June 01, 2024 <Back to Index>
PAGE SPONSOR |
Augustus De Morgan (27 June 1806 - 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. Augustus De Morgan was born in Madurai, Madras Presidency, India in 1806. His father was Colonel Augustus De Morgan, who held various appointments in the service of the East India Company. His mother descended from James Dodson, who computed a table of anti-logarithms, that is, the numbers corresponding to exact logarithms. Augustus De Morgan became blind in one eye a month or two after he was born. The family moved 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. His mathematical talents went unnoticed until he was fourteen, when a family-friend discovered him making an elaborate drawing of a figure in Euclid with ruler and compasses. She explained the aim of Euclid to Augustus, and gave him an initiation into demonstration. He received his secondary education from Mr. Parsons, a
fellow of Oriel College, Oxford, who appreciated classics
better than mathematics. His mother was an active and
ardent member of the Church of England, and desired that
her son should become a clergyman; but by this time De
Morgan had begun to show his non-conforming disposition. In 1823, at the age of sixteen, he entered Trinity College, Cambridge, where he came under the influence of George Peacock and William Whewell, who became his lifelong friends; 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. His Cambridge tutor was John Philips Higman. At college he played the flute for recreation and was
prominent in the musical clubs. His love of knowledge for
its own sake interfered with training for the great
mathematical race; as a consequence he came out fourth
wrangler. This entitled him to the degree of Bachelor of
Arts; but to take the higher degree of Master of Arts and thereby
become eligible for a fellowship it was then necessary to
pass a theological test. To the signing of any such test
De Morgan felt a strong objection, although he had been
brought up in the Church of England. In about 1875
theological tests for academic degrees were abolished in
the Universities of Oxford and Cambridge. 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 London University (now University College London) took shape. The two ancient universities of Oxford and Cambridge were so guarded by theological tests that no Jew or Dissenter outside the Church of England could enter as a student, still less be appointed to any office. 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, and 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 professors resigned, headed by De Morgan. Another professor of mathematics was appointed, who then 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 center of his labors 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; and he wrote one-sixth of the articles in the Penny Cyclopedia, published by the Society, and issued in penny numbers. When De Morgan came to reside in London he found a congenial friend in William Frend, notwithstanding his mathematical heresy about negative quantities. Both were arithmeticians and actuaries, and their religious views were somewhat similar. Frend lived in what was then a suburb of London, in a country house formerly occupied by Daniel Defoe and Isaac Watts. 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; a teaching University was demanded. 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; his students were required to sit down to them and bring him the results, which he looked over and returned revised before the next lecture. 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. During this period, he also promoted the work of the
self-taught Indian mathematician Ramchundra, who has been
called De Morgan's Ramanujam.
He supervised the publication in London of Ramchundra's
book on "Maxima and Minima" in 1859. In the introduction
to this book, he acknowledged being aware of the Indian
tradition of logic, although it is not known whether this
had any influence on his own work. In the autumn of 1837, he married Sophia Elizabeth, eldest daughter of William Frend and his wife, a granddaughter of Archdeacon Francis Blackburne. De Morgan had three sons and four daughters, including fairy tale author Mary de Morgan. His eldest son was the potter William De Morgan. His second son George acquired great distinction in mathematics 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. The first meeting was held in University College; De Morgan was the first president, his son the first secretary. It was the beginning of the London Mathematical Society. In 1866 the chair of mental philosophy in University
College fell vacant. James Martineau, a Unitarian
clergyman and professor of mental philosophy, was
recommended formally by the Senate to the Council; but in
the Council there were some who objected to a Unitarian
clergyman, and others who objected to theistic philosophy.
A layman of the school of Bain and Spencer was appointed.
De Morgan considered that the old standard of religious
neutrality had been hauled down, and forthwith resigned.
He was now 60 years of age. His pupils secured him a
pension of £500 p.a., but misfortunes followed. Two years
later his son George — the "younger Bernoulli", as
Augustus loved to hear him called, in allusion to the
eminent father-and-son mathematicians of that name — died.
