

see also Induction.

Lucia McKay

Bibliography

Bergamina, David, and editors of Life. Mathematics. New York: Time Incorporated, 1963.

Hogben, Lancelot. Mathematics in the Making. New York: Crescent Books, Incorporated, 1960.

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Proof

Proof

A proof is a logical argument demonstrating that a specific statement, proposition, or mathematical formula is true. It consists of a set of assumptions (also called premises) that are combined according to logical rules in order to establish a valid conclusion. This validation can take one of two forms. In a direct proof, a given conclusion can be shown to be true. In an indirect proof, a given conclusion can be shown not to be false and, therefore, presumably to be true.

Direct proofs

A direct proof begins with one or more axioms or facts. An axiom is a statement that is accepted as true without being proved. Axioms are also called postulates. Facts are statements that have been proved to be true to the general satisfaction of all mathematicians and scientists. In either case, a direct proof begins with a statement that everyone can agree with as being true. As an example, one might start a proof by saying that all healthy cows have four legs. It seems likely that all reasonable people would agree that this statement is true.

The next step in developing a proof is to develop a series of true statements based on the beginning axioms and/or facts. This series of statements is known as the argument of the proof. A key factor in any proof is to be certain that all of the statements in the argument are, in fact, true statements. If such is the case, one can use the initial axioms and/or facts and the statements in the argument to produce a final statement, a proof, that can also be regarded as true.

As a simple example, consider the statement: "The Sun rises every morning." That statement can be considered to be either an axiom or fact. It is unlikely that anyone will disagree it.

One might then look at a clock and make a second statement: "The clock says 6:00 a.m." If we can trust that the clock is in working order, then this statement can be regarded as a true statementthe first statement in the argument for this proof.

The next statement might be to say that "6:00 a.m. represents morning." Again, this statement would appear to be one with which everyone could agree.

The conclusion that can be drawn, then, is: "The Sun will rise today." The conclusion is based on axioms or facts and a series of two true statements, all of which can be trusted. The statement "The Sun will rise today" has been proved.

Indirect proofs

Situations exist in which a statement cannot be proved easily by direct methods. It may be easier to disprove the opposite of that statement. For example, suppose we begin with the statement "Cats do not meow." One could find various ways to show that that statement is not truethat it is, in fact, false. If we can prove that the statement "Cats do not meow" is false, then it follows that the opposite statement "Cats meow" is true, or at least probably true.

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proof (in mathematics)

proof, in mathematics, finite sequence of propositions each of which is either an axiom or follows from preceding propositions by one of the rules of logical inference (see symbolic logic). Mathematical proofs are quite distinct from inductive, statistical, heuristic, analogical, and other types of reasoning or persuasion that are sometimes accepted as proofs in other fields of science or human affairs. Proof theory has developed into one of the important branches of modern mathematical logic. Some schools of mathematical logic reject certain methods in proofs, such as use of the law of excluded middle (either p is true or p is false) or of mathematical definitions involving properties that are not effectively verifiable.

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proof

proof Informally, a form of deduction associated with a deductive logic. More formally, when applied to a formal system F, a proof is a sequence of well-formed formulas with each item in the sequence being either an axiom of F or being derived from previous items through the application of an inference rule of F.

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proof

proof the proof of the pudding is in the eating proverbial saying, early 14th century, meaning that the truth of an assertion will be demonstrated by how things actually turn out; proof here means ‘test’ rather than the more normal ‘verification, proving to be true.’

See also burden of proof.

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Proof

PROOF

Proof is the means of ascertaining the truth of an alleged fact or proposition. It may consist in presenting empirical evidence, documents, or witnesses. More often it is taken to mean a reasoning process. Proof is inductive if it proceeds from the singular to the universal, or from the less to the more universal. It is deductive (syllogism) if it proceeds from the more to the less universal, or from one universal to another coextensive universal. From the viewpoint of truth-value, deductive proof is either demonstration, in which certitude is attained, or dialectics, in which probability is attained. Reasoning from hypothesis and analogy yields dialectical conclusions. Statistical proofs are a mode of induction. The ancients assigned proper modes of proof also to rhetoric and poetics.

See Also: argumentation; demonstration; dialectics; induction.

[m. a. glutz]

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Proof

PROOF

The establishment of a fact by the use of evidence. Anything that can make a person believe that a fact or proposition is true or false. It is distinguishable from evidence in that proof is a broad term comprehending everything that may be adduced at a trial, whereas evidence is a narrow term describing certain types of proof that can be admitted at trial.

The phrase burden of proof includes two distinct concepts, the burden of persuasion and the burden of going forward. The burden of persuasion is the duty of a party to convince the trier of fact of all the elements of a cause of action. The burden of going forward refers to the need of a party to refute evidence introduced at trial that damages or discredits his or her position in the action. The burden of persuasion remains with the plaintiff or prosecutor throughout the action, whereas the burden of going forward can shift between the parties during the trial.

In a civil action, the requisite degree of proof is a preponderance of the evidence.The plaintiff must show that more probably than not the defendant violated his or her rights. In a criminal action, the prosecutor has the burden of establishing guilt beyond a reasonable doubt.

cross-references

Preponderance of Evidence.

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proof

proofaloof, behoof, goof, hoof, pouffe, proof, roof, shadoof, spoof, Tartuffe, underproof, woof •flameproof • rainproof • greaseproof •heatproof • windproof • bulletproof •childproof • shockproof • mothproof •stormproof • soundproof • foolproof •sunroof • ovenproof • rustproof •fireproof • showerproof •shatterproof • waterproof •weatherproof

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proof

proof / proōf/ • n. 1. evidence or argument establishing or helping to establish a fact or the truth of a statement: you will be asked to give proof of your identity| this is not a proof for the existence of God. ∎  Law the spoken or written evidence in a trial. ∎  the action or process of establishing the truth of a statement: it shifts the onus of proof in convictions from the police to the public. ∎ archaic a test or trial. ∎  a series of stages in the resolution of a mathematical or philosophical problem. 2. a trial print of something, in particular: ∎  Printing a trial impression of a page, taken from type or film and used for making corrections before final printing. ∎  a trial photographic print made for initial selection. ∎  each of a number of impressions from an engraved plate, esp. (in commercial printing) of a limited number before the ordinary issue is printed and before an inscription or signature is added. ∎  any of various preliminary impressions of coins struck as specimens. 3. the strength of distilled alcoholic liquor, relative to proof spirit taken as a standard of 100: [in comb.] powerful 132-proof rum. • adj. 1. able to withstand something damaging; resistant: the marine battle armor was proof against most weapons| [in comb.] the system comes with idiot-proof instructions. 2. denoting a trial impression of a page or printed work: a proof copy is sent up for checking. • v. [tr.] 1. make (fabric) waterproof: [as adj.] (proofed) the tent is made from proofed nylon. 2. make a proof of (a printed work, engraving, etc.): [as n.] (proofing) proofing could be done on a low-cost printer. ∎  proofread (a text): a book about dinosaurs was being proofed by the publisher. 3. activate (yeast) by the addition of liquid. ∎  knead (dough) until light and smooth. ∎  [intr.] (of dough) prove: shape into a baguette and let proof for a few minutes.

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proof

proof that which makes good a statement XIII; action of proving or testing XIV; something produced as a test XVI. Later ME. prōf (pl. prōves), superseding earlier prēf, prēve — OF. pr(o)eve, prueve (mod. preuve):- Late L. proba, f. probāre test, PROVE. The substitution of prōf for prēf was due to assim. to the vb.
Hence proof adj. of tested strength XVI; prob. from ellipsis of of in † armour of proof.

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Proof

Proof

Resources

In mathematics, along with the sciences, a proof is a logical argument demonstrating that a specific statement, proposition, or mathematical formula is true. It consists of a set of assumptions, or premises, which are combined according to logical rules (called rules of logic), to establish a valid conclusion. This validation can be achieved by direct proof that verifies the conclusion is true, or by indirect proof that establishes that it cannot be false.

The term proof is derived from the Latin probare, meaning to test. Greek philosopher and mathematician Thales Miletus (Turkey) (c.624c. 547 BC) is said to have introduced the first proofs into mathematics about 600 BC. A more complete mathematical system of testing, or proving, the truth of statements was set forth by Greek mathematician Euclid of Alexandra (c. 325c. 265 BC) in his geometry text, Elements, published around 300 BC. As proposed by Euclid, a proof is a valid argument from true premises to arrive at a conclusion. It consists of a set of assumptions (called axioms) linked by statements of deductive reasoning (known as an argument) to derive the proposition that is being proved (the conclusion). If the initial statement is agreed to be true, the final statement in the proof sequence establishes the truth of the theorem.

