Pierre de Fermat

Chapter 1 · The Councillor of Toulouse

Mathematics in the hours the court left over

1607 – 1665 · Beaumont-de-Lomagne · Bordeaux · Toulouse

Pierre de Fermat was born in 1607 near Montauban, the son of a prosperous leather merchant, and trained as a lawyer. In 1631 he bought a councillor's place in the parlement of Toulouse — the regional high court — and held it until his death thirty-four years later. He was, by all accounts, a conscientious judge in a court that dealt with capital cases.

This is not incidental to the mathematics; it shapes all of it. He was not a professor, had no students, and owed nobody a publication. Mathematics happened when the court rose, and he treated it as a private pleasure conducted by correspondence with a handful of people who could follow it.

The consequence is that almost nothing of his appeared in print in his lifetime. What he did instead was write letters — to Mersenne in Paris, who acted as the switchboard of European mathematics, and through him to Descartes, Roberval, Pascal, Frénicle and Wallis. He would announce a result, sometimes as a challenge, and very often decline to give the demonstration. That habit made him enemies, cost him credit, and has left historians arguing about what he actually had.

There is a further consequence. Because he wrote for correspondents rather than readers, his notation stayed old-fashioned. He used the algebra of Viète, with vowels for unknowns and consonants for givens, long after Descartes's better notation was available. His son Clément-Samuel gathered the papers and published them in 1679, fourteen years after his death.

Why this matters

The greatest mathematician of the century was an amateur with a demanding day job, which changes what 'amateur' can be taken to mean.

A judge who did mathematics after hours and refused to publish it. What would you ask him?

Ask Fermat

• “Why did you never publish anything?”
• “What did a councillor of the parlement actually do all day?”
• “Who was Mersenne, and why did everything go through him?”
• “Why announce a result and withhold the proof?”
• “Did you think of yourself as a mathematician at all?”

Chapter 2 · A Curve as a Relation

Plane and Solid Loci, about 1636

c. 1636 – 1638 · Toulouse · Paris

Around 1636 Fermat circulated a short manuscript, Ad locos planos et solidos isagoge — an introduction to plane and solid loci. Its opening move is one of the great simplifications in the history of the subject.

Take a fixed line. Measure a distance along it from a fixed point. From where you have got to, raise a second line — an ordinate — to reach the curve. Now every point of the curve corresponds to a pair of lengths, and the curve itself is nothing more than the RELATION those two lengths stand in. Write down the relation and you have written down the curve. He then works through the relations that give straight lines, circles, parabolas, ellipses and hyperbolas.

That is coordinate geometry, and Fermat had it before Descartes published La Géométrie in 1637. Descartes never conceded the point, and the two of them quarrelled bitterly — first over this, then over Fermat's method of tangents, which Descartes attacked as unsound and later accepted.

Two things about Fermat's version are worth noticing, because both are unfamiliar. His ordinate is not necessarily raised square to the axis; it is often oblique, and the geometry works perfectly well either way. And he does not admit quantities less than nothing, so his figures live in a single quarter of what a modern reader would draw. The idea that a point is a pair of signed numbers on two perpendicular axes is later than both of them, and neither man would recognise it as his.

Why this matters

The graph is the most-used object in school mathematics, and the man who thought of it first neither printed it nor drew it the way we do.

A curve reduced to a relation between two lengths. What would you ask him?

Ask Fermat

• “How does a curve turn into a relation between two lengths?”
• “Did you have this before Descartes published?”
• “Why are your ordinates not always square to the axis?”
• “What happens to your figures when a quantity would be less than nothing?”
• “What was the quarrel with Descartes about tangents?”

Chapter 3 · The Letters of 1654

How a stake should be divided

1654 · Toulouse · Paris

In the summer of 1654 Blaise Pascal wrote to Fermat about a problem a gambling acquaintance had put to him. Two players are partway through a match for a stake and must stop before either has won. How should the stake be divided?

Pacioli had answered it in 1494, in proportion to the rounds already won, and been wrong. Cardano and Tartaglia had tried and failed. The reason it was hard is that everybody looked backwards at the score, and the answer depends on what would have happened NEXT.

Fermat's method is to enumerate. If at most a certain number of further rounds could settle the match, write down every way those rounds might have gone — every sequence, whether or not the match would actually have continued that far — and count the sequences in which each player wins. Divide the stake in that proportion. Pascal thought at first that counting sequences which would never have been played was illegitimate, and Fermat had to argue him round; the argument is the interesting part of the exchange.

Pascal had a different route, working backwards recursively from the end, and the two answers agreed. Half a dozen letters settled a problem that had defeated everybody for a century and a half, and in doing so established that questions about chance have determinate answers reachable by counting.

That correspondence is generally taken as the beginning of probability as a mathematical subject. Neither man published it; it appeared in Fermat's collected works in 1679 and in Pascal's posthumously.

Why this matters

Probability begins in half a dozen private letters about a gambling problem, and it begins by looking forwards instead of back.

Two of the best minds in France, settling a gambling question by post. What would you ask him?

