Augustus DeMorgan

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Augustus De Morgan (27 June 1806, Madurai, India - 18 March 1871, London) was an Indian-born British mathematician and logician. He Professor of Mathematics at University College London between 1828 and 1866; and first president of the London Mathematical Society. Known for formulating the so-called De Morgan's laws, in his memory, and establishing a rigorous concept of the procedure, mathematical induction.

Biography

First years

Augustus De Morgan was born in Madurai, Madras, India, in 1806. His father was Lieutenant Colonel John De Morgan (1772-1816), who held various positions in the service of the East India Company. His mother, Elizabeth Dodson (1776-1856) was a descendant of James Dodson (known for being the author of tables of anti-logarithms). De Morgan went blind in one eye two months 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 British "no strings attached"≫.

His father died when De Morgan was ten years old. Mrs. De Morgan resided in various places in the South West of England, and her son received his primary education from her at various modest local schools. His mathematical talents went unnoticed until he was fourteen, when a family friend discovered him making a detailed drawing of a figure of Euclid with a straightedge and compass, explained to Augustus the ideas of the Greek geometer, and introduced him to the techniques of Euclid. demo.

He received his secondary education from Mr. Parsons, a Fellow of Oriel College, Oxford, who prized classical authors more than mathematics. Her mother was an active and ardent member of the Church of England, and she wished her son to become a clergyman; but by then de Morgan had begun to show his firm disagreement with his mother's plans.

University education

In 1823, at the age of sixteen, he entered Trinity College, Cambridge, where he studied under the influence of George Peacock and William Whewell, who became his lifelong friends; from the first derives his interest in the renewal of algebra, and from the second his dedication to the formalization of logic, the two main themes of his future working life. His tutor at the university was John Philips Higman (1793-1855).

In college he learned to play the flute, and was a prominent figure in music clubs. His love of knowledge itself interfered with the training necessary to begin a great academic career in mathematics; as a consequence, he earned the rank of fourth wrangler. This entitled him to the title of Bachelor of Arts; but to obtain the higher degree of Master of Arts and thus be eligible for a scholarship, at that time it was necessary to pass a theological test. His convictions led him to reject these tests, despite the fact that he had been raised in the Church of England. Around 1875 theological tests for obtaining academic degrees were abolished at the Universities of Oxford and Cambridge.

University of London

As he couldn't pursue any degree at his own university, he decided to follow the "Bar" (procedure established to qualify in legal practice), and settled in London; although he preferred to dedicate himself to teaching mathematics than to apply himself in reading the law. At the time, he was founding the University of London (now University College London). The two ancient universities of Oxford and Cambridge were so conditioned by theological evidence that no Jew or dissenter outside the Church of England could enter as a student, let alone be appointed to any academic office. A group of liberal-minded men resolved to overcome this obstacle by establishing a University in London on the principle of religious neutrality. De Morgan, at the age of 22, was appointed professor of mathematics. His introductory lecture "On the Study of Mathematics" is a discourse on the education of minds of permanent value, and has recently been reprinted in the United States.

The University of London was a new institution, and the relationships of the Board of Trustees, the Faculty Senate, and the student body were not well defined. A controversy arose between the professor of anatomy and his students, and as a consequence of the measures taken by the Council, several professors resigned, led by De Morgan. He was appointed a substitute math teacher, who drowned a couple of years later. De Morgan, who had shown himself to be a leader of the teachers, was invited to return to his charge, which thereafter became the permanent center of his work for thirty years.

The same group of reforming personalities headed by Lord Brougham, an eminent Scotsman in both science and politics who had founded the University of London, founded the Society for the Diffusion of Useful Knowledge at about the same time. Its objective was to disseminate all kinds of knowledge (especially scientific knowledge) through economic treatises and clearly written by the best authors of the time. One of its most prolific 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, also published by the Society, as well as those issued in other publications. When De Morgan moved to London, he found a kindred friend in William Frend, despite his mathematical heresy on negative quantities. Both had extensive knowledge of arithmetic, and their religious views were quite similar. Frend lived in what was then a London suburb, in a country house formerly occupied by Daniel Defoe and Isaac Watts. De Morgan, accompanied by his flute, was a welcome visitor.

The London University, where De Morgan was a professor, is a separate institution from the "University of London" (University of London). The University of London was founded about ten years later by the UK Government, in order to award degrees after the examination, without any prior qualification such as the period of 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 failed as a merely examining body; and it became an educational institution to use. De Morgan was a highly successful teacher of mathematics. He gave classes that lasted one hour, and at the end of each class he would pose a series of problems and illustrative examples of the subject on which he had lectured. He invited his students to sit with him and show him his results, which he had reviewed before the next class. In De Morgan's view, the deep understanding and assimilation of great principles was far more important than merely analytical skill in applying principles and means to handling particular cases.

