Fermat Essay, Research Paper

While Descartes was laying the foundations of analytical geometry, the same subject was occupying the attention of another and not less distinguished Frenchman. This was Pierre de Fermat, who was born near Montauban in 1601, and died at Castres on January 12, 1665. Fermat was the son of a leather-merchant, growing up being educated at home. He earned himself a name later in his life by making contacts with some of the most important people in mathematics during that time period. In this paper I will discuss his life works and the contacts he made who brought his work into light.

Fermat attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 gave a copy of his restoration of Apollonius’s Plane Loci to one of the mathematicians there. In Bordeaux he was in contact with Beaugrand(well-known mathematician) and during this time produced important work on maxima and minima, which he gave to Etienne d’Espagnet who clearly shared mathematical interests with Fermat.

From Bordeaux, Fermat went to Orleans where he studied law at the University. He received a degree in civil law and purchased offices of councilor at the parliament in Toulouse. So by 1631, Fermat was a lawyer and government official in Toulouse.

For the remainder of his life he lived in Toulouse working there, Beaumont-de-Lomagne, and the town of Castres. From May 14, 1631 until January 16, 1638, Fermat worked in the lower chamber of the parliament. In 1638, he was appointed to a higher chamber, then in 1652 he was promoted to the highest level at the criminal court. Still further promotions seem to indicate a meteoric rise through the profession. On the other hand, promotion was done mostly on seniority and the plague struck the region in the early 1650’s meaning that many of the older men died.

While all this was happening in Fermat’s life, every spare moment he got was devoted to mathematics. He had kept his friendship with Beaugrand after he moved and gained another mathematical friend, Pierre de Carcavi. Fermat met Carcavi while they were both councilors in Toulouse and they both shared a love for mathematics. Fermat shared all of his private discoveries that he had made with him.

In 1636, Carcavi went to Paris as royal Librarian and made contact with Mersenne (a priest who corresponded with eminent mathematicians and played a major role in communicating mathematical knowledge throughout Europe at a time when there were no scientific journals) and his group. Mersenne’s interest was aroused by Carcavi’s descriptions of Fermat’s discoveries on falling bodies, and he wrote to Fermat. Fermat replied on April 26, 1936 and in addition to telling Mersenne about his errors which he believed Galileo had made in his description of free fall, he also wrote about his work 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 generalized from Archimedes(ancient mathematician) to compute areas under spirals. It is somewhat ironical that this initial contact with Fermat and the scientific community came through his study of free fall since Fermat had little interest in physical applications of mathematics. Even with his results on free fall he was much more interested in proving geometrical theorems than in their relation to the real world. This first letter did however contain two problems on maxima which Fermat asked Mersenne to pass on to the Paris mathematicians and this was to be the typical style of his letters. He would challenge others to find results which he had already obtained.

Robervale(a well-known math teacher who was known widely for his discoveries on plane curves and for his method for drawing tangent to a curve) and Mersenne found that Fermat’s problems in this first, and subsequent letters to be extremely difficult and usually not soluble using current techniques. They asked him to divulge his techniques and Fermat sent 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 to the Paris mathematicians.

Fermat’s reputation as one of the leading mathematicians in the world came quickly but attempts to get his work published failed mainly because he never really wanted to put his work in polished form. However, bits and pieces of his work sprang up time to time. For example, Herigone( a server on many mathematical committees) added a supplement containing Fermat’s methods of maxima and minima to his major work Cursus mathematics. The widening correspondence between Fermat and other mathematicians did not find universal praise. Frenicle de Bessy( a well known amateur mathematician) found Fermat’s problems to be obsurd and took an immediate disliking towards everything he did. However, Fermat soon became engaged in a controversy with a more major mathematician than Frenicle de Bessy. Having been sent a copy of Descartes’ La Dioptrique by Beaugrand, Fermat 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 hime to give an opinion on La Dioptrique which Fermat described 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. To say that Descartes was not pleased was an understatement. Descartes soon found reason to feel even more angry since he viewed Fermat’s work on maxima, minima, and tangents as reducing the importance of his own work La Geometrie, which Descartes was most proud of and which he sought to show that his Discours de la method alone could give.

Descartes attacked Fermat’s method of maxima, minima, and tangents. Roberval and 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.

