Mathematician, born in Madura, N India. He studied at Cambridge, and became the first professor of mathematics at University College London (1828). He helped to develop the notion of different kinds of algebra, and collaborated with Boole in the development of symbolic logic.
Augustus De Morgan (June 27, 1806 – March 18, 1871) was an Indian-born British mathematician and logician. He formulated De Morgan's laws and was the first to introduce the term, and make rigorous the idea of mathematical induction. De Morgan crater on the Moon is named after him.
(This page makes extensive use of Alexander MacFarlane's Ten British Mathematicians, available freely at Project Gutenberg)
Biography
Childhood
Augustus De Morgan was born June 27, 1806 in Madura, Madras Presidency, India (now Madurai, Tamil Nadu, India); De Morgan, who held various appointments in the service of the East India Company. De Morgan moved his family to England when Augustus was seven months old. As his father and grandfather had both been born in India, De Morgan used to say that he was neither English, nor Scottish, nor Irish, but a Briton "unattached," using the technical term applied to an undergraduate of Oxford or Cambridge who is not a member of any one of the Colleges.
When De Morgan was ten years old, his father died. Mrs. De Morgan resided at various places in the southwest of England, and her son received his elementary education at various schools of no great account.
He received his secondary education from Mr. Parsons, a Fellow of Oriel College, Oxford, who could appreciate classics much better than mathematics. but by this time De Morgan had begun to show his non-grooving disposition, due no doubt to some extent to his physical infirmity.
University education
In 1823, at the age of sixteen he entered Trinity College, Cambridge, where he immediately came under the tutorial influence of George Peacock and William Whewell. from the former he derived an interest in the renovation of algebra, and from the latter an interest in the renovation of logic—the two subjects of his future life work.
At college the flute, on which he played exquisitely, was his recreation. His love of knowledge for its own sake interfered with training for the great mathematical race; To the signing of any such test De Morgan felt a strong objection, although he had been brought up in the Church of England.
London University
As no career was open to him at his own university, he decided to go to the Bar, and took up residence in London; but he much preferred teaching mathematics to reading law. About this time the movement for founding the London University took shape. A body of liberal-minded men resolved to meet the difficulty by establishing in London a University on the principle of religious neutrality. De Morgan, then 22 years of age, was appointed Professor of Mathematics. His introductory lecture "On the study of mathematics" is a discourse upon mental education of permanent value which has been recently reprinted in the United States.
The London University was a new institution, and the relations of the Council of management, the Senate of professors and the body of students were not well defined. A dispute arose between the professor of anatomy and his students, and in consequence of the action taken by the Council, several of the professors resigned, headed by De Morgan. Another professor of mathematics was appointed, who was accidentally drowned a few years later. De Morgan had shown himself a prince of teachers: he was invited to return to his chair, which thereafter became the continuous centre of his labours for thirty years.
The same body of reformers—headed by Lord Brougham, a Scotsman eminent both in science and politics who had instituted the London University—founded about the same time a Society for the Diffusion of Useful Knowledge. Its object was to spread scientific and other knowledge by means of cheap and clearly written treatises by the best writers of the time. One of its most voluminous and effective writers was De Morgan. He wrote a great work on The Differential and Integral Calculus which was published by the Society; When De Morgan came to reside in London he found a congenial friend in William Frend, notwithstanding his mathematical heresy about negative quantities. De Morgan with his flute was a welcome visitor;
The London University of which De Morgan was a professor was a different institution from the University of London. The University of London was founded about ten years later by the Government for the purpose of granting degrees after examination, without any qualification as to residence. The London University was affiliated as a teaching college with the University of London, and its name was changed to University College. The University of London was not a success as an examining body; De Morgan was a highly successful teacher of mathematics. It was his plan to lecture for an hour, and at the close of each lecture to give out a number of problems and examples illustrative of the subject lectured on; In De Morgan's opinion, a thorough comprehension and mental assimilation of great principles far outweighed in importance any merely analytical dexterity in the application of half-understood principles to particular cases.
De Morgan had a son George, who acquired great distinction in mathematics both at University College and the University of London. He and another like-minded alumnus conceived the idea of founding a Mathematical Society in London, where mathematical papers would be not only received (as by the Royal Society) but actually read and discussed. De Morgan was the first president, his son the first secretary. It was the beginning of the London Mathematical Society.
Retirement and death
In the year 1866 the chair of mental philosophy in University College fell vacant. De Morgan considered that the old standard of religious neutrality had been hauled down, and forthwith resigned. Two years later his son George -- the younger Bernoulli, as he loved to hear him called, in allusion to the two eminent mathematicians of that name, related as father and son -- died. Five years after his resignation from University College De Morgan died of nervous prostration on March 18, 1871, in the 65th year of his age.
