Augustus De Morgan

Augustus De Morgan was born in Madurai, in the Carnatic region of India in 1806.[2] [lower-alpha 1] His father was Lieut.-Colonel John De Morgan (1772–1816), who held various appointments in the service of the East India Company, and his mother, Elizabeth (née Dodson, 1776–1856), was daughter of John Dodson and granddaughter of James Dodson, who computed a table of anti-logarithms (inverse logarithms).[3] 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. [2] Mrs De Morgan resided at various places in the southwest of England, and her son received his primary 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 from one of Euclid's works with a ruler and compasses. [2]

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. He became an atheist.[4][5]

There is a word in our language with which I shall not confuse this subject, both on account of the dishonourable use which is frequently made of it, as an imputation thrown by one sect upon another, and of the variety of significations attached to it. I shall use the word Anti-Deism to signify the opinion that there does not exist a Creator who made and sustains the Universe.

— De Morgan 1838, p. 22

In 1823, at the age of sixteen, he entered Trinity College, Cambridge,[6] 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 college tutor was John Philips Higman, FRS (1793–1855).

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 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; 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 Ramanujan. He supervised the publication in London of Ramchundra's book Treatise on Problems of Maxima and Minima in 1859. In his preface to that edition, De Morgan wrote:

On examining this work I saw in it, not merely merit worthy of encouragement, but merit of a peculiar kind, the encouragement of which, as it appeared to me, was likely to promote native effort towards the restoration of the native mind in India.

In the same preface, he acknowledged his awareness of the Indian tradition of logic, and later wrote again, in 1860, of its significance:

"The two races which have founded the mathematics, those of the Sanscrit and Greek languages, have been the two which have independently formed systems of logic.[7]

Though the sophistication of Indian logical thought had been brought to the attention of Western mathematicians by a number of authors beginning in the late-18th century, it is not known whether this had any influence on De Morgan's own work. Mary Boole, however, claimed a profound influence – via her uncle George Everest – of Indian thought in general and Indian logic, in particular, on George Boole, as well as on De Morgan and Charles Babbage:

Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan, and George Boole on the mathematical atmosphere of 1830–65. What share had it in generating the Vector Analysis and the mathematics by which investigations in physical science are now conducted?[8]

Jonardon Ganeri has observed that it was this period of the mid-nineteenth century pointed to by Mary Boole that saw George Boole (1815–1864) and Augustus De Morgan (1806–1871) make their pioneering applications of algebraic ideas to the formulation of logic (algebraic logic and Boolean logic), and has suggested that these figures were likely to have been aware of the Indian system of logic, and in turn, that their awareness of the shortcomings of propositional logic as it was then formulated may have contributed to their willingness to look beyond their own logical tradition.[9]

Augustus was one of seven children, four of whom survived to adulthood.

• Eliza (1801–1836) married Lewis Hensley, a surgeon, living in Bath.
• Augustus (1806–1871)
• George (1808–1890), a barrister-at-law who married Josephine, daughter of Vice Admiral Josiah Coghill, 3rd Baronet Coghill
• Campbell Greig (1811–1876), a surgeon at the Middlesex Hospital

In the autumn of 1837, he married Sophia Elizabeth Frend (1809–1892), eldest daughter of William Frend (1757–1841) and Sarah Blackburne (1779–?), a granddaughter of Francis Blackburne (1705–1787), Archdeacon of Cleveland.[10]

De Morgan had three sons and four daughters, including fairytale author Mary de Morgan. His eldest son was the potter William De Morgan. His second son George acquired 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.

Augustus De Morgan.

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's work entitled Trigonometry and Double Algebra[11] 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 only natural numbers and symbols of operations such as +, ×, etc. are used. 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 a and b denote any natural numbers. An expression such as a − b may still be impossible, so 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 a + b√−1 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 √−1 is interpreted as denoting a quadrant. The expression a + b√−1 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 e^a√−1. De Morgan attempted it by reducing such an expression to the form b + q√−1, 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 a + b√−1 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 a + b√−1 + c√−1^√−1 although this contradicts the truth established by Euler that √−1^√−1 = e^−π/2. 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".

In Book II, Chapter II, following the above quoted passage about the theory of symbolic algebra, 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 ${\displaystyle 0}$, ${\displaystyle 1}$, ${\displaystyle +}$, ${\displaystyle -}$, ${\displaystyle \times }$, ${\displaystyle \div }$, ${\displaystyle ()}$⁽⁾, and letters; these only, all others are derived. As De Morgan explains, the last of these symbols represents writing a latter expression in superscript over and after a former. His inventory of the fundamental laws is expressed under fourteen heads, but some of them are merely definitions. The preceding list of symbols is the matter under the first of these heads. The laws proper may be reduced to the following, which, as he admits, are not all independent of one another, "but the unsymmetrical character of the exponential operation, and the want of the connecting process of ${\displaystyle +}$ and ${\displaystyle \times }$ ... renders it necessary to state them separately":

