Bernard Bolzano

Bernhard Bolzano (d. 5 Ekim 1781, Prag - ö. 18 Aralık 1848, Prag), İtalyan asıllı bir Çek filozof ve matematikçi.

Babası bir İtalyan göçmeni ve küçük bir esnaftı. Annesi de, Prag'da madeni eşya ile ilgilenen bir ailenin kızıydı. Bolzano, Prag Üniversitesi'nde, felsefe, fizik, matematik ve ilahiyat çalıştı. 1807 yılında Prag'da aynı üniversiteye din ve felsefe profesörü olarak atandı. 1816 yılına kadar bu üniversitede başarılı dersler verdi. 1816 yılında, Hristiyan kilisesince benimsenen inanç, duygu ve düşünceye ters düştüğü için, bu inançlarından dolayı suçlandı. 1820 yılındaAvusturya hükümeti Bolzano'nun bu yıkıcı ve kendileri için kırıcı olan konuşmalarından dolayı onu ülkeden uzaklaştırdı.

Bolzano, İtalya asıllı bir Çek filozofuydu. Aynı zamanda iyi bir mantıkçı ve çok iyi de bir matematikçiydi. Bolzano, 1820 yılında daha çok akılcılıkla suçlandı. Onun matematiğe dayalı bir felsefesi ve düşüncesi vardı. Bu nedenle Kant'ın idealizmine karşı çıktı. Kendisi aslında bir Katolik papazıydı. 1805 yılından sonra Prag Üniversitesi'nde din felsefesi okuttu. Matematikte, sonsuzluk ve sonsuz küçükler hesabı üzerinde çalıştı. "Sonsuzluk Üzerine Paradokslar" adlı kitabı 1851 yılında yayımlandı. Noktasal kümeler üzerine de çalışmaları olmuştur.

Bolzano'nun en acıklı yılları, 1819 ile 1825 yılları arasına rastlar. Prag Üniversitesi'nce, tam 7 yıl ders vermeme ve yayın yapmamak üzere cezalandırılır. Bu üniversitece profesörlüğü de elinden alınır. Tüm bu baskılara karşı onun yüksek kafası hiç durmadan çalışmıştır. Analizde, geometride, mantıkta, felsefede ve din üzerinde çok sayıda yayınını gerçekleştirmiştir. Bugün, analizde bildiğimiz ünlü Bolzano-Weierstrass teoremi'ni ilk kez "Fonksiyonlar" adlı kitabında o kullandı. Fakat, teoremin ispatını daha önceki çalışmalarında yaptığını ve kaynak olarak da bu çalışmasını verir. Fakat, sözü edilen bu çalışma ve kaynak bugüne kadar bulunamamıştır. Çok kullanılan ve kendisinin de çok kullandığı bir teoremin ispatının Bolzano tarafından verilmiş olması olasılığı çok fazladır. Zaten bu teoremin ispatı verilmeseydi, Bolzano tarafından bu kadar çok kullanılmazdı. Sonraki yıllarda bu teoremin ispatı tam olarak Weierstrass tarafından verilmiştir. Bu nedenle, bu teorem analizde Bolzano-Weierstrass teoremi olarak bilinir.

Bolzano'nun temel çalışmaları, sonsuzlar paradoksu üzerinedir. Bolzano'ya yayın yapma yasağı konduğu için, yaşamı sürecinde bu eserlerini ne yazık ki yayınlayamamıştır. "Sonsuzlar Paradoksları" adlı çalışması ancak onun ölümünden iki yıl sonra 1850 yılında basılmıştır. Bu çalışması, sonsuz terimli serilerin birçok özelliğini içerir. Diğer birçok matematikçide olduğu gibi yaşam sürecinde çok hırpalanan, şanssızlıklarla ve baskılarla horlanan Bolzano, 18 Aralık 1848 günü Prag'da öldü.

Bernard Bolzano's parents were Bernard Pompeius Bolzano and Maria Cecilia Maurer. His mother Maria, the daughter of a Prague merchant, was German speaking and a devout Roman Catholic. Bernard Bolzano senior, the father of the subject of this biography, was born in the north of Italy and had emigrated to Prague. He made his living as a dealer in art, but was a man of modest means. Well educated, he was also a pious Roman Catholic who showed great concern towards others. An indication of how seriously he put his beliefs into practice is the fact that he was the driving force behind the founding of an orphanage in Prague. Bernard, the subject of this biography, was born in the oldest part of the city of Prague, being the fourth of his parents twelve children. Despite the large family, Bernard and one of his brothers, Johann, were the only two to reach adulthood. His health, however, was delicate and he had to fight against respiratory problems throughout his life.

