[Philosophers]


Picture of FermatPierre de Fermat


French Mathematician

8-17-1601 to 12-12-1665


Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont- de- Lomagne. He was probably educated in his early years at the local Franciscan monastery.

He attended the University of Toulouse before moving to Bordeau in the second half of the 1620s. In Bordeau he began his first serious mathematical researches and in 1629 he had completed his restoration of Apollonius's "Plane Loci." In Bordeau he was in contact with Beaugrand and produced important work on maxima and minima.

From Bordeau Fermat went to Orléans where he studied law at the University. He received a degree in civil law and he purchased the offices of councillor at the parliament in Toulouse.

Fermat's meteoric rise through the government is evidenced by his multiple appointments between 1631 and 1653. News of his death due to the plague of the 1650s was exaggerated:

I informed you earlier of the death of Fermat. He is alive, and we no longer fear for his health, even though we had counted him among the dead a short time ago.
The following report, made to Colbert the leading figure in France at the time, has a ring of truth:-
Fermat, a man of great erudition, has contact with men of learning everywhere. But he is rather preoccupied; he does not report cases well and is confused.
Of course Fermat was preoccupied with mathematics. He kept his mathematical friendship with Beaugrand after he moved to Toulouse but there he gained a new mathematical friend in Carcavi.

Around 26 April 1636 wrote that he had found mistakes in Galileo's description of free fall, and that he was working on spirals and his restoration of Apollonius's Plane loci. His work on spirals had been motivated by considering the path of free falling bodies and he had used methods generalised from Archimedes' work On spirals to compute areas under the spirals. In addition Fermat wrote:-

I have also found many sorts of analyses for diverse problems, numerical as well as geometrical, for the solution of which Viète's analysis could not have sufficed. I will share all of this with you whenever you wish and do so without any ambition, from which I am more exempt and more distant than any man in the world.
His interest in physics only extended to proving geometrical theorems and their relation to the real world. Soon after he divulged his new methods in letters he sent which included "Method for determining Maxima and Minima and Tangents to Curved Lines," his restored text of Apollonius's "Plane Loci" and his algebraic approach to geometry "Introduction to Plane and Solid Loci."

Although his reputation as a leading mathematician came quickly his attempts to publish his works failed because Fermat never wanted to put it into a polished form. Some of his methods were published, for example Hérigone added a supplement containing Fermat's methods of maxima and minima to his major work Cursus mathematicus.

Fermat received a copy of Descartes' "La Dioptrique" from mathematician Beaugrand, but paid it little attention since he was in the middle of a correspondence with Roberval and Etienne Pascal over methods of integration and using them to find centers of gravity. Mersenne asked him for his opinion on "La Dioptrique" which Fermat did describing it as

"groping about in the shadows."
He claimed that Descartes had not correctly deduced his law of refraction since it was inherent in his assumptions. Descartes was furious, becuase he felt Fermat's work on maxima, minima and tangents reduced the importance of his own work "La Géométrie" which Descartes was most proud of and which he sought to show that his "Discours de la méthod" alone could give.

He attacked Fermat's method of maxima, minima and tangents. Roberval and Etienne Pascal became involved in the argument and eventually so did Desargues who Descartes asked to act as a referee. Fermat proved correct and eventually Descartes admitted this writing:-

... seeing the last method that you use for finding tangents to curved lines, I can reply to it in no other way than to say that it is very good and that, if you had explained it in this manner at the outset, I would have not contradicted it at all.
Descartes now tried to damage Fermat's reputation. He wrote to Fermat praising his work on determining the tangent to a cycloid (which is indeed correct), meanwhile writing to Mersenne claiming that it was incorrect and calling Fermat an inadequate mathematician and a thinker. Descartes was important and respected and was able to severely damage Fermat's reputation.

Fermat is best remembered for this work in number theory, in particular for Fermat's Last Theorem. This theorem states that

Xn + Yn = Zn
has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote, in the margin of Bachet's translation of Diophantus's "Arithmetica"
I have discovered a truly remarkable proof which this margin is too small to contain.
These marginal notes only became known after Fermat's son Samuel published an edition of Bachet's translation of Diophantus's Arithmetica with his father's notes in 1670.

It is believed that Fermat's 'proof' was wrong. The truth of Fermat's assertion was thought proved in June 1993 by the British mathematician Andrew Wiles, but Wiles withdrew the claim to have a proof when problems emerged later in 1993. In November 1994 Wiles again claimed to have a correct proof which has now been accepted.

Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries.

Fermat's correspondence with the Paris mathematicians restarted in 1654 when Blaise Pascal, Etienne Pascal's son, wrote to him to ask for confirmation about his ideas on probability. Fermant was interested in the topic of applying probability to number theory, in which Pascal was not interested.

In the 1650s one of Descartes' students was collecting his correspondence for publication and he turned to Fermat for help with the Fermat - Descartes correspondence. Fermat again looked at his arguments and objections to Descartes' optics. He now deduced that light always follows the shortest possible path. Fermat's principle, now one of the most basic properties of optics, did not find favour with mathematicians at the time.

In 1656 Huygens and Fermat had a correspondence; out of this came the topic of number theory. It did not interest Huygens but Fermat produced "New Account of Discoveries in the Science of Numbers," revealing more of his methods than he had done to others.

He described his method of infinite descent and gave an example on how it could be used to prove that every number of the form 4k+1 could be written as the sum of two squares.

Fermat was described in [3] as

Secretive and taciturn, he did not like to talk about himself and was loath to reveal too much about his thinking. ... His thought, however original or novel, operated within a range of possibilities limited by that [1600-1650] time and that [France] place.
Only later were Euler and other mathematicians able to divine his methods of solving complex number theory problems. Fermat remained secretive to the grave.

Some further problems Fermat worked on:

Find all solutions of Nx + 1 = y for N not a square.

Prove that the sum of two cubes cannot be a cube (a special case of Fermat's Last Theorem which may indicate that by this time Fermat realised that his proof of the general result was incorrect)

Prove that there are exactly two integer solutions of x + 4 = y

Prove that the equation x + 2 = y has only one integer solution.

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