1. Identity laws. ${\displaystyle a=0+a}$ ${\displaystyle =+a}$ ${\displaystyle =a+0}$ ${\displaystyle =a-0}$ ${\displaystyle =1\times a}$ ${\displaystyle =\times a}$ ${\displaystyle =a\times 1}$ ${\displaystyle =a\div 1}$ ${\displaystyle =0+1\times a}$
2. Law of signs. ${\displaystyle +(+a)=+a,}$ ${\displaystyle +(-a)=-a,}$ ${\displaystyle -(+a)=-a,}$ ${\displaystyle -(-a)=+a,}$ ${\displaystyle \times (\times a)=\times a,}$ ${\displaystyle \times (\div a)=\div a,}$ ${\displaystyle \div (\times a)=\div a,}$ ${\displaystyle \div (\div a)=\times a}$
3. Commutative law. ${\displaystyle a+b=b+a,}$ ${\displaystyle a\times b=b\times a}$
4. Distributive law. ${\displaystyle a(b+c)=ab+ac,}$ ${\displaystyle a(b-c)=ab-ac,}$ ${\displaystyle (b+c)\div a=(b\div a)+(c\div a),}$ ${\displaystyle (b-c)\div a=(b\div a)-(c\div a)}$
5. Index laws. ${\displaystyle a^{0}=1,}$ ${\displaystyle a^{1}=a,}$ ${\displaystyle (a\times b)^{c}=a^{c}\times b^{c},}$ ${\displaystyle a^{b}\times a^{c}=a^{b+c},}$ ${\displaystyle (a^{b})^{c}=a^{b\times c}}$

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 rules and no others—except they be formed by combination of these rules—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, ${\displaystyle (a+b)+c=a+(b+c),(ab)c=a(bc)}$ and to which was afterwards given the name 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 ${\displaystyle m^{n}}$ is not equal to ${\displaystyle n^{m}}$; 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.

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 at Dublin, and George Boole at Cork. De Morgan's work, 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 ${\displaystyle m}$, of the M's that are A's is ${\displaystyle a}$, and of the M's that are B's is ${\displaystyle b}$; then there are at least ${\displaystyle (a+b-m)}$ 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 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 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".

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 shall call each of these persons a paradoxer, and his system a paradox. 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. Many of the things brought forward would now be called crotchets, which is the nearest word we have to old paradox. But there is this difference, that by calling a thing a crotchet we mean to speak lightly of it; which was not the necessary sense of paradox. Thus in the 16th century many spoke of the earth's motion as the paradox of Copernicus and held the ingenuity of that theory in very high esteem, and some I think who even inclined towards it. In the seventeenth century the deprivation of meaning took place, in England at least.

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; mechanical contrivance sometimes, not very often, escapes this rule. All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors and learned in what had been before them. There is not one exception.

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.

I suspect that I know more of the English class than any man in Britain. I never kept any reckoning: but I know that one year with another?  and less of late years than in earlier time? – I have talked to more than five in each year, giving more than a hundred and fifty specimens. Of this I am sure, that it is my own fault if they have not been a thousand. Nobody knows how they swarm, except those to whom they naturally resort. They are in all ranks and occupations, of all ages and characters. They are very earnest people, and their purpose is bona fide, the dissemination of their paradoxes. A great many – the mass, indeed – are illiterate, and a great many waste their means, and are in or approaching penury. These discoverers despise one another.

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 ${\displaystyle \pi =3{\tfrac {1}{8}}}$. His mode of reasoning was a curious caricature of the reductio ad absurdum of Euclid. He said let ${\displaystyle \pi =3{\tfrac {1}{8}}}$, and then showed that on that supposition, every other value of ${\displaystyle \pi }$ must be absurd. Consequently, ${\displaystyle \pi =3{\tfrac {1}{8}}}$ is the true value. 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. A mathematical snail! This cannot be the thing so called which regulates the striking of a clock; for it would mean that I am to make Mr. Smith sound the true time of day, which I would by no means undertake upon a clock that gains 19 seconds odd in every hour by false quadrative value of ${\displaystyle \pi }$. 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 ${\displaystyle \pi =3{\tfrac {1}{8}}}$ and James Smith, Esq., of the Mersey Dock Board: and put hors de combat by pebbles from a sling. If Goliath had crept into a snail shell, David would have cracked the Philistine with his foot. There is something like modesty in the implication that the crack-shell pebble has not yet taken effect; it might have been thought that the slinger would by this time have been singing – And thrice [and one-eighth] I routed all my foes, And thrice [and one-eighth] I slew the slain.

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 developed the calculus of relations in his Syllabus of a Proposed System of Logic (1966: 208–46), first published in 1860. De Morgan was able to show that reasoning with syllogisms could be replaced with composition of relations.[12] The calculus was described as the logic of relatives by Charles Sanders Peirce, who admired De Morgan and met him shortly before his death. The calculus was further extended in the third volume of Ernst Schröder's Vorlesungen über die Algebra der Logik. Binary relations, especially order theory, proved critical to the Principia Mathematica of Bertrand Russell and Alfred North Whitehead. In turn, this calculus became the subject of much further work, starting in 1940, by Alfred Tarski and his colleagues and students at the University of California.