Augustus De Morgan Biography

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World of Mathematics on Augustus De Morgan

Augustus De Morgan entered the English mathematical scene during a period of inactivity and by the time of his death it had regained the stature it had since the time of Isaac Newton. Although De Morgan did not devote himself wholeheartedly to the pursuit of mathematics, he is credited for promoting its study by his publications and his teaching. He worked outside the established universities and was able to appeal to a wider audience than merely the mathematics graduates of Oxford and Cambridge.

De Morgan was born in June 1806 in Madurai, a picturesque town in southern India. Shortly after his birth he lost the sight of his right eye. The De Morgans moved back to England when he was seven months old. His father, a colonel in the Indian army, continued to spend time in India and died on St. Helena in 1816. De Morgan was educated at a series of private schools before enrolling at Trinity College, Cambridge, in 1823 and graduated as fourth wrangler.

After graduation, De Morgan entered Lincoln's Inn, one of the Inns of Court intended to prepare students for a legal career. He may have been discouraged by some aspects of the mathematics curriculum at Cambridge at the time, which was still recovering from the inertia of the 18th century. Partly in response to the quarrel between Isaac Newton and Gottfried Leibniz over the invention of calculus, the English mathematical community began to isolate itself from the continent in the 18th century, idolizing Newton and reluctant to change. The result was a petrification of both the foundations of the calculus and of its notation, an area in which Leibniz's version was clearly an improvement over Newton's. Cambridge was drifting along serenely, unaware of the progress of mathematics outside its walls until the arrival of a group of students who were known as the Analytical Society. They devoted themselves to the reform of mathematical education.

De Morgan soon found the legal profession unappealling and applied for the chair of mathematics at University College, London. He was offered the chair in 1828 on the strength of recommendations from his former tutors, who included some members of the Analytical Society. As a teacher, De Morgan was devoted to the presentation of ideas and principles rather than techniques, and his pupils included some of the most distinguished British mathematicians of the next generation. In addition, he also produced a series of textbooks on arithmetic, algebra, trigonometry, calculus, complex numbers, probability, and logic. These were written clearly and with attention to giving an intuitive understanding as well as one based on calculation.

In his own mathematical work, De Morgan made major contributions in the area of logic. The Aristotelian tradition of logic had become fossilized during the Middle Ages, and instruction in logic frequently resorted to memorizing a few lines of low Latin and a little caution about the misuse of rhetoric. If reasoning about mathematics was being carried out in ordinary language, there was not much advantage to studying mathematics in approaching questions of logic.

One of the main interests of the English mathematical community at that time was the status of the laws of algebra. It was clear that some of the laws applied to all the systems of numbers then known, but other laws did not apply beyond a restricted domain. The quaternions discovered by Sir William Rowan Hamilton, for example, (which are related to vectors in three dimensions) had an operation of multiplication which was not commutative (that is, a x b was not equal to b x a). An obvious question was that of which laws are automatically satisfied by any objects whatever, which could be considered laws of logic.

The crucial respect in which De Morgan sought to improve on the traditional logic of Aristotle was in the treatment of the logic of relations. In Aristotelian logic, all statements had to be analyzed into the form "A is (or is not) B," with the possible inclusion of "all" and "some." It was not clear how this could be used to handle statements like "A is taller than B" or "A is closer to B than C." De Morgan's work on the logic of relations did not become part of the mainstream, due to the shortcomings of his notation. More successful in the reform of logic was George Boole, author of The Mathematical Analysis of Logic and creator of a superior notation. Boole acknowledged his debt to De Morgan, whose name remains attached to two laws of Boolean algebra involving the negations of compound expressions.

De Morgan also wrote on probability. As far as the interpretation of probabilities was concerned, De Morgan fell in the "subjectivist" rather than the "frequentist" school. In other words, probability statements were reflections of the degree of belief attached to propositions rather than features of the natural world itself.

In 1831 De Morgan resigned his chair at University College. By 1836, following the death of his successor, however, De Morgan returned to his position. The next year he married Sophia Elizabeth Frend, who wrote De Morgan's biography after his death. Although De Morgan found his family life a comfort after some of the controversies in the academic world, his later years were saddened by the deaths of a couple of his children.

De Morgan wrote many articles for the popular press. This type of writing did not command much respect, and it is worth noting that De Morgan never became a Fellow of the Royal Society, of whom he criticized for being too open to social influence, as indicated by the proportion of nobility among its members. Much more to De Morgan's taste was the London Mathematical Society, which he cofounded and served as first president. This type of mathematical organization was more fitting for someone who had contributed 850 articles for one reference work alone.

De Morgan resigned a second time from University College in 1866 and on this occasion could not be tempted to return. He died in London on March 18, 1871. Not so much by his research as by his pedagogical efforts had De Morgan transformed the mathematical community in England into a setting for the discussion of the current topics of interest in mathematics both English and European.