Bolzano's upbringing was a major factor in the ideas that he taught later in his life. He was much influenced by his father's active attempts to care for his fellow men, and this was strengthened by the Piarist Gymnasium that Bolzano attended in Prague between 1791 and 1796. There he was taught by the Roman Catholic followers of Joseph Calasanz who was the Spanish founder of the Ordo Clericorum Regularium Pauperum Matris Dei Scholarum Piarum (Order of Poor Clerks Regular of the Mother of God of the Pious Schools). Although Spanish, Calasanz founded Order, usually called by the name Piarists, in Rome at the beginning of the 17^th century. Members of the Piarist Order were teachers who made a fourth vow to take special care of young people in addition to the usual three vows made by monks. It is fair to say that Bolzano left this environment more convinced of the moral beliefs, which had been foremost in his upbringing and in his schooling, than in the purely religious Christian beliefs.

Bolzano entered the Philosophy Faculty of the Charles University of Prague in 1796, studying philosophy, physics and mathematics. Bolzano was particularly influenced in his mathematical studies by reading Kaestner'sMathematische Anfangsgründe. Kaestner was concerned with philosophical questions in mathematics, was deeply interested in the philosophy of mathematics, and took great care to prove many results which were thought "obvious", so not requiring proof, by other mathematicians of the day. Bolzano, who soon developed a strong belief in this approach, wrote:-

My special pleasure in mathematics rested therefore particularly on its purely speculative parts, in other words I prized only that part of mathematics which was at the same time philosophy.

During the year 1799-1800 Bolzano undertook research in mathematics with Frantisek Josef Gerstner and also contemplated his future. In the autumn of 1800, against his father's wishes, he began three years of theological study at the Charles University. While pursuing his theological studies he prepared a doctoral thesis on geometry. He received his doctorate in 1804 writing a thesis giving his view of mathematics, and what constitutes a correct mathematical proof. In the preface he wrote:-

I could not be satisfied with a completely strict proof if it were not derived from concepts which the thesis to be proved contained, but rather made use of some fortuitous, alien, intermediate concept, which is always an erroneous transition to another kind.

Two days after receiving his doctorate Bolzano was ordained a Roman Catholic priest. However, as Russ points out in [50]:-

He came to realise that teaching and not ministering defined his true vocation.

In fact his years of study of theology had done nothing to strengthen his acceptance of the religious beliefs on which Christianity is founded. However, his professor at the Charles University had put forward an argument which Bolzano had found very persuasive, namely that faith in a doctrine was justified if it led to moral good. Later in his writings Bolzano argued for a more general form of this argument:-

Of all actions possible to you, choose always the one which, weighing all consequences, will most further the good of the totality, in all its parts.

This allowed Bolzano to accept the mystical elements of Christianity for the greater good of mankind, although he did not accept them to be historically true.

In order to understand the events of Bolzano's life, we need to fill in a little background about the situation in Bohemia. From early in the 17^th century, after Roman Catholic forces defeated the Bohemian Protestants, Bohemia had been ruled by the Habsburgs. It was absorbed into the Austrian Empire, German became the language of instruction in grammar schools and the Charles University, and Czech nationalism was suppressed. In 1781 Joseph II granted religious tolerance but the French Revolution brought in an era of free thinking which the rulers feared. To hold the Empire together, the rulers decided to move against all nationalistic organisations which they believed encouraged ideas of independence for various parts of the Empire. In 1804, Emperor Franz who ruled from Vienna, decided that one way to counter the impact of the enlightenment was to strengthen the hold of the Roman Catholic Church which tended to be very conservative and opposed to liberal thinking. He set up chairs of philosophy of religion in the universities as one of the means to achieve his aims.

Chairs in the universities were filled by competition and Bolzano entered two such competitions for chairs at the Charles University. One was for the chair of mathematics which became vacant on the death of Stanislaw Vydra, the other being for the new chair in the philosophy of religion which Emperor Franz had just established. Bolzano came top in both competitions, but the university preferred to give him the chair in the philosophy of religion since they were then able to give the mathematics chair to Ladislaw Jandera who had substituted for Vydra during his illness between 1801 and 1804. In many ways Bolzano was exactly the wrong person to fill this chair given the reasons for its creation, for he stood for all the ideas which Franz feared, being a free thinker who believed in social justice, pacifism, and equality for the Czech speaking Bohemians.

His appointment to the chair had to be confirmed in Vienna, and they certainly understood that he was not the Roman Catholic conservative that they had been hoping would be appointed. Confirmation of his appointment was delayed while Vienna considered what its best action should be, but in 1807 it was granted [10]:-

... he lectured on religion and moral philosophy with strong pacifistic and socialistic overtones. He used the pulpit to proclaim before hundreds of impressed students a kind of utopian socialism. In his sermons he tried to prove the essential equality of all human beings, attacked private property obtained without work, and exhorted his listeners to sacrifice everything in their struggle for human rights.