Mathematical work
De Morgan was a brilliant and witty writer, whether as a controversialist or as a correspondent. The one was Sir William Hamilton, 9th Baronet (that is, his title was inherited), a Scotsman, professor of logic and metaphysics at the University of Edinburgh; the knight, whose full name was William Rowan Hamilton, contributed to mathematics, especially geometric algebra, and first described the Quaternions. De Morgan was interested in the work of both, and corresponded with both; In one of his letters to Rowan, De Morgan says, "Be it known unto you that I have discovered that you and the other Sir W.
The correspondence of De Morgan with Hamilton the mathematician extended over twenty-four years; it contains discussions not only of mathematical matters, but also of subjects of general interest. It is marked by geniality on the part of Hamilton and by wit on the part of De Morgan. The following is a specimen: Hamilton wrote, "My copy of Berkeley's work is not mine; De Morgan replied, "Your phrase 'my copy is not mine' is not a bull.
De Morgan was full of personal peculiarities. He once printed his name: Augustus De Morgan, H - O - M - O - P - A - U - C - A - R - U - M - L - I - T - E - R - A - R - U - M.
He disliked the country, and while his family enjoyed the seaside, and men of science were having a good time at a meeting of the British Association in the country he remained in the hot and dusty libraries of the metropolis. He never sought to become a Fellow of the Royal Society, and he never attended a meeting of the Society;
Were the writings of De Morgan published in the form of collected works, they would form a small library. and to its Transactions De Morgan contributed four memoirs on the foundations of algebra, and an equal number on formal logic. The best presentation of his view of algebra is found in a volume, entitled Trigonometry and Double Algebra, published in 1849; it was revised and extended by De Morgan in the last years of his life, and was published posthumously by his widow. If you wish to read something entertaining, get De Morgan's Budget of Paradoxes out of the library. We shall consider more at length his theory of algebra, his contribution to exact logic, and his Budget of Paradoxes.
George Peacock's theory of algebra was much improved by D. Gregory, a younger member of the Cambridge School, who laid stress not on the permanence of equivalent forms, but on the permanence of certain formal laws. This new theory of algebra as the science of symbols and of their laws of combination was carried to its logical issue by De Morgan; Thus Chrystal founds his Textbook of Algebra on De Morgan's theory; De Morgan's theory is stated in his volume on Trigonometry and Double Algebra. In the chapter (of the book) headed "On symbolic algebra" he writes: "In abandoning the meaning of symbols, we also abandon those of the words which describe them. It is a mode of combination represented by + ; It is most important that the student should bear in mind that, with one exception, no word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra which may hereafter become the grammar of a hundred distinct significant algebras. The one exception above noted, which has some share of meaning, is the sign = placed between two symbols as in A = B.
Here, it may be asked, why does the symbol = prove refractory to the symbolic theory? De Morgan admits that there is one exception; but an exception proves the rule, not in the usual but illogical sense of establishing it, but in the old and logical sense of testing its validity.
De Morgan proceeds to give an inventory of the fundamental symbols of algebra, and also an inventory of the laws of algebra.
The last two may be called the rules of reduction. De Morgan professes to give a complete inventory of the laws which the symbols of algebra must obey, for he says, "Any system of symbols which obeys these laws and no others, except they be formed by combination of these laws, and which uses the preceding symbols and no others, except they be new symbols invented in abbreviation of combinations of these symbols, is symbolic algebra." they are formal laws, that is, arbitrarily chosen relations to which the algebraic symbols must be subject. He does not mention the law, which had already been pointed out by Gregory, namely, (a + b) + c = a + (b + c),(ab)c = a(bc) and to which was afterwards given the name of the law of association. Because the foundations of algebra are, after all, real not formal, material not symbolic. To the formalists the index operations are exceedingly refractory, in consequence of which some take no account of them, but relegate them to applied mathematics. To give an inventory of the laws which the symbols of algebra must obey is an impossible task, and reminds one not a little of the task of those philosophers who attempt to give an inventory of the a priori knowledge of the mind.
De Morgan's work entitled Trigonometry and Double Algebra consists of two parts; the former of which is a treatise on Trigonometry, and the latter a treatise on generalized algebra which he calls Double Algebra. But what is meant by Double as applied to algebra? The first stage in the development of algebra is arithmetic, where numbers only appear and symbols of operations such as + , , etc. The next stage is universal arithmetic, where letters appear instead of numbers, so as to denote numbers universally, and the processes are conducted without knowing the values of the symbols. The third stage is single algebra, where the symbol may denote a quantity forwards or a quantity backwards, and is adequately represented by segments on a straight line passing through an origin. the algebraic symbol denotes in general a segment of a line in a given plane; The expression then represents a line in the plane having an abscissa a and an ordinate b. Argand and Warren carried double algebra so far; but they were unable to interpret on this theory such an expression as . De Morgan attempted it by reducing such an expression to the form , and he considered that he had shown that it could be always so reduced. The remarkable fact is that this double algebra satisfies all the fundamental laws above enumerated, and as every apparently impossible combination of symbols has been interpreted it looks like the complete form of algebra.