This blow was followed by the death of a daughter. Five
years after his resignation from University College De
Morgan died of nervous
prostration on 18 March 1871. De Morgan was a brilliant and witty writer, whether as a controversialist or as a correspondent. In his time there flourished two Sir William Hamiltons who have often been confounded. 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 other was a knight (that is, won the title), an Irishman, professor at astronomy in the University of Dublin. The baronet contributed to logic, especially the doctrine of the quantification of the predicate; 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; but the correspondence with the Scotsman ended in a public controversy, whereas that with the Irishman was marked by friendship and terminated only by death. In one of his letters to Rowan, De Morgan says,
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,
De Morgan replied,
De Morgan was full of personal peculiarities. On the occasion of the installation of his friend, Lord Brougham, as Rector of the University of Edinburgh, the Senate offered to confer on him the honorary degree of LL. D.; he declined the honor as a misnomer. 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 (Latin for "man of few letters"). He disliked the provinces outside London, 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 said that he felt like Socrates, who declared that the farther he was from Athens the farther was he from happiness. He never sought to become a Fellow of the Royal Society, and he never attended a meeting of the Society; he said that he had no ideas or sympathies in common with the physical philosopher. His attitude was possibly due to his physical infirmity, which prevented him from being either an observer or an experimenter. He never voted at an election, and he never visited the House of Commons, or the Tower of London, or Westminster Abbey. Were the writings of De Morgan published in the form of collected works, they would form a small library, for example his writings for the Useful Knowledge Society. Mainly through the efforts of Peacock and Whewell, a Philosophical Society had been inaugurated at Cambridge; 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; and his earlier view of formal logic is found in a volume published in 1847. His most distinctive work is styled a Budget of Paradoxes; it originally appeared as letters in the columns of the Athenæum journal; it was revised and extended by De Morgan in the last years of his life, and was published posthumously by his widow. George Peacock's theory of algebra was much improved by D. F. 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; and his doctrine on the subject is still followed by English algebraists in general. Thus George Chrystal founds his Textbook of Algebra on De Morgan's theory; although an attentive reader may remark that he practically abandons it when he takes up the subject of infinite series. 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:
De Morgan proceeds to give an inventory of the fundamental symbols of algebra, and also an inventory of the laws of algebra. The symbols are 0, 1, +, −, ×, ÷, ()(), and letters; these only, all others are derived. His inventory of the fundamental laws is expressed under fourteen heads, but some of them are merely definitions. The laws proper may be reduced to the following, which, as he admits, are not all independent of one another:
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." From
his point of view, none of the above principles are rules;
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, and to which was afterwards
given the name of the law of association. If the
commutative law fails, the associative may hold good; but
not vice versa. It is an unfortunate thing for the
symbolist or formalist that in universal arithmetic is not equal to ; for then the
commutative law would have full scope. Why does he not
give it full scope? 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 called "double 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. Let and denote any numbers; then such an expression as may be impossible; so that in universal arithmetic there is always a proviso, provided the operation is possible. 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. Negative quantities are then no longer impossible; they are represented by the backward segment. But an impossibility still remains in the latter part of such an expression as which arises in the solution of the quadratic equation. The fourth stage is double algebra. The algebraic symbol denotes in general a segment of a line in a given plane. It is a double symbol because it involves two specifications, namely, length, and direction; and is interpreted as denoting a quadrant. The expression then represents a line in the plane having an abscissa and an ordinate . 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. In chapter 6 he introduced hyperbolic functions and discussed the connection of common and hyperbolic trigonometry. 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. Argand and some others guessed that it was although this contradicts the truth established by Euler that . 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; but a multiplier and an angle. In it the angles are confined to one plane. Hence the next stage will be a quadruple algebra, when the axis of the plane is made variable. And this gives the answer to the first question; 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. The moving spirit was Whewell, the Master of Trinity College, whose principal writings were a History of the Inductive Sciences, and Philosophy of the Inductive Sciences. Doubtless De Morgan was influenced in his logical investigations by Whewell; but other influential contemporaries were Sir William Rowan Hamilton of Edinburgh, and Professor Boole of Cork. De Morgan's work on Formal Logic, published in 1847, is principally remarkable for his development of the numerically definite syllogism. 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. But they go further and say in order that any relation about the A's and B's may follow of necessity, the middle term must be taken universally in one of the premises. 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 , of the M's that are A's is , and of the M's that are B's is ; then there are at least A's that are B's. Suppose that the number of souls on board a steamer was 1000, that 500 were in the saloon, and 700 were lost. It follows of necessity, that at least 700 + 500 - 1000, that is, 200, saloon passengers were lost. This single principle suffices to prove the validity of all the Aristotelian moods. It is therefore a fundamental principle in necessary reasoning. Here then De Morgan had made a great advance by introducing quantification of the terms. At that time Sir William Rowan Hamilton was teaching in Edinburgh a doctrine of the quantification of the predicate, and a correspondence sprang up. However, De Morgan soon perceived that Hamilton's quantification was of a different character; that it meant for example, substituting the two forms The whole of A is the whole of B, and The whole of A is a part of B for the Aristotelian form All A's are B's. Hamilton thought that he had placed the keystone in the Aristotelian arch, as he phrased it. Although it must have been a curious arch which could stand 2000 years without a keystone. 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. Identity and difference are the two relations which have been considered by the logician; but there are many others equally deserving of study, such as equality, equivalence, consanguinity, affinity, etc. In the introduction to the Budget of Paradoxes De Morgan explains what he means by the word.