The appointment of Bolzano was viewed with suspicion by the Austrian rulers in Vienna. He criticised discrimination wherever he saw it, principally by the German speaking Bohemians against their Czech fellow citizens, and also he criticised the anti-Semitism displayed by both the German and Czech Bohemians. Some members of the Roman Catholic Church were also unhappy because Bolzano's lectures contains elements of rationalism. He certainly had supporters within the Church, for example the important Archbishop of Prague and Dr Fessl who directed the seminary of Leitmeritz. In 1815 Bolzano was elected to the Royal Bohemian Society of Sciences which was bilingual society drawing its members mainly from the German speakers but also from Czech speakers. Bolzano published On the Condition of the Two Nationalities in Bohemia in 1816 in which he put into print his concerns that the Czech Bohemians were dominated by the German speaking Bohemians. The peasants were Czech speaking, the cities largely inhabited by German speakers but Bolzano saw the problems which were being created due to increasing industrialisation which saw Czech speakers moving into the cities. Bolzano's career continued to flourish, despite the fact that charges were bought against him at the Vienna court in 1816, and in 1818 he was elected Dean of the Faculty of Philosophy at Charles University.

Bolzano was suspended from his position in December 1819 after pressure from the Austrian government. In addition to being suspended from his professorship, he was put under house arrest, had his mail censored, and not allowed to publish. Between 1821 and 1825 he was tried by the Church, and despite his strong defence of his views, was required to recant his supposed heresies. He refused to do so and resigned his chair. From 1823 he had spent the summers living near the village of Techobuz in southern Bohemia, on the estate of his friends Josef and Anna Hoffmann, while he spent the winter living in Prague with his brother Johann. J J I Hoffmann was a mathematician who had reviewed several of Bolzano's earlier works (see [54]). Between 1830 and 1841 he lived throughout the year with the Hoffmanns, having much time to devote to study. When Anna Hoffmann took ill in 1841, Bolzano and the Hoffmanns moved to Prague where they all lived with Johann Bolzano (Anna died in 1842). There Bolzano again became an active member of the Royal Bohemian Society of Sciences and was president during 1842-43. He had suffered from respiratory problems for most of his life and these became more severe as he grew older. In the winter of 1848 he contracted a cold which, given the poor condition of his lungs, led to his death.

Although some of his books had to be published outside the Austrian Empire because of government censorship, he continued to write and to play an important role in the intellectual life of his country. In fact he had won a partial lifting of the publication ban, and he was only forbidden from publishing anything of a religious or political nature.

Bolzano wrote Beyträge zu einer begründeteren Darstellung der Mathematik. Erste Lieferung (1810), the first of an intended series on the foundations of mathematics. Bolzano wrote the second of his series but did not publish it. Instead he decided to:-

... make myself better known to the learned world by publishing some papers which, by their titles, would be more suited to arouse attention.

Pursuing this strategy he published Der binomische Lehrsatz ... (1816) and Rein analytischer Beweis ... (Pure Analytical Proof) (1817), which contain an attempt to free calculus from the concept of the infinitesimal. He is clear in his intention stating in the preface of the first of these that the work is:-

... a sample of a new way of developing analysis.

Although Bolzano did achieve exactly what he set out to achieve, he did not do so in the short term, his ideas only becoming well known after his death. In [50] Russ describes Bolzano's aims in the 1817 paper:-

In this work ... Bolzano ... did not wish only to purge the concepts of limit, convergence, and derivative of geometrical components and replace them by purely arithmetical concepts. He was aware of a deeper problem: the need to refine and enrich the concept of number itself.

The paper gives a proof of the intermediate value theorem with Bolzano's new approach and in the work he defined what is now called a Cauchy sequence. The concept appears in Cauchy's work four years later but it is unlikely that Cauchy had read Bolzano's work.

After 1817, Bolzano published no further mathematical works for many years. However, in 1837, he published Wissenschaftslehre , an attempt at a complete theory of science and knowledge. Between sometime before 1830 and the 1840s, Bolzano worked on a major work Grössenlehre. This attempted to put the whole of mathematics on a logical foundation was published in parts, while Bolzano hoped that his students would finish and publish the complete work. His work on paradoxes Paradoxien des Unendlichen, a study of paradoxes of the infinite, was published in 1851, three years after his death, by one of his students. The word set appears here for the first time. In this work Bolzano gives examples of 1-1 correspondences between the elements of an infinite set and the elements of a proper subset.

Most of Bolzano's works remained in manuscript and did not become noticed and therefore did not influence the development of the subject. Many of his works were not published until 1862 or later. Bolzano's theories of mathematical infinity anticipated Georg Cantor's theory of infinite sets. It is also remarkable that he gave an example of a function which is nowhere differentiable yet everywhere continuous.

Attempts to publish Bolzano's manuscripts are described in our article Bernard Bolzano's manuscripts.