Did this end the matter and increase Fermat’s standing? Not at all since Descartes tried to damage Fermat’s reputation. For example, although he wrote to Fermat praising his work on determining the tangent to a cycloid, Descartes wrote to Mensenne claiming that it was incorrect and saying that Fermat was inadequate as a mathematician and a thinker. Descartes was important and respected and thus was able to severely damage Fermats reputation.

The period from 1643 to 1654 was one when Fermat was out of touch with his scientific colleagues in Paris. There are a number of reasons for this. Firstly, pressure of work kept him from devoting so much time to mathematics. Second, the Fonde, a civil war in France, took place in 1648 and Toulouse was greatly affected. Finally, there was the plague of 1651 which must have had great consequences both on life in Toulouse and its near fatal consequences on Fermat himself. It was at this time when Fermat started his work on what he is best known for.

Fermat is best remembered for his work in number theory, in particular for Fermat’s Last Theorem. This theorem states that x^n+y^n=z^n. 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 work with his father’s notes in 1670.

It is now believed that Fermat’s proof was wrong although it is impossible to be completely certain. The truth of Fermat’s assertion was proved in June 1993 by the British mathematician Andrew Wiles, but Wles withdrew the claim to have proof when problems emerged later in 1993. In November of 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. This is where many believed Fermat’s genious lied. His ability to create almost unsolvable problems which would generate applications that we still use today.

Fermat’s correspondence with the Paris mathematicians restarted in 1654 when Blaise Pascal, son of Etienne Pascal, wrote to ask for confirmation about his ideas on probability. Blaise knew of Fermat through his father, who had died three years before, and was well aware of Fermat’s outstanding mathematical abilities. Their short correspondence set up a theory of probability and from this they are now regarded as joint founders of the subject. Fermat however, feeling his isolation and still wanting to adopt his old style of challenging mathematics, tried to change the topic from probability to number theory. Pascal was not interested and therefore once again Fermat’s hope of being published was rejected.

Fermat’s problems in number theory did not prompt too much interest as most mathematicians seemed to think that number theory was not an important topic. The second of two of his problems, namely to find all solutions of Nx^2+1=y^2 for N not a square, was however solved by Wallis and Brouncker. Brouncker produced rational solutions which led to arguments. Frenicle de Bessy was interested in number theory but his feeling toward Fermat we have already discussed.

Fermat posed further problems, namely that the sum of two cubes cannot be a cube (a special case of Fermat’s Last Theorem which may indicate that his proof of the general result was incorrect), that there are exactly two integer solutions of x^2+4+y^3. and that the equation x^2+2=y^3 has only one integer solution. He posed problems directly to the English. Everyone failed to see that Fermat had been hoping his specific problems would lead them to discover, as he had done, deeper theoretical results.

Around this time one of Descartes’students was collecting his correspondence for publication and he turned to Fermat for help with the Fermat-Descartes corrspondence. This led Fermat to look again at the argumanets he had used 20 years before and he looked again at his objections to Descartes’ optics. In particular, he had been unhappy with Descartes’ description of refraction that Snell and Descartes had proposed. However, Fermat had now deduced it from a fundamental property that he proposed, namely that light always follows the shortest possible path. Fermat’s principle, now one of the most basic properties of optics, did not favor with mathematicians at the time.

In 1656, Fermat had started a correspondence with Huygens. This grew out of Huygens interest in probability and the correspondence was soon manipulated by Fermat onto topics of number theory. This topic did not interest Huygens but Fermat tried hard and in New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of his methods than he had done to others.

Fermat described his method of infinite descent and gave an example on how it could be used to prove that every prime of the form 4k+1 could be written as the sum of two squares. For suppose some number of the form 4k+1 could not be written as two squares. Then there is a smaller number of the form 4k+1 which cannot be written as the sum of two squares. Continuing the argument will lead to a contradiction. What Fermat failed to explain in this letter is how the smaller number is constructed from the larger. One assumes that Fermat did know how to make this step, but again his failure to disclose the method made mathematicians lose interest. It was not until Euler took up these problems that the missing steps were filled in.

In conclusion, Pierre de Fermat was one of the most influential mathematicians of all history even though not too many people could understand his work. By coming up with problems that seemed to have no answer, he generated theories and applications that we still use today. Although almost none of his work was ever published, we can still have a great understanding of most of his work through the letters he sent to the Paris mathematicians and the people who worked closely with him.

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