Each proof begins with one or more axioms, which are statements that are accepted as facts. Also known as postulates, these facts may be well known mathematical formulae for which proofs have already been established. They are followed by a sequence of true statements known as an argument. The argument is said to be valid if the conclusion is a logical consequence of the conjunction of its statements. If the argument does not support the conclusion, it is said to be a fallacy. These arguments may take several forms. One frequently used form can be generally stated as follows: If a statement of the form if p then q is assumed to be true, and if p is known to be true, then q must be true. This form follows the rule of detachment; in logic, it is called affirming the antecedent; and the Latin term modus ponens can also be used. However, just because the conclusion is known to be true does not necessarily mean the argument is valid. For example, a mathematics student may attempt a problem, make mistakes or leave out steps, and still get the correct answer. Even though the conclusion is true, the argument may not be valid.

The two fundamental types of proofs are direct and indirect. Direct proofs begin with a basic axiom and reach their conclusion through a sequence of statements (arguments) such that each statement is a logical consequence of the preceding statements. In other words, the conclusion is proved through a step-by-step process based on a key set of initial statements that are known or assumed to be true. For example, given the true statement that either John eats a pizza or John gets hungry and that John did not get hungry, it may be proved that John ate a pizza. In this example, let p and q denote the propositions:

p: John eats a pizza.

q: John gets hungry.

Using the symbols / for intersection and ~ for not, the premise can be written as follows: p/ q: Either John eats a pizza or John gets hungry. and ~q: John did not get hungry. (Where ~q denotes the opposite of q).

One of the fundamental laws of traditional logic, the law of contradiction, tells one that a statement must be true if its opposite is false. In this case, one is given q: John did not get hungry. Therefore, its opposite (q: John did get hungry) must be false. But the first axiom tells us that either p or q is true; therefore, if q is false, p must be true: John did eat a pizza.

In contrast, a statement may also be proven indirectly by invalidating its negation. This method is known as indirect proof, or proof by contradiction. This type of proof aims to directly validate a statement; instead, the premise is proven by showing that it cannot be false. Thus, by proving that the statement ~p is false, one indirectly proves that p is true. For example, by invalidating the statement cats do not

KEY TERMS

Axiom A basic statement of fact that is stipulated as true without being subject to proof.

Direct proof A type of proof in which the validity of the conclusion is directly established.

Euclid Greek mathematician who proposed the earliest form of geometry in his Elements, published circa 300 BC

Hypothesis In mathematics, usually a statement made merely as a starting point for logical argument.

meow, one can indirectly prove the statement cats meow. Proof by contradiction is also known as reductio ad absurdum. A famous example of reductio ad absurdum is the proof, attributed to Pythagoras (582500 BC), that the square root of 2 is an irrational number.

Other methods of formal proof include proof by exhaustion (in which the conclusion is established by testing all possible cases). For example, if experience tells one that cats meow, then one will conclude that all cats meow. This is an example of inductive inference, whereby a conclusion exceeds the information presented in the premises (there is no way of studying every individual cat). Inductive reasoning is widely used in science. Deductive reasoning, which is prominent in mathematical logic, is concerned with the formal relation between individual statements, and not with their content. In other words, the actual content of a statement is irrelevant. If the statement if p then q is true, q would be true if p is true, even if p and q stood for, respectively, The moon is a philosopher and Triangles never snore.

Resources

BOOKS

Azzouni, Jody. Tracking Reason: Proof, Consequence, and Truth. Oxford, UK, and New York: Oxford University Press, 2006.

Cupillari, Antonella. The Nuts and Bolts of Proof. Amsterdam, Netherlands, and Boston, MA: Elsevier Academic Press, 2005.

Hedman, Shawn. A First Course in Logic: An Introduction in Model Theory, Proof Theory, Computability, and Complexity. Oxford, UK: Oxford University Press, 2004.

Lay, Steven R. Analysis: With an Introduction to Proof. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

Lloyd, G.E.R. Early Greek Science: Thales to Aristotle. New York: W. W. Norton, 1970.

Solow, Daniel. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes. New York: Wiley, 2002.

Sundstrom, Theodore A. Mathematical Reasoning: Writing and Proof. Upper Saddle River, NJ: Prentice Hall, 2003.

Randy Schueller

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Proof

Proof

A proof is a logical argument demonstrating that a specific statement, proposition, or mathematical formula is true. It consists of a set of assumptions, or premises, which are combined according to logical rules, to establish a valid conclusion. This validation can be achieved by direct proof that verifies the conclusion is true, or by indirect proof that establishes that it cannot be false.

The term proof is derived from the Latin probare, meaning to test. The Greek philosopher and mathematician Thales is said to have introduced the first proofs into mathematics about 600 b.c. A more complete mathematical system of testing, or proving, the truth of statements was set forth by the Greek mathematician Euclid in his geometry text, Elements, published around 300 b.c. As proposed by Euclid, a proof is a valid argument from true premises to arrive at a conclusion. It consists of a set of assumptions (called axioms) linked by statements of deductive reasoning (known as an argument) to derive the proposition that is being proved (the conclusion). If the initial statement is agreed to be true, the final statement in the proof sequence establishes the truth of the theorem .

Each proof begins with one or more axioms, which are statements that are accepted as facts. Also known as postulates, these facts may be well known mathematical formulae for which proofs have already been established. They are followed by a sequence of true statements known as an argument. The argument is said to be valid if the conclusion is a logical consequence of the conjunction of its statements. If the argument does not support the conclusion, it is said to be a fallacy. These arguments may take several forms. One frequently used form can be generally stated as follows: If a statement of the form "if p then q" is assumed to be true, and if p is known to be true, then q must be true. This form follows the rule of detachment; in logic, it is called affirming the antecedent; and the Latin term modus ponens can also be used. However, just because the conclusion is known to be true does not necessarily mean the argument is valid. For example, a math student may attempt a problem, make mistakes or leave out steps, and still get the right answer. Even though the conclusion is true, the argument may not be valid.

The two fundamental types of proofs are direct and indirect. Direct proofs begin with a basic axiom and reach their conclusion through a sequence of statements (arguments) such that each statement is a logical consequence of the preceding statements. In other words, the conclusion is proved through a step by step process based on a key set of initial statements that are known or assumed to be true. For example, given the true statement that "either John eats a pizza or John gets hungry" and that "John did not get hungry," it may be proved that John ate a pizza. In this example, let p and q denote the propositions:

p: John eats a pizza.

q: John gets hungry.

Using the symbols / for "intersection" and ~ for "not," the premise can be written as follows: p/q: Either John eats a pizza or John gets hungry. and ~q: John did not get hungry. (Where ~q denotes the opposite of q).

One of the fundamental laws of traditional logic, the law of contradiction, tells us that a statement must be true if its opposite is false. In this case, we are given ~q: John did not get hungry. Therefore, its opposite (q: John did get hungry) must be false. But the first axiom tells us that either p or q is true; therefore, if q is false, p must be true: John did eat a pizza.

In contrast, a statement may also be proven indirectly by invalidating its negation. This method is known as indirect proof, or proof by contradiction. This type of proof aims to directly validate a statement; instead, the premise is proven by showing that it cannot be false. Thus, by proving that the statement ~p is false, we indirectly prove that p is true. For example, by invalidating the statement "cats do not meow," we indirectly prove the statement "cats meow." Proof by contradiction is also known as reductio ad absurdum. A famous example of reductio ad absurdum is the proof, attributed to Pythagoras, that the square root of 2 is an irrational number .

Other methods of formal proof include proof by exhaustion (in which the conclusion is established by testing all possible cases). For example, if experience tells us that cats meow, we will conclude that all cats meow. This is an example of inductive inference, whereby a conclusion exceeds the information presented in the premises (we have no way of studying every individual cat). Inductive reasoning is widely used in science. Deductive reasoning, which is prominent in mathematical logic, is concerned with the formal relation between individual statements, and not with their content. In other words, the actual content of a statement is irrelevant. If the statement" if p then q" is true, q would be true if p is true, even if p and q stood for, respectively, "The Moon is a philosopher" and "Triangles never snore."

Resources

books

Dunham, William. Journey Through Genius. New York: Wiley, 1990.

Fawcett, Harold P,. and Kenneth B. Cummins. The Teaching ofMathematics from Counting to Calculus. Columbus: Charles E. Merril, 1970.

Kline, Morris. Mathematics for the Nonmathematician. New York: Dover, 1967.

Lloyd, G.E.R. Early Greek Science: Thales to Aristotle. New York: W. W. Norton, 1970.

Salmon, Wesley C. Logic. 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1973.


Randy Schueller

KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Axiom

—A basic statement of fact that is stipulated as true without being subject to proof.

Direct proof

—A type of proof in which the validity of the conclusion is directly established.

Euclid

—Greek mathematician who proposed the earliest form of geometry in his Elements, published circa 300 b.c.