As an example of the mathematics that Bolzano was working on while he was professor of the philosophy of religion, here is a description of what he recorded in his notebook Miscellanea mathematica in 1816. The description is by Dauben reviewing this material when it was first published in 1996:-

Bolzano opens this notebook of Miscellanea mathematica with notes on irrational and transcendental numbers and functions. But he was reading and recording his ideas on a host of other subjects as well, including the problem of how best to approach the proper mathematical understanding of zero; Legendre's work on surfaces, convexity, concavity, and conditions for congruity; analysis of other geometric concepts, including lengths, areas, volumes, and spheres; trigonometric formulas and spherical trigonometry; imaginary and exponential numbers; definition of the differential and discussion of the infinite and various opinions about it, as well as aspects of maxima and minima. ... Other topics covered here include various approaches to the calculus(including the method of exhaustion), and grounds for asserting the certainty of mathematics.

In addition to his mathematical work, Bolzano was important as a philosopher and as a logician. We mention briefly two major works. First there is Lehrbuch der Religionswissenschaft (Textbook of the Science of Religion) (1834) in which he bases his religious philosophy on ethics [35]:-

He criticises Kant's categorical imperative and his doctrine of postulates, and advocates a version of utilitarianism.

His second major work which we mention is Wissenschaftslehre (Theory of Science) published in four volumes in 1837. The first two volumes cover his ideas on the philosophy of logic, the third volume presents a theory of scientific discovery, while the final volume presents his methodology of writing textbooks [17]:-

Bolzano's theory of science (Wissenschaftslehre) contains a great amount of very valuable information concerning the development of logic from its beginnings in Aristotle till the post-Kantian period. In a critical exposition, Bolzano presents views of his predecessors and compares them with his own point of view.

Pierre de Fermat

Pierre de Fermat (Fransızca telaffuz: [pjɛːʁ dəfɛʁˈma]) (17 Ağustos 1601; Beaumont-de-Lomagne – 12 Ocak 1665; Castres), Bask kökenli Fransızhukukçu ve matematikçi. İlk öğrenimini doğduğu şehirde yapmıştır. Yargıç olmak için çalışmalarına Toulouse’de devam etmiştir. Fermat, memurluğunun yoğun işlerinden geriye kalan zamanlarında matematikle uğraşmıştır. Arşimet'in eğildiği diferansiyel hesaba geometrik görünümle yaklaşmıştır. Sayılar teorisinde önemli sonuçlar bulmuş, olasılık ve analitik geometriye de katkılarda bulunmuştur.

Günümüzde hatırlanmasının en önemli sebebi Fermat'nın son teoremi'dir. Modern sayılar kuramının kurucusu olarak kabul edilen 17. yüzyıl matematikçisi Pierre de Fermat'nın adını taşıyan bu teorem, şu şekilde ifade edilebilir:

Herhangi x, y, ve z pozitif tam sayıları için,

ifadesini sağlayan ve 2'den büyük bir doğal sayı n yoktur. Fermat, bu problemi çözmüş, kanıtı da Eski Yunanlı matematikçi Diaphontos'unArithmetika adlı kitabının kendindeki kopyasının sayfalarından birinin kenarına 1637'de şöyle yazmıştı:

x, y, z ve n pozitif tamsayılar ve n>2 olmak koşuluyla, xⁿ + yⁿ = zⁿdenkleminin çözümü yoktur. Ben bunun kanıtını buldum, ama kanıtı bu kenar boşluğuna sığdırmak olanaksız.

Ancak bu kanıt bulunamamıştır. Fermat'tan sonra matematikçiler bu önermenin bir türlü içinden çıkamamışlardır. Fermat'ın bıraktığı defterler arasında teoremin kanıtına rastlayamadıkları gibi, kendileri de ne doğruluğunu ne yanlışlığını kanıtlayabilmişlerdir. Yıllar boyunca (300 yıl sonrasına kadar) bu konuda yapılan çalışmalar sonucu bu teoremin Shimura-Taniyama Konjektürü'nün bir özel durumu olduğu anlaşılmış, ardından da 1993'te İngilizmatematikçi Andrew Wiles, eski öğrencilerinden Richard Taylor'ın da yardımıyla ve cebirsel geometrinin çok karmaşık araçlarını kullanarak teoremi kanıtlamanın bir yolunu bulmuş ve bu kanıtı 1995'te Annals of Mathematics adlı dergide yayımlamıştır. Shimura-Taniyama Konjektürü'nün böylelikle ispatlanması sonucu Fermat'nın Son Teoremi de 1995'te ispatlanmış oldu.

Asal sayılar üzerinde çok durmuştur. Onun bu konuda çeşitli teoremleri vardır. Örneğin,

şeklinde yazılan bir asal sayı p, yalnızca bir tek şekilde iki karenin toplamı olarak yazılabilir.

Mesela en ufak asal sayılar p:

 ve 

dir. Bu teoremi daha sonra Euler kanıtlamıştır.

Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont- de- Lomagne. There is some dispute [14] about the date of Pierre's birth as given above, since it is possible that he had an elder brother (who had also been given the name Pierre) but who died young. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. Although there is little evidence concerning his school education it must have been at the local Franciscan monastery.

He 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 he gave a copy of his restoration ofApollonius's Plane loci to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat.

From Bordeaux 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. So by 1631 Fermat was a lawyer and government official in Toulouse and because of the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat.

For the remainder of his life he lived in Toulouse but as well as working there he also worked in his home town of Beaumont-de-Lomagne and a nearby town of Castres. From his appointment on 14 May 1631 Fermat worked in the lower chamber of the parliament but on 16 January 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 fairly meteoric rise through the profession but promotion was done mostly on seniority and the plague struck the region in the early 1650s meaning that many of the older men died. Fermat himself was struck down by the plague and in 1653 his death was wrongly reported, then corrected:-

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. Fermat met Carcavi in a professional capacity since both were councillors in Toulouse but they both shared a love of mathematics and Fermat told Carcavi about his mathematical discoveries.

In 1636 Carcavi went to Paris as royal librarian and made contact with Mersenne 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 26 April 1636 and, in addition to telling Mersenne about errors which he believed that Galileo had made in his description of free fall, he also told Mersenne about his work on spirals and his restoration ofApollonius'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.

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 Fermat's letters, he would challenge others to find results which he had already obtained.

Roberval and Mersenne found that Fermat's problems in this first, and subsequent, letters were extremely difficult and usually not soluble using current techniques. They asked him to divulge his methods and Fermat sentMethod 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.

His reputation as one of the leading mathematicians in the world came quickly but attempts to get his work published failed mainly because Fermat never really wanted to put his work into a polished form. However 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. The widening correspondence between Fermat and other mathematicians did not find universal praise. Frenicle de Bessy became annoyed at Fermat's problems which to him were impossible. He wrote angrily to Fermat but although Fermat gave more details in his reply,Frenicle de Bessy felt that Fermat was almost teasing him.

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 Étienne Pascal over methods of integration and using them to find centres of gravity. Mersenne asked him to give an 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. To say that Descartes was not pleased is 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 Géométrie which Descartes was most proud of and which he sought to show that his Discours de la méthode alone could give.

Descartes attacked Fermat's method of maxima, minima and tangents. Roberval and Étienne 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 (which is indeed correct), Descartes wrote to Mersenne 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 Fermat's 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. Secondly the Fronde, a civil war in France, took place and from 1648 Toulouse was greatly affected. Finally there was the plague of 1651 which must have had great consequences both on life in Toulouse and of course its near fatal consequences on Fermat himself. However it was during this time that Fermat worked on number theory.

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 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 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, Étienne Pascal's son, wrote to him to ask for confirmation about his ideas on probability. Blaise Pascal 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 the 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 mathematicians, tried to change the topic from probability to number theory. Pascal was not interested but Fermat, not realising this, wrote to Carcavi saying:-

I am delighted to have had opinions conforming to those of M Pascal, for I have infinite esteem for his genius... the two of you may undertake that publication, of which I consent to your being the masters, you may clarify or supplement whatever seems too concise and relieve me of a burden that my duties prevent me from taking on.

However Pascal was certainly not going to edit Fermat's work and after this flash of desire to have his work published Fermat again gave up the idea. He went further than ever with his challenge problems however:-

Two mathematical problems posed as insoluble to French, English, Dutch and all mathematicians of Europe by Monsieur de Fermat, Councillor of the King in the Parliament of Toulouse.

His problems did not prompt too much interest as most mathematicians seemed to think that number theory was not an important topic. The second of the two problems, namely to find all solutions of Nx² + 1 = y² for N not a square, was however solved by Wallis and Brouncker and they developed continued fractions in their solution. Brouncker produced rational solutions which led to arguments. Frenicle de Bessy was perhaps the only mathematician at that time who was really interested in number theory but he did not have sufficient mathematical talents to allow him to make a significant contribution.

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 by this time Fermat realised that his proof of the general result was incorrect), that there are exactly two integer solutions of x² + 4 = y³ and that the equation x² + 2 = y³ 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 correspondence. This led Fermat to look again at the arguments 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 of light and he now settled on a principle which did in fact yield the sine law 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 find favour 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 the sum of 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.

Fermat is described in [9] 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.

Carl B Boyer, writing in [2], says:-

Recognition of the significance of Fermat's work in analysis was tardy, in part because he adhered to the system of mathematical symbols devised by François Viète, notations that Descartes' "Géométrie" had rendered largely obsolete. The handicap imposed by the awkward notations operated less severely in Fermat's favourite field of study, the theory of numbers, but here, unfortunately, he found no correspondent to share his enthusiasm.

Augustin Louis Cauchy

Augustin Louis Cauchy, 21 Ağustos 1789 Paris'te; 23 Mayıs 1857 Sceaux'de), Fransız matematikçisi.