Augustus De Morgan.

During this period, he also promoted the work of the self-taught Indian mathematician Ramchundra, who was called the Ramanujan of De Morgan. He oversaw the London publication of Ramchundra's book 'Maximums and Minims'. in 1859. In the introduction to this book, he acknowledged being aware of the Indian tradition of logic, although whether this had any influence on his own work is not known.

He was also the tutor of Ada Lovelace, with whom he later maintained written correspondence.

Last years

In 1866 the chair of mental philosophy at University College became vacant. James Martineau, a Unitarian clergyman and professor of mental philosophy, was formally recommended by the Senate to the Council; but in the Council there were some who opposed a Unitarian cleric, and others who opposed theistic philosophy. A layman from the school of Alexander Bain and Herbert Spencer was named. De Morgan considered that the old rule of religious neutrality had been breached, and he promptly resigned. He was then 60 years old. His students secured him a pension of £500 a year, but his misfortunes continued. Two years later, his son George (the & # 34; Younger Bernoulli & # 34; , as Augustus loved to hear him called, after the eminent mathematicians father and son of this name) died. This coup was followed by the death of a daughter. Five years after his resignation from University College, De Morgan died of a nervous condition on March 18, 1871.

Family

Augustus was one of seven De Morgan siblings, four of whom survived to adulthood.

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

De Morgan had three sons and four daughters, including fairy tale author Mary De Morgan. Her eldest child was the potter William De Morgan. Her second son, George De Morgan, gained great mathematical prestige at University College and the University of London . With another like-minded alumnus he conceived the idea of founding the London Mathematical Society, where mathematical papers would not only be archived (as at the Royal Society), but would actually be read and debated. The first meeting was held at University College; De Morgan was the first president and his son the first secretary. It was the beginning of the London Mathematical Society.

Personality

Photograph by Augustus de Morgan (1898)

De Morgan was a brilliant and witty writer, whether as a polemicist or as a correspondent. In his time he met two Sir William Hamilton , who are often confused. One of these was Sir William Hamilton, 9th Baronet (ie the title was inherited from him), a Scotsman, Professor of Logic and Metaphysics at the University of Edinburgh; the other was a gentleman (that is, he won the title), an Irishman, a professor of astronomy at the University of Dublin. The baron contributed to logic, especially to the doctrine of the quantification of the predicate; the gentleman, whose full name was William Rowan Hamilton, contributed to mathematics, especially geometric algebra, being the first to describe quaternions. De Morgan took an interest in their work, and corresponded with both; but the correspondence with the Scotchman ended in public controversy, while with the Irishman he maintained a friendship which ended only on his death. In one of his letters to Rowan, De Morgan says:

It is notorious that I have discovered that you and the other WH are reciprocal polars regarding me (intellectually and morally, for me the Scottish baron is a Polar bearAnd you, I was thinking, it's a Polar knight). When I send some research for Edinburgh, the WH of that spider says it's apart from him. When I send it to you, assimilate it, generalizing it to the naked eye, getting it this way applicable to society in general, and makes me the second discoverer of a known theorem.

De Morgan's correspondence with Hamilton the mathematician spanned more than twenty-four years; contains discussions not only of mathematical subjects, but also of topics of general interest. It was characterized by the genius of Hamilton and the ingenuity of De Morgan. The following is a sample: Hamilton wrote,

My copy of Berkeley's work is not mine; like Berkeley, you know, I'm an Irishman.

DeMorgan responded:

His phrase "my copy is not mine" is not a trigger. In English it is perfectly correct to use the same word in two different ways in a single sentence, especially when it is usual. The inconsistencies of language are no nonsense, because they express a meaning. But the incongruity of ideas (as in the case of the Irishman who was pulling off a rope, and when he realized that it was over, he shouted that someone had cut the other end...) that is a genuine nonsense.

De Morgan was full of personal quirks. On the occasion of the inauguration of his friend, Lord Brougham, as Chancellor of the University of Edinburgh, the Senate offered to confer on him the honorary degree of LL. D.; Morgan declined the honor for misnomering himself. He once had his name printed like this:

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, "man of few letters")

He did not like to travel outside of London, and while his family enjoyed themselves by the seaside, and men of science had a good time at British Association meetings in every corner of England, he remained in the sweltering libraries and dusty streets of the metropolis. He said that he felt like Socrates, who declared that the further he was from Athens, the further he was from happiness. He never tried to become a member of the Royal Society or attend one of its meetings; he said that he had no ideas or sympathies in common with the study of physics. His attitude was possibly due to his physical limitations, since they prevented him from being a good observer or experimenter. He never voted in an election, and never visited the House of Commons, the Tower of London, or Westminster Abbey.