Hypothesis

—In mathematics, usually a statement made merely as a starting point for logical argument.

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Proof

Proof

DAVID AUBURN
2001

INTRODUCTION
AUTHOR BIOGRAPHY
PLOT SUMMARY
CHARACTERS
THEMES
STYLE
HISTORICAL CONTEXT
CRITICAL OVERVIEW
CRITICISM
SOURCES
FURTHER READING

INTRODUCTION

Proof (2001), a play by David Auburn, won the Pulitzer Prize in 2001, as well as several other major awards for drama. The play is set in Chicago, where Robert, a former genius of a mathematician who suffered from mental illness, has recently died. Robert appears in the play talking with his daughter Catherine, a depressed college drop-out who stayed at home and cared for her father over the last few years of his life. As preparations are made for the funeral and Catherine's sister Claire returns from New York, Catherine forms a tentative friendship with Hal, a mathematician who is one of her father's former students.

The plot moves into high gear when Hal discovers in one of the notebooks that Robert left behind a proof of a mathematical theorem that mathematicians had thought impossible. It is a sensational discovery, but Catherine stuns Hal by claiming she wrote the proof. But did she? The handwriting in the notebook looks very like her father's. As the mystery develops and resolves, the playwright explores issues such as what the link may be between genius and madness and whether either or both can be inherited. But Proof is also a story about human relationships, suggesting that developing trust and love can be as difficult, and just as uncertain, as establishing the truth of a mathematical proof.

AUTHOR BIOGRAPHY

David Auburn was born in Chicago in 1969. Raised in Ohio and Arkansas, he attended the University of Chicago where he studied political philosophy. At the time, Auburn did not know he wanted to be a writer, but he joined a student group that performed comedy sketches. Auburn started writing some of the sketches and found he had a talent for it. He then started to write longer pieces. After Auburn graduated in 1991, he won a writing fellowship offered by Steven Spielberg's Amblin Productions, and he moved to Los Angeles to learn the craft of screenwriting. When the fellowship ended, Auburn moved to New York where he wrote plays and had some of them performed in tiny theaters. During the day, Auburn worked as a copywriter for a chemical company. In 1994, Auburn was accepted into the playwriting program at Juillard, where he also studied acting. Auburn soon gave up acting to concentrate on playwriting. His work at Juillard led to his first major play, Skyscraper (1997). In this play, the lives of a group of people are changed as they discover their connections with each other during the demolition of a crumbling skyscraper in Chicago.

In 1998, the Dramatists' Play Service published several of Auburn's one-act plays under the title Fifth Planet and Other Plays. The title play charts the friendship between two observatory workers as it waxes and wanes over the course of a year. Other plays in the collection included Are You Ready ? in which the fates of three people drawn to the same restaurant are altered in an instant; Damage Control, about a politician and his aide during a crisis; Three Monologues, depicting a young woman's solitude; We Had a Very Good Time, in which a married couple travels to a dangerous foreign country, and What Do You Believe about the Future, in which ten characters answer the question posed by the play's title.

Proof, Auburn's most successful play, premiered at the Manhattan Theatre Club in May 2000 and opened at Broadway's Walter Kerr Theatre on October 24, 2000. The play won the Pulitzer Prize for Drama in 2001, the Joseph Kesselring Prize, the Drama Desk Award, and the Tony Award for Best Play of 2001. Auburn has written a screenplay based on the play, and the film was in production as of 2004.

Also in 2004, Auburn had his play The Journals of Mihail Sebastian debut at the Keen Company in New York on March 6. A one-man show, it is adapted from the writings of Mihail Sebastian, a Romanian novelist and playwright, whose journals recalling anti-Semitism in Romania during World War II were published in 1996. The expressionistic play covers six years in Sebastian's life, with the journal being created over the course of the evening.

PLOT SUMMARY

Act 1, Scene 1

Proof begins at one o'clock in the morning on the porch of a house in Chicago. Catherine sits in a chair, exhausted, and is startled when she realizes her father, Robert, is there. Robert gives her a bottle of champagne and wishes her happy birthday. He wants her to celebrate her birthday with friends, but she says she has none. Robert expresses concern about her, saying that she sleeps until noon, eats junk food, and does not work. He tells her to stop moping. She has potential and there is still time. It transpires that Robert did his best work by the time he was in his mid-twenties. After that, he became mentally ill. Catherine is worried that she will inherit the illness.

It then transpires that Robert died a week before, of heart failure, and the funeral is the next day.

Hal, a former student of Robert's, enters. He has been working on Robert's notebooks, but Catherine says there is nothing valuable in them. Hal invites her to hear him play in a rock band, but she is not interested. He speaks about how he admired her father, who helped him through a difficult period in his doctoral studies. This was four years ago, when Robert's illness went into remission. Catherine, fearing that Hal may be taking one of her father's notebooks from the house without permission, demands to see his backpack. She finds nothing there, but as he is about to leave, a notebook falls from his jacket pocket. She accuses him of stealing it and calls the police. He protests that in the notebook, Robert wrote something appreciative about Catherine on her birthday four years ago. Hal was going to wrap the notebook and give it to her.

Act 1, Scene 2

The next morning, Catherine and Claire, who has arrived from New York, are drinking coffee. Claire tries to be kind, but Catherine is not receptive. Claire quizzes Catherine about Hal and about why she called the police, but Catherine resents the questioning. Hal enters unexpectedly, and there is a moment of confusion as Catherine berates her sister. Hal quickly exits, leaving Claire saying that decisions must be made. She wants Catherine to stay with her in New York.

Act 1, Scene 3

That night, there is a party following the funeral. Catherine is on the porch when Hal, who has been playing in the band, approaches her. He compliments her on her dress and talks about how mathematicians consider they are past their peak after the age of twenty-three. He refers to them as men, but Catherine mentions Sophie Germain, an eighteenth-century Frenchwoman who did important work on prime numbers. Catherine apologizes for her behavior the day before, and Hal confides that he thinks his work in mathematics is trivial. They talk about how elegant Robert's work was. Catherine then surprises Hal by kissing him. Hal reminds her of when they first met, four years ago, and they kiss again.

Act 1, Scene 4

Hal and Catherine have spent the night together, and the next morning she gives him a key to the bottom drawer of her father's desk. Claire enters with a hangover. She tells Catherine that she would like her to move to New York. Catherine says she would prefer to stay in Chicago, but Claire replies that she has already sold the house. They quarrel. Catherine complains that Claire never helped to take care of their father; Claire replies that she worked fourteen-hour days so she could pay off the mortgage on the house. She says that Robert should have been sent to an institution, but Catherine disagrees. Hal returns with a notebook. Inside it, he says, is a proof of a theorem about prime numbers. If it checks out, it will show that when Robert was supposedly insane, he was doing some of the most important math work in the world. Catherine stuns him by saying that it was she who wrote it.

Act 2, Scene 1

It is a September afternoon four years earlier. Robert and Catherine talk on the porch. Catherine says she has enrolled as a math major at Northwestern. She tells him that since he has been well for nearly seven months, he does not need her there all the time. Robert is not happy about her decision and says she should have discussed it with him. Hal enters. At this time, he is Robert's graduate student, and he brings a draft of his dissertation. Robert says he will look it over and tells Hal to come by his office in a week. Then, he realizes that it is Catherine's birthday, and he had forgotten it. He is annoyed with himself, but Catherine tells him not to worry. They agree to go out to dinner. As Catherine goes out to dress, Robert begins writing in his notebook.

Act 2, Scene 2

Catherine, Hal, and Claire discuss the newly discovered notebook. Catherine insists that she wrote the proof, working on it for years after she dropped out of school. Hal and Claire are skeptical. Claire thinks the proof is written in her father's handwriting. She suggests that Catherine talk them through it to convince them, but Hal says that would not prove anything, since her father might have written it and explained it to Catherine later. Catherine is unhappy that they do not believe her. She says she trusted Hal and wanted him to be the first person to see the proof. He still cannot believe that she wrote it, since to do so she would have to have been as good as her father. After Catherine snaps at him, he exits. Catherine and Claire struggle over the notebook and Catherine throws it to the floor.

Act 2, Scene 3

The next day, Claire berates Hal for taking advantage of Catherine and sleeping with her. She refuses to let Hal talk to her, but she does let him take the notebook. She tells Hal to figure out what is in there and advise the family about what to do.

Act 2, Scene 4

It is winter, three and a half years earlier. Robert is on the porch in the cold, writing in a notebook. When Catherine, who is a student at Northwestern, arrives, he tells her that he is working again. He feels he has regained all his intellectual brilliance and is excited about what he will be able to produce. He wants her to collaborate with him and hands her his notebook, which Catherine reads slowly. It is confused, rambling nonsense. She puts her arm around him and takes him inside the house.