Cauchy, 1789’da Paris’te doğdu. 1814 yılında, karmaşık fonksiyonlar kuramını geliştirdi. Bugün, Cauchy teoremi adıyla bilinen ünlü teoremi ifade ederek ispatladı. Bu alanda integraller ve bunların hesaplama yöntemleri yine Cauchy tarafından verildi. Bu sahadaki eseri 1827 yılında basıldı. 1815 yılında,Fermat'ın bir teoreminin ispatını verdi. 1816 yılında sıvılar üzerinde dalgaların yayılmasının kuramını içeren yaptıyla Akademi ödülünü aldı. 1815 yılında Polytechnique’te analiz öğretmeni ve profesör oldu. Sorbonne'a ve College de France'a girdi. Her işte başarılı oluyordu. Akademiye haftada iki çalışma sunuyordu. Geliştirdiği ve yaptığı çalışmaları öğrenmek için Avrupa’nın her yanından matematikçiler geliyordu. 1816 yılında Akademiye başkan seçildi.

1816 yılından itibaren cebir ve mekanik dersleri vermeye başladı. 1830 devriminden sonra bağlılık andını kabul etmediği için görevinden ayrıldı veTorino'ya giderek kendisi için açılan matematik kürsüsünde çalışmaya başladı. 1833'te Bordeaux Dükü'nün fen eğitimini yönetmek üzere Prag'a çağrıldı.1838'de Paris'e döndü. Paris Fen Fakültesi matematiksel gökbilim profesörlüğüne atandı ve 1852 yılına dek bu görevine devam etti. Cauchy, arı ve uygulamalı matematiğin bütün bölümleriyle ilgilendi. Ama tarihe çözümleme üstüne yaptığı çalışmalarla geçti. 1821'de yayımlanan Cours d’analyse adlı kitabında çözümlemenin ana ilkelerini gözden geçirdi ve bunları yapıcı bir biçimde eleştirdi; böylece elementer fonksiyonların ve serilerin incelenmesine kesinlik kazandırdı.

Cauchy her şeyden önce, karmaşık bir değişkenin fonksiyonları kuramının yaratıcısıdır. Bu konuda çıkış noktası karmaşık bölgelerde integrallemeydi (1814 - 1830): eğrisel integrali tanımladı, bunun temel özelliklerini kanıtladı ve kalanlar hesabını ortaya attı. İkinci grup çalışmasında (1830 - 1846) fonksiyonların serilere açılımını ve karmaşık diferansiyelleme ya da analitiklik kavramlarını inceledi. Yaptığı cebir çalışmaları (yerine koyma hesabı,determinantlar ve matrisler kuramı, gruplar ve cebirsel genişlemeler kuramının oluşturulması) XIX. yüzyıl tarihsel hareketine, cebirsel yapıların bulunması ve incelenmesi biçiminde geçti. Cauchy mekanik alanında esneklik kuramının matematikle ilgili yönünü düzenledi. Gökbilim hesaplarını kolaylaştırdı vehatalar kuramını geliştirdi.

Fonksiyonlar kuramında da çok yenilikleri olan Cauchy, Cauchy - Riemann denklemleri, Cauchy teoremi, Cauchy integral formülü ve Cauchy esas değeri buluşları sayılabilir. Bu saydığımız bağıntılar oldukça geniş buluşlardır. Karmaşık analizde çok uygulaması olan çok derin konuları içine almaktadır. İstenildiği kadar da genişletilip ilmin diğer dallarına uygulanabilirliği vardır.

Paris was a difficult place to live in when Augustin-Louis Cauchy was a young child due to the political events surrounding the French Revolution. When he was four years old his father, fearing for his life in Paris, moved his family to Arcueil. There things were hard and he wrote in a letter [4]:-

We never have more than a half pound of bread - and sometimes not even that. This we supplement with the little supply of hard crackers and rice that we are allotted.

They soon returned to Paris and Cauchy's father was active in the education of young Augustin-Louis. Laplace and Lagrange were visitors at the Cauchy family home and Lagrange in particular seems to have taken an interest in young Cauchy's mathematical education. Lagrange advised Cauchy's father that his son should obtain a good grounding in languages before starting a serious study of mathematics. In 1802 Augustin-Louis entered the École Centrale du Panthéon where he spent two years studying classical languages.

From 1804 Cauchy attended classes in mathematics and he took the entrance examination for the École Polytechnique in 1805. He was examined by Biot and placed second. At the École Polytechnique he attended courses by Lacroix, de Prony and Hachette while his analysis tutor was Ampère. In 1807 he graduated from the École Polytechnique and entered the engineering school École des Ponts et Chaussées. He was an outstanding student and for his practical work he was assigned to the Ourcq Canal project where he worked under Pierre Girard.