Collection of paradoxes. "Paradoxers"

The original title of this work in English is "A Budget of Paradoxes". It was published posthumously in 1872 by his wife, a year after the mathematician's death. The book reflects very well the ironic character of De Morgan, who never ceases to be amazed by the abundance of revolutionary discoverers (he calls them false "paradoxers"), who claim to have solved insoluble problems, such as squaring the circle or the trisection of the angle. In the introduction to the book, De Morgan explains what he means by the word "paradox":

A large number of people, since the rise of the mathematical method, have attacked their direct and indirect consequences. I'm gonna call each of these people a "paradoxer," and their critique system, a "paradox." I use the word in the old sense: a paradox is something that is outside the general opinion, either because of its matter, method or conclusion. Many of the things now established, were called at the time with "corchetes", which is the closest expression we have to the old paradox. But there is a difference: that placing a text in square brackets we want to indicate that it is going to be spoken lightly in it; what does not necessarily coincide with the sense of paradox. Thus, in the 16th century, many spoke of the movement of the earth as "Copernic paradox" and were convinced of the ingenuity of that theory. I even think that some still think the same way; in the seventeenth century there was a decay of the intellect, at least in England.

How can you tell the fake paradoxer from the real one? De Morgan supplies the following proof:

The way in which a true "paradoxer" will show itself, in terms of sense or meaning, will not depend on what he maintains, but on whether or not he has sufficient knowledge of what has been done by others, especially as to how to do it, step before the invention of knowledge by himself... New knowledge, for any purpose, must come from the study of previous knowledge, in all matters affecting thought; mechanical artifices sometimes, not very often, escape this rule. All men who are now called discoverers, in every matter governed by thought, have been men versed in the minds of their predecessors and have learned from what had happened before them. There's no exception. I remember that just before a meeting of the American Association in Indianapolis in 1890, the local newspapers announced a great discovery that was to be presented to the wise gathered there: a young man from somewhere in the country, had squared the circle. While the meeting was ongoing, I watched a young man who was around with a roll of paper in his hand. He addressed me, and complained that the document had not been received with his discovery. I asked him if his object in the presentation of the document was not to get it read, printed and published, so that everyone could be informed of the result; accessing the written easily. But, I said, many men have worked on this issue, and their results have been fully tested, and are printed for the benefit of anyone who knows how to read. Have you reported your results? To that question there was no feeling, but yes the sick smile of the false paradoxer.

His "Collection" consists of a review of a large collection of paradoxical books, which De Morgan had accumulated in his own library, partly formed by his purchases in bookstores, by the books that were sent to him for his review, and by the books sent by their own authors. It includes the following classification: Squarers of circles; angle trisectors; cube duplicators; Perpetual Motion Constructors; Gravity Nullifiers; Earth Immobilizers; and Builders of the Universe. He is of the opinion that specimens of all these kinds can still be found in the New World and in the new century. De Morgan gives his personal knowledge of the 'paradoxers':

I suspect I know more about this kind of English than any man in Britain. I never took the exact account; but I know that one year with another (something less in the last few years than in previous times), I have spoken with more than five in each year, which results in more than one hundred and fifty specimens. Something I'm sure, it's my fault that it wasn't a thousand. No one knows how they polish, except those they resort to naturally. They are in all ranks and occupations, of all ages and characters. They are very serious people, and their purpose is to spread their paradoxes in good faith. A large part - the majority, in fact - are illiterate, and a large number have lost their means of life, and are or approach to hardship. These discoverers despise each other.

A paradoxer that De Morgan made the same "gift" that Achilles made Hector - dragging him around the walls over and over - was James Smith, a successful Liverpool merchant, who claimed to have found that ${displaystyle pi =3{tfrac {1}{8}}}$ . His mode of reasoning was a curious caricature of the "reductio ad absurdum" from Euclides. He said that ${displaystyle pi =3{tfrac {1}{8}}}$ and then it was exposing that in such case any other value of $pi$ It must be absurd. Accordingly, ${displaystyle pi =3{tfrac {1}{8}}}$ is the true value. What follows is a sample of "drawn by De Morgan around the walls of Troy":

Mr. Smith keeps writing me long letters, insisting I have to answer them. In his last 31-sided, tightly written notes paper, I am informed, with reference to my obstinate silence, that although I myself and other people think I am a mathematical Goliath, I have decided to behave like a mathematical snail, remaining within my shell. A mathematical "caracol"! This should not refer to the so-called object that regulates the bells of a watch; it would mean that I will make Mr. Smith sounds at the right time of the day, as I would in no way wear a watch that curiously earns 19 seconds in every hour because of a false square value. But he dares to tell me that the slings of the funda of simple truth and common sense ultimately will break my shell and put me out of combat. The confusion of the images is fun: Goliath hiding in a snail to avoid ${displaystyle pi =3{tfrac {1}{8}}}$ and James Smith of the Mersey Pier Board, putting him out of combat with the pebbles of a funnel. If Goliath had slipped into a snail shell, David could have crushed the Philistine simply with his foot. There is something similar to the modesty in the implication that the stone that breaks the shell has not yet effected; it could have been thought that the mushroom during this time has been chanting: "And three times [and an eighth], I enchanted all my enemies. And three times [and an eighth], I killed the dead. "