Act 2, Scene 5

Back in the present, Claire and Catherine prepare to leave for New York. At first, they appear to be getting on well, but when Claire tells her how much she will love New York, Catherine gives sarcastic replies, and the two women quarrel. Claire exits, upset. Hal enters. He is excited. The proof has checked out. Catherine is not surprised and tells him to publish it. He now believes that it is her work because it uses new mathematical techniques that he thinks Robert would not have known. He wants Catherine to talk about her work so he can understand it better. Catherine is upset that he did not trust her in the first place. He hands her the book. She says that doing the proof was just a matter of connecting the dots. Her father knew nothing of her work. Hal asks her to go through it with him, and she picks up the book, finds a section, looks at him, and begins speaking.

CHARACTERS

Catherine

Catherine is Robert's twenty-five-year-old daughter. A college dropout, she has spent several years at home caring for her mentally ill father. A few years earlier, when his illness went into remission for almost a year, she enrolled as a sophomore at Northwestern University in Evanston, Illinois. She dropped out of that program and returned to look after her father when he again became ill. Their relationship, although sometimes antagonistic on the surface, was sustained by strong mutual affection.

Catherine is worried that she may inherit her father's illness, and the signs of mental instability are already there. Although she is a highly intelligent woman, she has no direction in life and often, according to her father, sleeps till noon. Some days she does not even get out of bed. She is obviously suffering from depression, and her attitude about life is bitter. Claire, her sister, wants her to move to New York so she can keep an eye on her and arrange for the best medical treatment, but Catherine resents her interference. Evidence of her unstable mental condition emerges in Claire's report of her aggressive behavior toward the police officers who came to the house after Catherine reported a burglary in progress (which was her extreme reaction to Hal's attempt to smuggle out one of her father's notebooks).

Hal attempts to befriend Catherine. She then takes the lead and seduces him. Wanting to show affection and trust, she allows him to discover the amazing mathematical proof that she has written in one of her father's notebooks. She is upset when Hal does not believe she wrote it and feels that her trust in him has been betrayed. Eventually, Hal is convinced that she wrote the proof, and the mathematical genius that Catherine inherited from her father is finally revealed and acknowledged. It appears that Catherine and Hal may be on their way to a rewarding relationship, both professionally and personally.

Claire

Claire is Catherine's efficient, practical, and successful sister. Unlike Catherine, she has inherited none of her father's erratic genius. Instead, she has made a career in New York as a currency analyst. She made enough money to pay off the mortgage on the family home in Chicago, even when she was living in a studio apartment in Brooklyn, New York. Claire lives with her boyfriend, Mitch, who also has a successful career, and they plan to marry in January. Claire and Catherine have never gotten along well, and when Claire returns from New York for their father's funeral, they quarrel. Claire feels responsible for Catherine's welfare and wants her to move to New York, but Catherine resents what she sees as Claire's interference in her life. It transpires that they have quarreled in the past over how to care for their father. Claire thought he should be sent to an institution, but Catherine believed it was important for him to remain near the university. Claire has little understanding of Catherine and regards her as mentally ill, but she means well and takes her family responsibilities seriously.

Hal

Hal, whose full name is Harold Dobbs, is a twenty-eight-year-old mathematician who teaches at the University of Chicago. He also plays drums in a rock band made up of mathematicians. Hal is a former student of Robert's, whom he admires immensely, not only for the brilliance of his achievements in mathematics but because Robert helped him through a bad patch in his doctoral studies. Hal first met Catherine briefly four years earlier, and when he meets her again, he tries to make friends with her. He seems rather shy and inexperienced with women, and it is she who seduces him rather than the other way round. After they spend the night together, he is ready to fall in love with her. Hal also confides in Catherine that he is dissatisfied with the progress of his career. His academic papers are being rejected by journals, and he feels that his work is trivial. Although he does not openly acknowledge it, this is one of the underlying reasons that he is examining Robert's notebooks. If he can discover something important, it will boost his career and perhaps make a name for himself. He is thrilled when he finds the proof in Robert's notebook and takes some convincing by Catherine that it is her work. This harms their relationship, since Catherine is annoyed that he does not believe her. When Hal is convinced, he reacts with humility rather than jealousy. He tries to repair their relationship and asks Catherine to go over the proof with him so he can ask questions and understand it better.

Robert

Robert was a famous mathematician who has just died of a heart attack in his fifties. He is already dead when the play begins, but he appears in the first scene in Catherine's imagination and returns in two later scenes, which flash back to earlier years. Robert was a mathematical genius. When he was in his early twenties, he made major contributions to game theory, algebraic geometry, and nonlinear operator theory. According to Hal, his former graduate student, he invented the mathematical techniques for studying rational behavior. While he was still in his twenties, Robert was afflicted by a serious mental illness, which dogged the remainder of his life. He became so incapacitated that his daughter Catherine had to stay at home to care for him. Robert had a deep affection for Catherine. He realized the sacrifices she made in caring for him, and he believed that she saved his life. Robert was also worried that she appeared to be wasting her life. Four years before his death, Robert's illness went into remission, and he was able to teach again for one academic year. During that year, Robert thought he was back at his best and would once more be able to do exciting, pioneering work in mathematics. He even asked Catherine if she would collaborate with him, but she soon found out that his notebooks were full of nonsense; his mind was confused, and he was lapsing into insanity.

THEMES

Genius and Madness

Robert and Catherine, the two mathematical geniuses, are brilliant but mentally unstable, and they are contrasted with the other two characters, Hal and Claire, who lack the genius of the other two but are better adjusted to the world.

MEDIA
ADAPTATIONS

Proof was adapted to film and set to release in the United States some time in 2005. The screenplay is written by David Auburn and Rebecca Miller. Directed by John Madden, the film stars Anthony Hopkins as Robert, Hope Davis as Claire, Jake Gyllenhaal as Hal, and Gwyneth Paltrow as Catherine. Paltrow reprises her role as Catherine, which she played on stage in London's West End.

Robert revolutionized the field of mathematics when he was in his early twenties, but he has waged a long battle with mental illness. The implication is that the illness is somehow connected with his genius. Another implication, in addition to the fact that genius, at least in this case, appears to be inherited, is that insanity may be inherited too. Catherine worries about this possibility, and although Robert tries to reassure her that it is not the case, she too shows signs of mental instability. She is too depressed to function effectively, and her life is not moving in a positive direction. She is bitter and finds it hard to trust the good intentions of others. And yet she is as brilliant as her father. Genius is therefore presented as a fragile thing; it can produce great intellectual achievements but may be inimical to personal happiness and stability. There is a price to pay for being an extraordinary individual.

The genius of Robert and Catherine is contrasted to the more pedestrian figures of Hal and Claire. Hal is a hard worker, a competent mathematician, and probably a good teacher, but he lacks the spark of genius. His work, as he says himself, is trivial. The big ideas elude him, and always will. This is why he combs through Robert's notebooks, hoping that some spark of genius will fly out from the pages, enabling him to bask in reflected glory. Claire too is competent and practical, "very quick with numbers," and this has enabled her to have a successful career as a currency analyst. But making money in the big city is a far cry from genius, which Claire acknowledges in her father but does not understand. She is too well adjusted to the world to have any interest in the beauty of abstractions.

Thus through the four characters the play contrasts the mundane and the ordinary, on which the day-to-day world turns, with the exceptional and the extraordinary, which is the rare stuff of genius that creates the peaks of human achievements.

Love and Trust

The certainty of a mathematical proof, which can be followed logically and established as absolutely true beyond any doubt, is a sharp contrast to the fragility and uncertainty of human life and relationships. Unlike in mathematics, truth in life is a harder thing to understand and grasp. Much of it, the play suggests, depends on trust. Catherine and Robert trust each other, and Robert believes that his daughter's love for him saved his life. There is never any doubt of the strength of the bond between father and daughter. But the other central relationship in the play, that between Catherine and Hall, is more problematic. It develops tentatively, and issues of trust soon surface. The truth is hard to determine. Catherine is suspicious of Hal's motives in going through Robert's notebooks, thinking that he may want to publish some of her father's work under his own name. Hal vigorously denies this, but she does not believe him, and perhaps Hal may not be willing to acknowledge even to himself that his motivation may not be entirely disinterested. He knows, after all, that his career has stalled, and a major discovery such as he seeks might give it a boost.

TOPICS FOR
FURTHER
STUDY

  • Women have made valuable contributions to mathematics in the nineteenth and twentieth centuries. Research the work of two female mathematicians and briefly describe their achievements.
  • What do you think Catherine means when she refers in the play to "proofs like music?" What might mathematics and music, which on the surface seem so different, have in common?
  • What signs does Catherine show that she is suffering from depression? What is depression? How is it recognized? What are the causes of it? How is it treated?
  • In the script, the playwright uses the word "beat" as a cue for the actors. "Beat" means a pause in the dialogue, a moment of silence. It can indicate a moment of confusion or awkwardness or a change of mood in the characters. Examine act 1, scene 1, after Hal enters. From there to the end of the scene there are eleven beats during Hal's conversation with Catherine. Imagine you are playing Hal or Catherine. What is each character feeling during each beat? Describe what the actors would need to convey at each beat.

