Archive for April, 2009


Peter Abelard (2 of 4)

There he taught openly but was also given  as a private pupil the young Heloise, niece of one of the clergy of the cathedral of Paris, Canon Fulbert.  Abelard and Heloise fell in love and had a son whom they called Astralabe.  They then married secretly.  To escape her uncle’s wrath Heloise withdrew into the convent of Argenteuil outside Paris.  Abelard suffered castration at Fulbert’s instigation. 

In shame he embraced the monastic life at the royal abbey of Saint-Denis near Paris and made the unwilling Heloise become a nun at Argenteuil. At Saint-Denis Abelard extended his reading in theology and tirelessly criticized the way of life followed by his fellow monks.  His reading of the Bible and of the Fathers of the Church led him to make a collection of quotations that seemed to represent inconsistencies of teaching by the Christian Church. 

He arranged his findings in a compilation entitled Sic et Non (“Yes and No”); and for it be wrote a preface in which, as a logician and as a keen student of language, he formulated basic rules with which students might reconcile apparent contradictions of meaning and distinguish the various senses in which words had been used over the course of many centuries.  He also wrote the first version of his book called Theologia, which was formally condemned as heretical and burned by a council held at Soissons in 1121.

Abelard’s dialectical analysis of the mystery of God and the Trinity was held to be erroneous, and he himself was placed for a while in the abbey of Saint-Medard under house arrest.  When he returned to Saint-Denis he applied his Sic et Non methods to the subject of the abbey’s patron saint; he argued that St. Denis of Paris, the martyred apostle of Gaul, was not identical with Denis of Athens (also known as Dionysius the Areopagile), the convert of St. Paul.  (D.E.L)


Peter Abelard (1 of 4)


Peter Abelard, the most famous and the most controversial teacher of his age, was a logician, moral philosopher, and theologian.  He was also a gifted and original poet, the founder of a famous convent for women, and the tragic lover of Heloise.  He was twice condemned for heresy by ecclesiastical councils in France.

The outline of Abelard’s career is well-known, largely because he described so much of it in his famous Historia calamitatum (“History of  Troubles”)  he was born the son of a knight in 1079 at Le Pallet in Brittany south of the Loire River.  He sacrificed his rights of inheritance and the prospect of a military career in order to study philosophy, particularly logic, in France.  He provoked bitter quarrels with two of his masters, Roscelin of Compiegne and William of Champeaux. 

The two men represented opposite poles of philosophy.  Roscelin was a Nominalist who asserted that universals are nothing more than mere words;  William in Paris upheld a form of Platonic Realism according to which universals exists.  Abelard in his own logical writings brilliantly elaborated an independent philosophy of language.  While showing how words could be used significantly, he stressed that language itself is not able to demonstrate the truth of things (res) that lie in the domain of physics.

Abelard was a peripatetic both in the manner in which he wandered from school to school at Paris, Lelun, Corbell and elsewhere and as one of the exponents of Aristotelian logic who were called the Peripatetics.  In 1113 or 1114 he went north to Laon to study theology under Anselm of Laon, the leading biblical scholar of the day.  He quickly developed a strong contempt for Anselm’s teaching, which he found vacuous, and returned to Paris.  – (D.E.L)

Photo courtesy:  records.viu


Niels Henrik Abel (3 of 3)

This theorem forms the basis for the later theory of Abelian integrals and Abelian functions.  Abel was accepted with restrained civility in Paris, for his work was still unknown.  He submitted his memoir for presentation to the Academy of Sciences, hoping that it would establish his reputation; but he waited in vain.  Before leaving Paris thinking he had a persistent cold, Abel consulted a physician, who informed him he had tuberculosis.

Abel returned to Norway heavily in debt.  He subsisted by tutoring, by receiving a small grant from the university, and in 1828, by accepting a substitute teaching position.  His poverty and ill health did not decrease his production; he wrote a great number of papers, principally on equation theory and elliptic functions.  Among them are the theory of the Abelian equations with Abelian groups.  He rapidly developed the theory of elliptic functions in competition with Karl Gustav Jacobi.

By this time Abel’s fame had spread to all mathematical centers, and strong efforts were made to secure a suitable position for him by a group from the French academy, who addressed Bernadotte, the king of Norway-Sweden; Crelle worked to secure a professorship for him in Berlin.  

In the fall of 1828, Abel became seriously ill, and his condition deteriorated on a sled trip at Christmas time to visit his fiancée at Froland, where he died on April 6, 1829.  The French Academy of Science published this memoirs in 1841.


Niels Henrik Abel (2 of 3)

While waiting for the royal decree to be issued, in 1824 the published at his own expense his proof of the impossibility of solving algebraically the general equation of the fifth degree, which he hoped would bring him recognition.  He sent the pamphlet to Gauss, who dismissed it, failing to recognize that the famous problem had indeed been settled.

Abel spent the winter of 1825-26 with Norwegian friends in Berlin, where he met August Leopold Crelle, civil engineer and self-taught enthusiast of mathematics, who became his close friend and mentor.  With Abel’s warm encouragement, Crelle founded the Journal fűr die reine und angewandte Mathematik (“Journal for Pure and Applied Mathematics”), the first volume of which (1826) contains papers by Abel, including a more elaborate version of his work on the quintic equation.  Other papers dealt with equation theory, functional equations, integration in finite forms, and problems from theoretical mechanics.

Abel’s early mathematical training had been in the formal school typified by Euler.  In Berlin new directions in mathematics stimulated him to do further independent work.  Soon distracted socially, however, Abel travelled throughout Europe.

Arriving in Paris in the summer of 1826, he called on the foremost mathematicians and completed a memoir on transcendental functions.  In this major work he presented a theory of integrals of algebraic functions, in particular the result know as Abel’s theorem: there is a finite number, or genus, of independent integrals of this nature. 


Niels Henrik Abel ( 1 of 3)


Niels Henrik Abel was a Norwegian mathematician, recognized only after his death as a pioneer in the development of modern mathematics.

Abel was born on August 5, 1802, on the island of Finnǿy, near Stavanger, Norway, where his father was a poor Protestant minister.  The family soon moved to the parish of Gjerstad, near the town of Risǿr (southeast Norway), where the boy grew up.  In 1815, when he entered the cathedral school in Oslo, his mathematical talent was recognized by a teacher who introduced him to the classics in mathematical literature and proposed original problems for solution.  Thoroughly challenged, Abel studied the works of the 17th-century English mathematician and physicist Isaac Newton and the contemporary mathematicians Leonhard Euler (German), Joseph-Louis Lagrange (French), and Carl Friedrich Gauss (German) and learned to detect gaps in their mathematical reasoning.

Although when Abel’s father died in 1820 the family was left in straightened circumstances, the boy was able to enter the University of Christiania (Oslo) in 1821 because his teacher contributed and raised funds.

On graduation from the University in 1822, Abel continued his studies with further subsidies obtained by his teacher.  His first papers, published in 1823 in the new periodical Magazin for Naturvidenskaberne, were on functional equations and integrals, his solution of an integral equation being the first.  Abel’s friends urged the Norwegian government to grant him a fellowship for study in Germany and France. 


Photo courtesy:  media-2


April 2009

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