In 1810 Cauchy took up his first job in Cherbourg to work on port facilities for Napoleon's English invasion fleet. He took a copy of Laplace's Mécanique Céleste and one of Lagrange's Théorie des Fonctions with him. It was a busy time for Cauchy, writing home about his daily duties he said [4]:-

I get up at four o'clock each morning and I am busy from then on. ... I do not get tired of working, on the contrary, it invigorates me and I am in perfect health...

Cauchy was a devout Catholic and his attitude to his religion was already causing problems for him. In a letter written to his mother in 1810 he says:-

So they are claiming that my devotion is causing me to become proud, arrogant and self-infatuated. ... I am now left alone about religion and nobody mentions it to me anymore...

In addition to his heavy workload Cauchy undertook mathematical researches and he proved in 1811 that the angles of a convex polyhedron are determined by its faces. He submitted his first paper on this topic then, encouraged by Legendre and Malus, he submitted a further paper on polygons and polyhedra in 1812. Cauchy felt that he had to return to Paris if he was to make an impression with mathematical research. In September of 1812 he returned to Paris after becoming ill. It appears that the illness was not a physical one and was probably of a psychological nature resulting in severe depression.

Back in Paris Cauchy investigated symmetric functions and submitted a memoir on this topic in November 1812. This was published in the Journal of the École Polytechnique in 1815. However he was supposed to return to Cherbourg in February 1813 when he had recovered his health and this did not fit with his mathematical ambitions. His request to de Prony for an associate professorship at the École des Ponts et Chaussées was turned down but he was allowed to continue as an engineer on the Ourcq Canal project rather than return to Cherbourg. Pierre Girard was clearly pleased with his previous work on this project and supported the move.

An academic career was what Cauchy wanted and he applied for a post in the Bureau des Longitudes. He failed to obtain this post, Legendre being appointed. He also failed to be appointed to the geometry section of the Institute, the position going to Poinsot. Cauchy obtained further sick leave, having unpaid leave for nine months, then political events prevented work on the Ourcq Canal so Cauchy was able to devote himself entirely to research for a couple of years.

Other posts became vacant but one in 1814 went to Ampère and a mechanics vacancy at the Institute, which had occurred when Napoleon Bonaparte resigned, went to Molard. In this last election Cauchy did not receive a single one of the 53 votes cast. His mathematical output remained strong and in 1814 he published the memoir on definite integrals that later became the basis of his theory of complex functions.

In 1815 Cauchy lost out to Binet for a mechanics chair at the École Polytechnique, but then was appointed assistant professor of analysis there. He was responsible for the second year course. In 1816 he won the Grand Prix of the French Academy of Sciences for a work on waves. He achieved real fame however when he submitted a paper to the Institute solving one of Fermat's claims on polygonal numbers made to Mersenne. Politics now helped Cauchy into the Academy of Sciences when Carnot and Monge fell from political favour and were dismissed and Cauchy filled one of the two places.

In 1817 when Biot left Paris for an expedition to the Shetland Islands in Scotland Cauchy filled his post at the Collège de France. There he lectured on methods of integration which he had discovered, but not published, earlier. Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral. His text Cours d'analyse in 1821 was designed for students at École Polytechnique and was concerned with developing the basic theorems of the calculus as rigorously as possible. He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitésimal while in 1829 in Leçons sur le Calcul Différentiel he defined for the first time a complex function of a complex variable.

Cauchy did not have particularly good relations with other scientists. His staunchly Catholic views had him involved on the side of the Jesuits against the Académie des Sciences. He would bring religion into his scientific work as for example he did on giving a report on the theory of light in 1824 when he attacked the author for his view that Newton had not believed that people had souls. He was described by a journalist who said:-

... it is certain a curious thing to see an academician who seemed to fulfil the respectable functions of a missionary preaching to the heathens.

An example of how Cauchy treated colleagues is given by Poncelet whose work on projective geometry had, in 1820, been criticised by Cauchy:-

... I managed to approach my too rigid judge at his residence ... just as he was leaving ... During this very short and very rapid walk, I quickly perceived that I had in no way earned his regards or his respect as a scientist ... without allowing me to say anything else, he abruptly walked off, referring me to the forthcoming publication of his Leçons à 'École Polytechnique where, according to him, 'the question would be very properly explored'.

Again his treatment of Galois and Abel during this period was unfortunate. Abel, who visited the Institute in 1826, wrote of him:-

Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.

Belhoste in [4] says:-

When Abel's untimely death occurred on April 6, 1829, Cauchy still had not given a report on the 1826 paper, in spite of several protests from Legendre. The report he finally did give, on June 29, 1829, was hasty, nasty, and superficial, unworthy of both his own brilliance and the real importance of the study he had judged.

By 1830 the political events in Paris and the years of hard work had taken their toll and Cauchy decided to take a break. He left Paris in September 1830, after the revolution of July, and spent a short time in Switzerland. There he was an enthusiastic helper in setting up the Académie Helvétique but this project collapsed as it became caught up in political events.