The relationship between Hal and Catherine moves in an awkward dance of mistrust followed by attempts at trust. In act 1, scene 1, Catherine thinks he is stealing a notebook, and he is, but not for the purpose she thinks. In act 1, scene 4, she tries to show her regained trust when she gives him the key to the drawer which contains her proof. But then when she claims the proof is hers, the tables are turned; it is now Hal who mistrusts Catherine, refusing to believe that she is capable of such work of genius. In turn, she once more becomes suspicious of him, saying the reason he wants to take the proof is to show off to his colleagues: "You can't wait to show them your brilliant discovery," she says. Mistrust again fills the air, on both sides. The proof that sits harmlessly in the notebook may embody a beautiful, irrefutable truth, but for the people arguing over it, such truth is elusive, not only about who wrote the proof, but also in terms of the truthfulness of their relationship.

The uncertainty continues into the final scene. Hal has overcome his doubts about whether Catherine wrote the proof, but she is still dealing with the hurt feelings that arose because he did not trust her word at first. She now plays devil's advocate and makes a telling comment that plays on the contrast between mathematical certainty and the uncertain, ambiguous world of human activities and relations. Even though Hal has carefully elaborated his reasons for concluding that the work is hers, she says that none of the arguments he has produced prove anything. "You should have trusted me," she says. It seems that trust is the only way that certainty can be established in this uncertain world; it is the only thing that can guide people through the complexity of human relationships, although the play leaves no doubt about how easy it is to undermine trust and how hard it is to maintain it. To Hal's credit, he does not try to argue with Catherine. Like a fine mathematical proof ("streamlined, no wasted moves," as Hal says of Robert's work), he takes the surest way to the goal, acknowledging that she is correct: he should have trusted her. It is on that basis of trust that he and Catherine can go forward together.

STYLE

Exposition

The exposition of a play is the introductory material, which creates the tone, introduces the characters, perhaps suggests the theme, and gives the background information necessary in order to understand the play. In this play, the exposition is done with great economy and skill. So much is accomplished in the first eight pages of the script, amounting to less than half of the first scene, in which Robert and Catherine talk to each other. In this short time, the audience learns that there is affection as well as frustration between father and daughter; that Robert is a mathematician and a genius who did his best work while he was in his twenties and who is now mentally ill; that Catherine is depressed, has no friends, and does not like her sister; that she has some mathematical knowledge and can banter with her father about mathematical concepts; that she is worried about inheriting his illness. All this is accomplished within a couple hundred mostly short lines of dialogue.

Theatrical Surprise

The playwright shows that he is a master of the theatrical surprise, a moment when something is revealed that the audience up to that point had not known or guessed. Halfway through the first scene, for example, Robert says that the only reason he can admit that he is crazy is because he is also dead. This is a startling moment and also a surprisingly humorous one (the intensity and sadness of the play is offset many times by humor). Another aspect of this strategy of surprise is the fact that in a number of scenes, a new piece of information is produced near the end to give a twist to the interactions of the characters. This occurs for example at the end of the first scene, when Hal reveals the real reason he tried to sneak out with the notebook. It also occurs in act 1, scene 3, when Hal reveals that he and Catherine have met before. The most stunning use of this technique occurs at the end of act 1, when Catherine claims that she is the author of the proof.

HISTORICAL CONTEXT

Sophie Germain

Sophie Germain, the French mathematician so admired by Catherine in Proof, was born into a middle-class family in Paris in 1776. She first became interested in mathematics when she was thirteen. Confined to her home because the French Revolution had broken out, she taught herself mathematics in her father's library. Her family tried to discourage her, considering that mathematics was not an appropriate field of study for a girl. But Sophie persisted. She obtained lecture notes from the École Polytechnique, an academy founded in 1794 that trained mathematicians and scientists but refused to enroll women. Becoming interested in the work of J. L. Lagrange, she submitted a paper to him under the pseudonym Antoine-August Le Blanc, a man who was a former student at the academy. Lagrange was impressed by the paper and wanted to meet the author. Overcoming his surprise at discovering a young female mathematician, he agreed to become her mentor. This gave Germain entry into the circle of mathematicians and scientists that had up to then been closed to her.

In 1804, Germain began corresponding with the German mathematician, Carl Friedrich Gauss (as Catherine tells Hal in Proof), one of the most brilliant mathematicians of all time. Germain shared with him her work in number theory. It was three years before Gauss discovered that the bright young correspondent whom he had been mentoring was a woman. A dozen years later, Germain wrote to the mathematician Legendre, presenting the work in number theory that was to become her greatest contribution to mathematics. In 1816, Germain was awarded a prize by the French Academy of Sciences for her work in explaining mathematically the vibration of elastic surfaces. That Germain continued work in this area was another of her lasting contributions to mathematical theory. Germain died in 1831, before she could accept an honorary degree from the University of Göttingen that Gauss had convinced the university to award. Germain's contribution to mathematics was all the more remarkable because, like Catherine in Proof, she lacked formal academic training.

Trends on Broadway

When Proof was first produced in 2000, it was the latest in a number of plays that took their inspiration from intellectual disciplines such as mathematics and physics. The aim of the playwrights seemed to be to give the audience some substantial food for thought as well as an evening's entertainment. The fashion began with British playwright Tom Stoppard. Stoppard's Hapgood (1988; revised 1994) used the intricacies and paradoxes of quantum physics as metaphors for the world of espionage during the cold war. In 1994, Stoppard wrote Arcadia, another play in which the audience found themselves immersed in quantum physics, as well as chaos theory. Like Proof, Arcadia features a young woman with an extraordinary grasp of mathematical theory. Also like Proof, it alludes to a nineteenth-century woman who made an impact on mathematical theory. This was not Sophie Germain but Ada Byron, Lady Lovelace, the daughter of the poet Lord Byron, who worked with mathematician Charles Babbage in developing the theory of a new calculating machine. The mathematical plan she wrote is now considered to be the first computer program.

Other plays which drew on quantum physics included Michael Frayn's Copenhagen (2000), a sophisticated investigation of a meeting between physicists Werner Heisenberg and Niels Bohr in 1941, and Hypatia by Mac Wellman, which was based on the life and death of Hypatia, a fifth-century mathematician and philosopher. According to Bruce Weber, whose New York Times article, "Science Finding a Home Onstage," is about the contemporary fashion of writing plays with scientific content:

This flowering use of science as narrative material and scientific concepts as metaphors for the stage.… provides evidence that science is re-entering the realm of popular culture, not just in imaginative, futuristic fiction but also in other mainstream and alternative forms: from historical reconstruction and theoretical abstraction to fluffy romance and contemporary realism.

CRITICAL OVERVIEW

As might be expected of a Pulitzer prize-winning play, Proof was received enthusiastically by audiences and most reviewers. The stunning revelation at the end of act 1, when Catherine announces that it was she, not her father, who wrote the proof, was regularly greeted with gasps by the audience. Bruce Weber, in the New York Times, called Proof "an exhilarating and assured new play … that turns the esoteric world of higher mathematics literally into a back porch drama, one that is as accessible and compelling as a detective story." Weber admired the pacing of the play, and further noted that it "presents mathematicians as both blessed and bedeviled by the gift for abstraction that ties them achingly to one another and separates them, also achingly, from concrete-minded folks like you and me." Weber also appreciated the spirit of the play in which there was no meanness; the characters struggling to deal with the devastating effects of mental illness were all "good people."

Weber reported again on the play over a year later, noting in the New York Times that it was still playing to sold-out houses at the Walter Kerr Theatre. A change of cast had not diminished its appeal, but rather shown that the characters and their relationships could be given "new and distinct emotional shadings."

In Variety, Robert Hoffler wrote of the play's "rich, aching melancholia" and praised its ambitious structure and its sense of humor: "The mercurial nature of the mathematician's art is refracted everywhere, usually in ways that offer a humorous counterpoint to somber loss." In Library Journal, Robert W. Melton was equally enthusiastic, describing Proof as a "wonderful" play: "[its] deft dialog, its careful structure, and the humanity of the central characters are themselves proof of a major new talent in the American theater."

One dissenting voice was that of Robert Brustein, in The New Republic, who complained that although the playwright had a competent grasp of his material, the plot was too thin. The author "runs out of material so quickly that, by the middle of the second act, the play jerks to a halt and starts running in place."