How can the sound paradoxer be distinguished from the false paradoxer? De Morgan supplies the following test:
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. You will still find specimens of all these classes in the New World and in the new century. 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. He found . His mode of reasoning was a curious caricature of the reductio ad absurdum of Euclid. He said let , and then showed that on that supposition, every other value of π must be absurd. Consequently is the true value. The following is a specimen of De Morgan's dragging round the walls of Troy:
In the region of pure mathematics, De Morgan could detect easily the false from the true paradox; but he was not so proficient in the field of physics. His father-in-law was a paradoxer, and his wife a paradoxer; and in the opinion of the physical philosophers De Morgan himself scarcely escaped. His wife wrote a book describing the phenomena of spiritualism, table - rapping, table - turning, etc.; 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. From this alternative he left out ordinary material causes. Faraday delivered a lecture on Spiritualism, in which he laid it down that in the investigation we ought to set out with the idea of what is physically possible, or impossible; De Morgan did not believe this. De Morgan discovered relation algebra in his Syllabus
of a Proposed System of Logic (1966: 208-46), first
published in 1860. This algebra was extended by Charles
Sanders Peirce (who admired De Morgan and met him shortly
before his death), and re-exposited and further extended
in vol. 3 of Ernst Schröder's Vorlesungen über die
Algebra der Logik. Relation algebra proved critical
to the Principia Mathematica of Bertrand Russell
and Alfred North Whitehead. 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. De Morgan later in his life became interested in the phenomena of Spiritualism. In 1849 he had investigated clairvoyance and was impressed by the subject. He later carried out paranormal investigations in his own home with the medium Maria Hayden. The result of these investigations was later published by his wife Sophia. De Morgan believed that his career as a scientist might have been affected if he had revealed his interest in the study of spiritualism so he helped to publish the book anonymously. The book was published in 1863 titled From Matter to Spirit: The Result of Ten Years Experience in Spirit Manifestations. According to (Oppenheim, 1988) De Morgan's wife Sophia was a convinced spiritualist but De Morgan shared a third way position on spiritualist phenomena which Oppenheim defined as a "wait - and - see position", he was neither a believer or a skeptic, instead his viewpoint was that the methodology of the physical sciences does not automatically exclude psychic phenomena and that such phenomena may be explainable in time by the possible existence of natural forces which as yet physicists had not identified. In the preface of From Matter to Spirit (1963) De Morgan stated:
John Beloff in Parapsychology:
A Concise History (1997) wrote that De Morgan was
the first notable scientist in Britain to take an interest
in the study of spiritualism and his studies had
influenced the decision of William Crookes to also study
spiritualism. De Morgan was also an atheist and because of this
had debarred him from a position at Oxford and Cambridge. Beyond his great mathematical legacy, the headquarters of the London Mathematical Society is called De Morgan House and the student society of the Mathematics Department of University College London is called the August De Morgan Society. The crater De Morgan on the Moon is named after him. |