Political events in France meant that Cauchy was now required to swear an oath of allegiance to the new regime and when he failed to return to Paris to do so he lost all his positions there. In 1831 Cauchy went to Turin and after some time there he accepted an offer from the King of Piedmont of a chair of theoretical physics. He taught in Turin from 1832. Menabrea attended these courses in Turin and wrote that the courses [4]:-

were very confused, skipping suddenly from one idea to another, from one formula to the next, with no attempt to give a connection between them. His presentations were obscure clouds, illuminated from time to time by flashes of pure genius. ... of the thirty who enrolled with me, I was the only one to see it through.

In 1833 Cauchy went from Turin to Prague in order to follow Charles X and to tutor his grandson. However he was not very successful in teaching the prince as this description shows:-

... exams .. were given each Saturday. ... When questioned by Cauchy on a problem in descriptive geometry, the prince was confused and hesitant. ... There was also material on physics and chemistry. As with mathematics, the prince showed very little interest in these subjects. Cauchy became annoyed and screamed and yelled. The queen sometimes said to him, soothingly, smilingly, 'too loud, not so loud'.

While in Prague Cauchy had one meeting with Bolzano, at Bolzano's request, in 1834. In [16] and [18] there are discussions on how much Cauchy's definition of continuity is due to Bolzano, Freudenthal's view in [18] that Cauchy's definition was formed before Bolzano's seems the more convincing.

Cauchy returned to Paris in 1838 and regained his position at the Academy but not his teaching positions because he had refused to take an oath of allegiance. De Prony died in 1839 and his position at the Bureau des Longitudes became vacant. Cauchy was strongly supported by Biot and Arago but Poisson strongly opposed him. Cauchy was elected but, after refusing to swear the oath, was not appointed and could not attend meetings or receive a salary.

In 1843 Lacroix died and Cauchy became a candidate for his mathematics chair at the Collège de France. Liouville and Libri were also candidates. Cauchy should have easily been appointed on his mathematical abilities but his political and religious activities, such as support for the Jesuits, became crucial factors. Libri was chosen, clearly by far the weakest of the three mathematically, and Liouville wrote the following day that he was:-

deeply humiliated as a man and as a mathematician by what took place yesterday at the Collège de France.

During this period Cauchy's mathematical output was less than in the period before his self-imposed exile. He did important work on differential equations and applications to mathematical physics. He also wrote on mathematical astronomy, mainly because of his candidacy for positions at the Bureau des Longitudes. The 4-volume text Exercices d'analyse et de physique mathématique published between 1840 and 1847 proved extremely important.

When Louis Philippe was overthrown in 1848 Cauchy regained his university positions. However he did not change his views and continued to give his colleagues problems. Libri, who had been appointed in the political way described above, resigned his chair and fled from France. Partly this must have been because he was about to be prosecuted for stealing valuable books. Liouville and Cauchy were candidates for the chair again in 1850 as they had been in 1843. After a close run election Liouville was appointed. Subsequent attempts to reverse this decision led to very bad relations between Liouville and Cauchy.

Another, rather silly, dispute this time with Duhamel clouded the last few years of Cauchy's life. This dispute was over a priority claim regarding a result on inelastic shocks. Duhamel argued with Cauchy's claim to have been the first to give the results in 1832. Poncelet referred to his own work of 1826 on the subject and Cauchy was shown to be wrong. However Cauchy was never one to admit he was wrong. Valson writes in [7]:-

...the dispute gave the final days of his life a basic sadness and bitterness that only his friends were aware of...

Also in [7] a letter by Cauchy's daughter describing his death is given:-

Having remained fully alert, in complete control of his mental powers, until 3.30 a.m.. my father suddenly uttered the blessed names of Jesus, Mary and Joseph. For the first time, he seemed to be aware of the gravity of his condition. At about four o'clock, his soul went to God. He met his death with such calm that made us ashamed of our unhappiness.

Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations and Cauchy sequences. He produced 789 mathematics papers, an incredible achievement. This achievement is summed up in [4] as follows:-

... such an enormous scientific creativity is nothing less than staggering, for it presents research on all the then-known areas of mathematics ... in spite of its vastness and rich multifaceted character, Cauchy's scientific works possess a definite unifying theme, a secret wholeness. ... Cauchy's creative genius found broad expression not only in his work on the foundations of real and complex analysis, areas to which his name is inextricably linked, but also in many other fields. Specifically, in this connection, we should mention his major contributions to the development of mathematical physics and to theoretical mechanics... we mention ... his two theories of elasticity and his investigations on the theory of light, research which required that he develop whole new mathematical techniques such as Fourier transforms, diagonalisation of matrices, and the calculus of residues.

His collected works, Oeuvres complètes d'Augustin Cauchy (1882-1970), were published in 27 volumes.