CRITICISM

Bryan Aubrey

Aubrey holds a Ph.D. in English and has published many articles on modern drama. In this essay, Aubrey discusses the parallels between the mathematicians Robert, Catherine, and Hal and the lives and creativity of real life mathematicians, especially John Forbes Nash Jr.

In researching Proof, Auburn consulted with a number of mathematicians and also read the biographies of prominent mathematicians, aspects of whose lives find their way into the play. When Hal tells Catherine that some of the older mathematicians he encounters at conferences are addicted to amphetamines, which they take to make their minds feel sharp, he is amplifying the well-known story of mathematician Paul Erdös who began taking amphetamines so he could keep up the fast pace of his mathematical work. When friends persuaded him to stop taking the amphetimines for a month, Erdös complained that he had not been able to do any creative work during that time and promptly resumed taking the drugs.

Andrew Wiles is another mathematician whose story finds an echo in Proof. Wiles, a professor of mathematics at Princeton University, worked for many years to prove Fermat's Last Theorem when the conventional wisdom was that such a proof was impossible. In 1993, Wiles announced at a conference that he had proved the theorem. It transpired that he had been working on it in solitude, in an office in his attic, for seven years, telling no one of what he was doing. This surely inspired the picture presented in Proof of Catherine, who also works in solitude and in secret, and then suddenly, out of the blue, unveils a ground-breaking mathematical proof.

But the mathematician whose life story is most closely linked to Proof is John Forbes Nash, Jr, who is the subject of A Beautiful Mind (1998), a biography by Sylvia Nasar which was made into a popular movie in 2001. Nash was a mathematical genius. In 1949, when he was twenty-one years old and a graduate student at Princeton University, he wrote a slim, twenty-seven-page doctoral thesis on game theory (a theory of how people behave when they expect their actions to influence the behavior of others) that revolutionized the field of economics. Nash became a professor at the Massachusetts Institute of Technology (MIT) when he was only twenty-three and quickly went on to solve a series of mathematical problems that other mathematicians had deemed impossible. He seemed destined to become one of the greatest mathematicians in the history of the discipline. Then, in 1959, when Nash was thirty years old, his behavior, which had always been eccentric, became bizarre and irrational. He heard strange voices and became obsessed with the idea of world government. He accused a colleague of entering his office to steal his ideas. He turned down the offer of a chair at the University of Chicago with the explanation that he was going to become Emperor of Antarctica. Nash was admitted to McLean Hospital in Belmont, Massachusetts, where he was diagnosed as a paranoid schizophrenic.

Schizophrenia is a severe mental disorder that distorts thinking and perception. It leads to a loss of contact with reality and bizarre, sometimes antisocial behavior as the sufferer withdraws into his own inner world. Schizophrenia is difficult to treat and there is no cure. Nash spent the next thirty years afflicted with the disease, which would occasionally go into temporary partial remission before returning. His career was destroyed although he made a surprise recovery during the 1990s. He resumed living a normal life and studying mathematics and was awarded the Nobel Prize in 1994.

The parallels between the real life of Nash and the fictional life of Robert in Proof are many, and they prompt questions of whether genius and insanity are linked and whether both are inherited. Robert is clearly a Nash-like figure. Hal reminds Catherine in act 1, scene 1 that when Robert was in his early twenties he had made major contributions to three fields: game theory, algebraic geometry, and nonlinear operator theory. These are exactly the same fields, according to Nasar, in which the young Nash made his impact. Nasar also points out that in the early days of his illness, Nash seemed to have a heightened awareness of life:

He began to believe that a great many things he saw—a telephone number, a red necktie, a dog trotting along the sidewalk, a Hebrew letter, a sentence in the New York Times—had a hidden significance, apparent only to him.… He believed he was on the brink of cosmic insights.

WHAT
DO I READ
NEXT?

  • Auburn's Fifth Planet and Other Plays (2001) contains several one-act plays that Auburn wrote before Proof. The plays are Fifth Planet, Are You Ready, Damage Control, Miss You, Three Monologues, What Do You Believe in the Future, and We Had a Very Good Time.
  • Wit (1999) is a Pulitzer Prize–winning play by Margaret Edson. The protagonist is a female scholar of English literature who specializes in the work of the seventeenth-century poet, John Donne. She is now dying of cancer and uses the experience to explore mortality, the value of human relationships, and how life should be lived.
  • Copenhagen (2000) is a Tony Award–winning play by Michael Frayn about Werner Heisenberg (a physicist who was head of the Nazi's attempts to develop a nuclear bomb), Danish physicist Niels Bohr, and his wife Margrethe. Bohr and Heisenberg met in Copenhagen in 1941, and what they discussed has been a matter of dispute ever since. By adapting to the theater principles drawn from quantum physics, Frayn cleverly shows it is impossible to reach an objective understanding of what the two men discussed that day.
  • The Mind-Body Problem (1993), by Rebecca Goldstein, is a coming-of-age novel set in Princeton's mathematics community, about a young Jewish woman who marries a world-renowned mathematician.
  • Strange Brains and Genius: The Secret Lives of Eccentric Scientists and Madmen (1998), by Clifford A. Pickover, examines the connection between genius and madness in a highly eclectic way. Pickover profiles many eccentric scientists, from Nikola Tesla to the Unabomber, Ted Kaczynski (who was a mathematician), as well as some writers and artists.

This is echoed by Robert, as he recalls his mental state soon after he became ill. He tells Catherine about the clarity with which he saw things, and he believed that his mind was even sharper than before:

If I wanted to look for information—secrets, complex and tantalizing messages—I could find them all around me. In the air. In a pile of fallen leaves some neighbor raked together. In box scores in the paper, written in the steam coming up off a cup of coffee. The whole world was talking to me.

Although the play does not mention the exact nature of Robert's illness, the hallucinations and delusions he suffered from make it clear that he, like the real-life Nash, was schizophrenic. Robert was no doubt mistaken when he claimed that his mind had become sharper, because during his illness his mental processes no longer bore any relation to reality. As with Nash, the insights he thought he had contained meanings known only to him and were useless for objectively verifiable mathematical knowledge. Just as Nash believed that powers from outer space, or foreign governments, were communicating with him through cryptic messages in the New York Times that only he could decode, so too Robert used to borrow large numbers of books from libraries because he thought that aliens were sending him messages through the Dewey decimal numbers on the books, and he was trying to work out the code.

Was Nash's insanity, or that of Robert in Proof, somehow related to their genius? The idea that creativity and madness are linked is an old one. Plato wrote in his dialog Ion that the poet was inspired with a kind of divine mania, and cultural history turns up many examples of exceptionally creative people who have been afflicted with mental illness of one kind or another, including the philosopher Friedrich Nietzsche, the artist Vincent van Gogh, and the writer Virginia Woolf. In more modern times, American poets Sylvia Plath and Robert Lowell suffered from mental illness. (In 1959, Lowell was a patient at McLean Hospital in Belmont when Nash was admitted.)

The most common type of mental illness amongst creative artists is manic-depression, also known as bipolar disorder. This is not the same as schizophrenia. Although manic-depression can produce delusions, it is mainly characterized by extreme mood swings, ranging from great elation to deep depression. Research suggests that creative artists, poets in particular, are two to three times more likely to suffer from manic-depression than scientists. For the poet or writer, it is possible that manic-depression can enhance creativity, since the mood swings may offer more acute insight into the peaks and troughs of human experience, which in turn can lend the artist's work a profundity that might escape those who live on a more even emotional keel. Creative people who suffer from manic depression are often able to function quite normally between episodes, which is usually not the case with schizophrenia.

It would seem that schizophrenia, far from being somehow linked with creativity, is in fact inimical to it, since the feeling of heightened awareness it may produce translates only into delusional perceptions, not deeper insights into truth. Although there does seem to be a certain unusual quality to the minds and personalities of many great scientists and philosophers, madness does not describe it. Nasar points out many examples of men of genius, including Immanuel Kant, Ludwig Wittenstein, Isaac Newton, and Albert Einstein, who had emotionally detached, eccentric, solitary, inward-looking personalities that may have been useful in promoting the kind of creativity that these disciplines require. Such people—Nash was one of them before his illness—are able to think not only more profoundly but also in different ways than less gifted individuals. Nash was used to solving problems in ways that had not occurred to others. He developed this habit of thinking "out of the box" at an early age. His sister reported that Nash's mother was once told that her son, then in elementary school, was having trouble with math, because he could see ways of solving mathematical problems that were different from the methods the teachers were used to.

When Nash was a mature mathematician, his mind not only worked faster than anyone else's, he continued to approach mathematical problems in unusual ways that would unlock new possibilities that astonished his colleagues. Nasar reports that Donald Newman, a mathematician who knew Nash at MIT in the 1950s, said of him that "everyone else would climb a peak by looking for a path somewhere on the mountain. Nash would climb another mountain altogether and from that distant peak would shine a searchlight back onto the first peak." Sometimes when Nash presented his unexpected results to professional audiences, there would be some who said they could not possibly believe them, so novel was Nash's approach to the problem.