If the above theory is true, the next stage of development ought to be triple algebra and if truly represents a line in a given plane, it ought to be possible to find a third term which added to the above would represent a line in space. De Morgan and many others worked hard at the problem, but nothing came of it until the problem was taken up by Hamilton. We now see the reason clearly: the symbol of double algebra denotes not a length and a direction; double algebra is nothing but analytical plane trigonometry, and this is why it has been found to be the natural analysis for alternating currents. But De Morgan never got this far; he died with the belief "that double algebra must remain as the full development of the conceptions of arithmetic, so far as those symbols are concerned which arithmetic immediately suggests."
When the study of mathematics revived at the University of Cambridge, so did the study of logic. Doubtless De Morgan was influenced in his logical investigations by Whewell; The followers of Aristotle say that from two particular propositions such as Some M's are A's , and Some M's are B's nothing follows of necessity about the relation of the A's and B's. De~Morgan pointed out that from Most M's are A's and Most M's are B's it follows of necessity that some A's are B's and he formulated the numerically definite syllogism which puts this principle in exact quantitative form. Suppose that the number of the M's is m, of the M's that are A's is a, and of the M's that are B's is b;
Here then De Morgan had made a great advance by introducing quantification of the terms. However, De Morgan soon perceived that Hamilton's quantification was of a different character; As a consequence he had no room for De Morgan's innovations. He accused De Morgan of plagiarism, and the controversy raged for years in the columns of the Athenæum, and in the publications of the two writers.
The memoirs on logic which De Morgan contributed to the Transactions of the Cambridge Philosophical Society subsequent to the publication of his book on Formal Logic are by far the most important contributions which he made to the science, especially his fourth memoir, in which he begins work in the broad field of the logic of relatives. This is the true field for the logician of the twentieth century, in which work of the greatest importance is to be done towards improving language and facilitating thinking processes which occur all the time in practical life.
In the introduction to the Budget of Paradoxes De Morgan explains what he means by the word. "A great many individuals, ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. I use the word in the old sense: a paradox is something which is apart from general opinion, either in subject matter, method, or conclusion.
How can the sound paradoxer be distinguished from the false paradoxer? De Morgan supplies the following test: "The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself... New knowledge, when to any purpose, must come by contemplation of old knowledge, in every matter which concerns thought;
"I remember that just before the American Association met at Indianapolis in 1890, the local newspapers heralded a great discovery which was to be laid before the assembled savants -- a young man living somewhere in the country had squared the circle. To this there was no assent, but the sickly smile of the false paradoxer" [Note: De Morgan did not say this (how could he? Rather, as pointed out on the discussion page, this paragraph (and the rest of the article) is copied verbatim from a lecture given in 1916]
The Budget consists of a review of a large collection of paradoxical books which De Morgan had accumulated in his own library, partly by purchase at bookstands, partly from books sent to him for review, partly from books sent to him by the authors. He gives the following classification: squarers of the circle, trisectors of the angle, duplicators of the cube, constructors of perpetual motion, subverters of gravitation, stagnators of the earth, builders of the universe.
De Morgan gives his personal knowledge of paradoxers.
A paradoxer to whom De Morgan paid the compliment which Achilles paid Hector -- to drag him round the walls again and again -- was James Smith, a successful merchant of Liverpool. The following is a specimen of De Morgan's dragging round the walls of Troy: "Mr. Smith continues to write me long letters, to which he hints that I am to answer. In his last of 31 closely written sides of note paper, he informs me, with reference to my obstinate silence, that though I think myself and am thought by others to be a mathematical Goliath, I have resolved to play the mathematical snail, and keep within my shell. But he ventures to tell me that pebbles from the sling of simple truth and common sense will ultimately crack my shell, and put me hors de combat. The confusion of images is amusing: Goliath turning himself into a snail to avoid and James Smith, Esq., of the Mersey Dock Board: and put hors de combat by pebbles from a sling.
In the region of pure mathematics De Morgan could detect easily the false from the true paradox; and in the opinion of the physical philosophers De Morgan himself scarcely escaped. and De Morgan wrote a preface in which he said that he knew some of the asserted facts, believed others on testimony, but did not pretend to know whether they were caused by spirits, or had some unknown and unimagined origin. De Morgan could not understand this.
Relations
De Morgan discovered relational algebra in his (1966: 208-46), first published in 1860. This algebra was extended by Charles Peirce (who admired De Morgan and met him shortly before his death), and re-exposited and further extended in vol. In turn, this algebra became the subject of much further work, starting in 1940, by Alfred Tarski and his colleagues and students at the University of California.
Legacy
Beyond his great mathematical legacy, the student society of the Mathematics Department of University College London is called the August De Morgan Society.
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