Ask Fermat

• “How should the stake be divided, and why?”
• “Why does it matter what would have happened next rather than what has happened?”
• “Pascal objected to counting rounds that would never be played. Was he right?”
• “Did you think you were founding a new subject?”
• “What was it like to argue mathematics by post?”

Chapter 4 · The Theory of Numbers

Assertions, mostly true, mostly unproved

1636 – 1665 · Toulouse

Fermat's deepest work was on whole numbers, and it is the part of his output that most infuriated everybody.

He would announce, in a letter, that every prime one greater than a multiple of four can be written as the sum of two squares. Or that if a number is prime, then any other number raised to the power one less than it, and divided by it, leaves a remainder of one. Or that a particular equation has no solutions in whole numbers. These assertions are almost all true and almost all unproved — by him, in writing, where anybody could check.

He had genuine methods. The most powerful he called infinite descent: to show something is impossible, assume a solution exists, and construct from it a smaller solution of the same kind; since whole numbers cannot descend for ever, the assumption fails. That technique is his, and it is still used.

His contemporaries were not grateful. Whole numbers were unfashionable — Pascal and Descartes both thought the questions trivial — so his correspondents largely declined to engage, and he complained about it. The next mathematician to take him seriously was Euler, seventy years later, who worked through the assertions one by one over decades and proved most of them. Gauss finished much of the rest.

The result is that Fermat set the agenda for a subject for two hundred years, mostly by writing down things he did not explain.

Why this matters

A century of number theory consisted of proving things Fermat had merely asserted, which is a strange and instructive kind of influence.

A man who announced results and left the work to others. What would you ask him?

Ask Fermat

• “What is your method of infinite descent?”
• “Why did your contemporaries think whole numbers were beneath them?”
• “Did you actually have proofs for the things you asserted?”
• “How would you feel about Euler spending decades proving your claims?”
• “What is the most beautiful thing you found about whole numbers?”

Chapter 5 · The Margin Was Too Narrow

A note that took three hundred and fifty years

c. 1637 – 1994 · Toulouse · Princeton

Somewhere around 1637, reading his copy of Diophantus, Fermat wrote in the margin beside a problem about squares that it is impossible to write a cube as the sum of two cubes, or a fourth power as the sum of two fourth powers, or any power beyond the second as the sum of two like powers. He added that he had discovered a truly marvellous demonstration of this, which the margin was too narrow to contain.

He never mentioned it again in any surviving letter, which for a man who announced everything to his correspondents is telling. His son found the note when preparing the collected works and printed it with the rest.

It then defeated everybody for three hundred and fifty years. Euler settled the case of cubes. Sophie Germain made the first general progress. Kummer, in the nineteenth century, developed whole new machinery to attack it and disposed of a large class of cases. It became the most famous unsolved problem in mathematics, and generated a great deal of mathematics that had nothing to do with it.

Andrew Wiles proved it in 1994, after seven years working largely in secret, using the theory of elliptic curves and modular forms — mathematics that did not exist in any form until the twentieth century and could not have been imagined in the seventeenth. The proof runs to over a hundred pages.

Almost no historian believes Fermat had a valid proof. The most likely reconstruction is that he had an argument that works for fourth powers and thought, briefly, that it would generalise. He was thirty and he was reading in the margin of a book.

He died at Castres in January 1665, two days after last sitting in court.

Why this matters

A casual note in a book margin generated three and a half centuries of mathematics — and its author was almost certainly mistaken.

A sentence written in a margin, and three hundred and fifty years of consequences. What would you ask him?

Ask Fermat

• “Did you really have a proof, or did you think you did?”
• “Why did you never mention it in a single letter?”
• “What do you make of a proof that runs to a hundred pages?”
• “Does it matter that a wrong claim produced so much good mathematics?”
• “What would you most want proved, if you could choose one?”

What Fermat changed

Fermat devised coordinate geometry around 1636, before Descartes published, and every graph drawn since rests on the idea that a curve is a relation between two lengths. His 1654 letters with Pascal founded the mathematical theory of probability by looking forwards at what remained to be played rather than back at the score. His assertions about whole numbers set the agenda for the theory of numbers for two centuries, and his method of infinite descent is still in use. The note in his Diophantus margin became the most famous unsolved problem in mathematics and was settled only in 1994.

A debate that continues

Whether he had a valid proof of the marginal assertion is the standing question, and the consensus is firmly that he did not — Wiles's proof needed twentieth-century machinery, and Fermat never mentioned the claim again. His priority over Descartes in coordinate geometry is accepted on date and was never accepted by Descartes. His habit of announcing results without demonstrations means that in several cases nobody can be certain what he actually possessed.

Keep exploring — ask Fermat

• “Does a result belong to whoever finds it or whoever proves it?”
• “How much can be inferred from what somebody did NOT write down?”
• “Is a problem that generates good mathematics valuable even if its claim is false?”

Related lives

• René Descartes — Who Conserved The Quantity Of Motion
• Blaise Pascal — Who Carried A Barometer Up A Mountain
• Leonhard Euler — Who Gave Us The Notation

Related themes

Coordinate geometry · The beginnings of probability · Number theory