Auburn clearly incorporated this dimension of Nash's mind into the character of Robert in Proof. When Hal says to Catherine that hard work was not the secret of Robert's success, she contradicts him but immediately explains that the work went on almost unseen, and Robert's success resulted from his taking an unusual starting point:

He'd attack a question from the side, from some weird angle, sneak up on it, grind away at it. He was slogging. He was just so much faster than anyone else that from the outside it looked magical.

Hal's immediate response, about the beauty and the elegance of Robert's work, also corresponds to what mathematicians said about Nash's work. It is quite common for mathematics to be described in this way, as if it somehow partakes in the essential beauty and order of the universe. The French mathematician Henri Poincaré wrote about the aesthetic feeling known by all mathematicians when they recognized these qualities revealed in their work, describing it as "the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance."

A final aspect of Nash's life finds its way into Proof in Catherine's worries that she may inherit her father's illness, even though the depression she suffers from is not related to the symptoms of schizophrenia. Catherine is right to be concerned, since expert opinion considers that although the cause of schizophrenia is unknown, there is a genetic factor in the disease. It can be inherited and, indeed, Nash's own son, John Charles Nash, was diagnosed, like his father, as a paranoid schizophrenic. Like his father also, John Charles Nash was a mathematician, brilliant but without his father's spark of genius. Unlike schizophrenia, genius is not transmitted through genes, and there are numerous examples of geniuses whose offspring have been distinguished only by their mediocrity. So for Catherine in Proof to inherit both Robert's genius and his mental illness would be a very unlikely event in real life, although of course, as Proof shows, it can be turned into excellent drama. Nash himself discovered this when at the age of seventy-three his biographer, Nasar, took him to see a performance of the play. An article in the Los Angeles Times by John Clark contains Nasar's description of how Nash reacted:

'He loved it,' says Nasar, who admits she was a little nervous about his response. 'It was so much fun to see him laugh and react to Proof because [the father] is clearly inspired by Nash's story, and to witness John Nash seeing this on the stage in front him—it was adorable.'

Source:

Bryan Aubrey, Critical Essay on Proof, in Drama for Students, Thomson Gale, 2005.

Curt Guyette

Guyette, a longtime journalist, received a bachelor's degree in English writing from the University of Pittsburgh. In this essay, Guyette discusses how Auburn highlights the uncertain nature of human existence by contrasting it with the certainty found in mathematics.

In his Puliter Prize–winning play Proof, Auburn brings into high relief the uncertain nature of life by contrasting it with the world of mathematics, where the truth or falsity of an idea can be proved with absolute certainty. In the world of numbers, two plus two always equals four; there is no doubt involved. But in matters of flesh and blood, especially in the way people relate to the world around them, there is no formula for absolute knowledge.

The tenuous nature of reality as perceived through human eyes is vividly depicted in the play's very first scene. Catherine, the troubled daughter of Robert, a brilliant mathematician of world renown, is having a revealing conversation with her father early in the morning of her twenty-fifth birthday. During the conversation, it is revealed that Robert suffers from mental illness. By its very nature, mental illness radically distorts a person's perceptions of the world. It is also the nature of such an illness that the person afflicted with it is deluded into thinking that his perceptions are completely grounded in reality. As Robert tells Catherine, "Crazy people don't sit around wondering if they're nuts." As their conversation continues, he reinforces the point by saying, "Take it from me. A very good sign that you're crazy is an inability to ask the question, 'Am I crazy?"'

Robert is, in fact, an expert on the subject. After displaying mathematical genius in his early twenties, his career had been cut short by a debilitating mental illness. This is a man who, after rocking the math world with his proofs, began attempting to decipher the Dewey decimal codes of library books because he was convinced that they held hidden secret messages. Consequently, Catherine is wary of accepting the insights of a certified crazy person. It is not until midway through this first scene that the audience discovers that Richard is actually dead and that the action playing out in front of them is only a figment of Catherine's imagination, calling into question her sanity. As a result, from the outset, the audience itself is forced to ask the question: What is true and what is not—and how do you prove the conclusions arrived at?

This theme is carried throughout the play as Auburn compels the audience to keep wondering what the truth is. There is a particularly poignant scene near the end of act 2 when Robert makes another appearance, this time in a flashback. After suffering through years of mental illness, he has experienced months of clarity. His recovery has been so significant that Catherine, who had given up pursuit of her own career in mathematics in order to care for him, was able to return to school. She pays a visit to her dad and finds him sitting outside in the freezing cold, working. He tells her that his "machinery," meaning his brain, is once again firing on all cylinders. He is exhilarated to the point of being overheated and has gone out into the December day in order to cool off. Trying to describe for his incredulous daughter the incredible feeling that he is experiencing, Robert tells her that it is not as if a light has suddenly turned on in his mind, but rather the whole "power grid" that has been activated after years of dormancy. "I'm back!' he tells her. "I'm back in touch with the source—the font, the—whatever the source of my creativity was all those years ago. I'm in contact with it again." She reads what he has been scribbling in his notebook and in an instant it becomes painfully clear that what has returned is not the spark of genius but insanity.

The play's most significant questions are raised about Catherine, who is the main focus of uncertainty. Has she inherited her father's genius? Does she suffer from the same mental illness that afflicted him? Have both the incandescent brilliance and the dark demons been passed from father to daughter? Again, unlike the world of mathematics, the answers to those questions are anything but clear-cut. It is part of Auburn's genius that he constructed a play guaranteed to hold the audience's interest by inserting the compelling elements of a mystery into what is, at its heart, the story of complex human relations. In an interview with Mel Gussow of the New York Times, Auburn notes that the genesis of this play can be traced to two ideas. One involved writing about two sisters "quarreling over the legacy of something left behind by their father." The other had to do with someone whose parent suffered from mental illness and began to wonder whether she, too, might be starting to succumb to madness. To pull the audience along, Auburn tells Gussow that he wanted to use what Alfred Hitchcock referred to as a "Maguffin," or plot device involving an object of mysterious origin. In this case, Auburn chose to insert the discovery of a mathematical proof into the story. That proof, whose existence is revealed at the end of act 1, provokes two essential questions: Is it indeed a brilliant breakthrough and, if so, who produced it—Robert or, as she herself claims, Catherine?

The character asking those questions is Hal, a former student of Robert's who has gone on to become a mathematics professor. He also has had a romantic eye on Catherine for many years. The question of the proof's validity is relatively easy to solve. Writing about this play in The Chronicle of Higher Education, David Rockmore explains that this is fundamental to the concept of a proof. "Assuming that a person knows the language and has the background," writes Rockmore, "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." Determining whether Catherine is the source of this brilliant piece of work, or is instead merely suffering from the same sort of insane delusions that afflicted her father, is a much more difficult task. As Rockmore, a professor of mathematics at Dartmouth College, observes, "In statements about life, proofs of similarly absolute certainty are difficult, if not impossible, to derive."

Consequently, Auburn does not wrap his play up into a neat and tidy package. In that sense, it mirrors life. As the play approaches the final curtain, Hal comes to believe that it was indeed Catherine who produced the proof. It is Catherine herself who keeps the mystery alive, telling Hal:

You think you've figured something out? You run over here so pleased with yourself because you changed your mind. Now you're certain. You're so … sloppy. You don't know anything. The book, the math, the dates, the writing, all that stuff you decided with your buddies, it's just evidence. It doesn't finish the job. It doesn't prove anything.

That is the way life is. Very few things are completely provable beyond a shadow of doubt. But absent proof, there is always possibility. And so, it is entirely appropriate that this play ends on an optimistic note. There is the promise that Catherine is indeed every bit as brilliant a mathematician as her father. There is also the very real possibility that she will not be overtaken by madness and will instead be able to keep a firm grasp on reality. As the curtain falls with her and Hal sitting side by side, there is no proof positive that the two will find happiness and build a life together. There is, however, hope.

Source:

Curt Guyette, Critical Essay on Proof, in Drama for Students, Thomson Gale, 2005.

John Simon

In the following excerpt, Simon heaps praise on Proof, saying all parts of the play "spell J-O-Y."

Manhattan theater club does it again! David Auburn's Proof is what Copenhagen ought to be: a play about scientists whose science matters less than their humanity. Here, those of us who want their dramatic characters to be real people need not feel excluded. Robert, a world-famous mathematician who went crazy; Catherine, his mathematically brilliant but too-depressed-to-work daughter; Hal, a young math teacher going through Robert's hundred-plus confused notebooks: and Claire, Robert's older daughter and a successful actuary, are above all fascinating individuals. Robert isn't any less human even for being, through most of the play, dead. All four—whether loving, hating, encouraging or impeding one another—are intensely alive, complex, funny, human.

The very first scene in Proof is masterly: a birthday dialogue between father and daughter, in which Catherine, alive, is barely living, and her celebrated father is sparklingly trying to rouse her into action although he is (I hate to give it away but must) dead—Catherine's fantasy. Yet this mysterious, droll, and electrifying scene is really exposition in disguise: something generally a bore, but here so splendidly reconceived as to fascinate—as indeed all of Proof does.

So here we have Robert, the near-genius mathematician who went mad and eventually died, and Catherine, who gave up a potentially great mathematical career to look after him and, in the process, let herself run down, perhaps irreversibly. Here, too, is Claire, the narrowly practical daughter, who wants to save Catherine from what may be incipient madness by dragging her from Chicago to New York and supervising her life—benignly as she sees it, but horribly as Catherine does. And here is Hal, revering Robert's work and secretly in love with Catherine, bumbling and bungling everything. Out of this curious quartet, Auburn creates emotionally and intellectually enveloping music.

The performances are perfect: Larry Bryggman's lovable but exasperating Robert; Johanna Day's officious yet well-meaning Claire: Ben Shenkman's clumsy but gradually maturing Hal. As for Mary-Louise Parker, her Catherine is a performance of genius. Is there another young actress as manifold, incisive, sexy, and effortlessly overpowering? Add to this Daniel Sullivan's superb direction and the classy production values (by John Lee Beatty, Jess Goldstein, and Pat Collins), and it all spells J-O-Y. Instead of taking up more time reading, you are urged to run and get your tickets immediately.

Source:

John Simon, "Proof Positive," in New York, June 5, 2000, p. 106.

Robert Brustein

In the following excerpt, Brustein places Proof within the company of other recent one-word title plays and says it is "not exactly the brilliant debut that some have been claiming."

Proof, which recently re-opened at the Walter Kerr Theatre after a run at the Manhattan Theatre Club, is the latest in a string of plays with one-word titles that represent the theater's belated tribute to the conceptual mind. Tom Stoppard probably started the whole fashion with Arcadia, a period comedy that features, among other things, dialogues on English gardening and Newtonian physics. But the trend has exploded in the last few years to include Yasmina Reza's Art, an argument provoked by a post-modern painting, Margaret Edson's Wit, an infirmity play surrounded by a frame of metaphysical poetry, and Michael Frayn's Copenhagen, a scientific discourse on the subject of quantum theory, indeterminacy, and atomic fission. These are the major examples of a genre with terse titles and prolix personae that has now managed to occupy the middle (or the middlebrow) ground of the Western stage.

I am still trying to figure out why this development leaves me somewhat less ecstatic than it does my critical colleagues. Obviously we should encourage anything that raises the intellectual level of our benighted theater; and it is also true that some of these plays (notably Wit) have a lot more going for them than mental pyrotechnics. Yet the danger of this kind of Cliffs Notes approach to playwriting is that the dramatist, simply by dropping names or equations, will feel relieved of the obligation to investigate the emotional and spiritual aspects of the material, and the spectator will leave the theater feeling a lot more intelligent than he actually is. "Tell me where is fancy bred," Shakespeare wrote, "Or in the heart or in the head." There is no doubt that this playwright, at least, located the seat of the imagination (which he called "fancy") in the non-cerebral parts of the human body.

Proof is David Auburn's first major production; and if it is not exactly the brilliant debut that some have been claiming, it certainly represents the work of a writer with a fairly decent grasp on his not terribly fanciful material. The "proof" of the title is a breakthrough mathematical equation regarding prime numbers, the authorship of which is a subject of dispute. Catherine (Mary-Louise Parker) is the daughter of an intermittently psychotic and recently deceased professor at the University of Chicago (Larry Bryggman), whose ghost comes to visit her from time to time. She has a fling with one of her father's graduate students (Ben Shenkman) after she finds him rifling through her father's notebooks. She is in conflict with her rather unimaginative sister (Johanna Day), who has come to sell the family house and move Catherine to New York. And when this relatively under-educated woman claims to be the author of the theorem in question (I am ruining what is intended to be a stunning first act revelation), there is some debate as to whether she is really treading in her father's demented footsteps.

We never learn the actual nature of the discovery, or why it constitutes such a great contribution to human knowledge. By his own admission, Auburn does not know or care much about mathematical theory. But what makes this play problematic is not its author's ignorance regarding prime numbers. It is the thinness of his plot. He runs out of material so quickly that, by the middle of the second act, the play jerks to a halt and starts running in place. Proof sometimes looks like a rather austere stage version of Good Will Hunting, insofar as it features a whiz kid central character who is also an idiot savant. But if Good Will Hunting was concerned with questions of class, Proof focuses on questions of gender—how "Shakespeare's sister" could have written all his plays if she hadn't been forced to shine unappreciated on the ocean floor, and so on.

David Auburn's writing may not be terribly electric or dynamic. But Daniel Sullivan's direction muffs the few opportunities that the playwright offers to hoist the action out of the quotidian. With its spectral visitations from the heroine's father and its non-linear treatment of time, Proof is, after all, something of a ghost story. But the production remains mired in domesticity. It is relentlessly realistic, with John Lee Beatty contributing another in his gallery of Edward Hopper brick structures, and Neil A. Mazzela's lighting failing to distinguish between the gritty present and the ethereal past.

Where the evening does prosper is in the acting, especially in Mary-Louise Parker's Catherine. I first saw this fine actress in 1988 playing Emily to Eric Stolz's George in the Lincoln Center production of Our Town. Young as she was at the time, she made it instantly clear that she was born for the stage, a promise that she confirmed nine years later playing L'il Bit in How I Learned to Drive. Here she turns the twenty-eight-year-old Catherine into a restless, angry ragdoll of a woman with a frazzled slouch, who manages to accomplish one of the speediest costume changes in recorded history. (She goes up a whole flight of stairs, then appears seconds later on stage in a completely new set of rumpled clothes.) That she can also create such texture out of her underwritten role is an even more impressive feat of stage magic.

Source:

Robert Brustein, "Or in the Heart or in the Head," in New Republic, September 13, 2000, pp. 28–29.

SOURCES

Auburn, David, Proof, Faber and Faber, 2001.

Barbour, David, "Proof Positive" in Entertainment Design, Vol. 43, November 2000, p. 19.

Brustein, Robert, Review of Proof, in the New Republic, November 13, 2000, pp. 28–29.

Clark, John, "So Smart It Hurts," in Los Angeles Times, December 16, 2001.

Foster, John Evan, Review of Proof, in Theatre Journal, Vol. 53, No. 3, October 2001, Performance Review Sec., pp. 503–04.

Gussow, Mel, "With Math, a Playwright Explores a Family in Stress," in the New York Times, May 29, 2000, Sec. E, Col. 2, p. 1.

Hoffler, Robert, Review of Proof, in Variety, Vol. 380, No. 11, October 30, 2000, p. 34.

Melton, Robert W., Review of Proof, in Library Journal, April 1, 2001, p. 100.

Nasar, Sylvia, A Beautiful Mind, Simon & Schuster, 1998.

Poincaré, Henri, "Mathematical Creation," in The Creative Process: A Symposium, edited by Brewster Ghiselin, New American Library, 1960, p. 40.

Rockmore, Daniel, "Uncertainty Is Certain in Mathematics and Life," in the Chronicle of Higher Education, June 23, 2000, Opinion & Arts Sec., p. 89.

Weber, Bruce, Review of Proof, in New York Times, May 24, 2000, p. B3.

——, Review of Proof, in New York Times, October 27, 2001.

——, "Science Finding a Home Onstage," in New York Times, June 2, 2000, p. B1.

FURTHER READING

Billington, Michael, Review of Proof, in Guardian, May 16, 2002.

This review of the British production of Proof at London's Donmar Warehouse censures the playwright for not explaining what the crucial mathematical theory is. Billington calls this the weak point of the play.

Feingold, Michael, Review of Proof, in Village Voice, June 6, 2000.

A review that is generous in its praise. Feingold points out that Auburn has no interest in explaining the finer points of mathematics; it is simply a given that for three of the four characters, mathematics is something they love, and the play is more of a love story than anything else—love of mathematics, love of father and daughter, and the growing love of Hal and Catherine.

Heilpern, John, Review of Proof, in New York Observer, June 19, 2000, p. 5.

A laudatory review that praises the play's evocation of love between father and daughter, the fragility of life, and the discovery of love. The only flaw Heilpern sees is that the mystery of who wrote the proof is too easily resolved.

Parker, Christian, "A Conversation with David Auburn," in Dramatist Magazine, December 10, 2001.

In this interview, Auburn talks about how he became interested in writing plays and how his career developed.

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