In the region of pure mathematics, De Morgan could easily distinguish the false from the true paradox; but he was not that expert in the field of physics. His mother-in-law and his wife could somehow be considered "paradoxers"; and in the opinion of the physicists of his time, De Morgan himself barely escaped this denomination. His wife wrote a book describing the phenomena of spiritualism, mesa-rap, turntables, etc.; and De Morgan drew up a preface in which he asserted that he knew of some of the facts asserted, and that he believed others through testimony, but that he did not claim to know whether they were caused by spirits, or had some unknown and unimaginable origin. From this alternative he left out ordinary material causes. Faraday gave a lecture on spiritualism, in which he made it clear that research must establish the idea of what is physically possible or impossible; De Morgan did not believe in this.

Spiritism

As a mature man, De Morgan became interested in the phenomena of spiritualism. In 1849 he had investigated clairvoyance and was impressed by the experience. Paranormal investigations were later carried out in his own home with the medium Maria Hayden. The result of these investigations was later published by his wife Sophia. De Morgan thought that his career as a scientist might have been affected if his interest in the study of spiritualism had been revealed, so he helped edit the book anonymously. It was published in 1863, under the title:& #34;From Matter to Spirit: The Result of Ten Years Experience in Spirit Manifestations." spirits.")

According to Oppenheim (1988), De Morgan's wife, Sophia, was a convinced spiritualist, but De Morgan held a different position on spiritualistic phenomena that Oppenheim defines as a "wait-and-see position" 3. 4;; he was neither a believer nor a skeptic. His view was that the methodology of the physical sciences does not automatically exclude psychic phenomena and that these phenomena may be further explainable by the possible existence of natural forces that physicists had not yet identified.

In the preface to "From Matter to Spirit" (1863) De Morgan said:

Thinking that it is very likely that the universe may contain a few agents - it is said that half a million- of whom no one knows anything, I cannot but suspect that a small proportion of these agents, - say five thousand - can be jointly competent for the production of all the spiritist phenomena, or may be at their height. The physical explanations I have seen are easy, but miserably insufficient: the spiritist hypothesis is sufficient, but difficultly ponderable. Time and thought will decide, the second asking the first more test results.

In Parapsychology: A Concise History (1997), John Beloff wrote that De Morgan was the first notable scientist in Britain interested in the study of spiritualism, and his activities would have influenced William Crookes's decision to also study spiritualism. He also claims that De Morgan was an atheist and that this deprived him of attaining a position at Oxford or Cambridge.

Mathematical work

Formal Logic (1847)

When the study of mathematics was revived at Cambridge University, so was the study of logic. The spirit of this movement was Whewell, the Master of Trinity College, whose principal writings were a "History of the Inductive Sciences" and a "Philosophy of the Inductive Sciences. inductive sciences". No doubt De Morgan was influenced in his logical investigations by Whewell; but other influential contemporaries included Sir William Rowan Hamilton of Dublin, and Professor Boole of Cork. De Morgan's work on formal logic, published in 1847, is chiefly notable for its development of the numerically definite syllogism:

Aristotle's followers claim that from two particular propositions, as "some M are A."and "some M are B."a relationship between A and B is not necessarily followed. But they go further, and postulate that in order for any relationship between the A and B to be necessarily followed, the average term must be universally taken on one of the premises. De Morgan said, some M are A and some M are B necessarily follows: Some A are B in certain conditions it formulated through exact numerical expressions, giving shape to the so-called numerically defined. Assuming the M number is $m$ of the M who are A is $a$ and the M that are B is $b$ ; then there is at least ${displaystyle (a+b-m)}$ Which is also B. For example, if the number of people on board a boat is supposed to be 1000, that 500 were in the living room, and that 700 are lost, it is necessarily deduced that at least 700 + 500 - 1000, that is, 200 passengers from the living room have been lost.

This single principle is sufficient to demonstrate the validity of all Aristotelian modes of reasoning. Therefore, it is a fundamental principle in deductive reasoning.

Lo and behold, De Morgan had made a breakthrough with the introduction of quantification terms. At that time, the philosopher Sir William Hamilton was teaching in Edinburgh a doctrine of the quantification of the predicate, and the correspondence between the two arose. However, De Morgan soon perceived that Hamilton's quantification was of a different character; which meant, for example, that the expressions The whole of A is the whole of B, and The whole of A is part of B, could replace the Aristotelian form Every A is B. Hamilton thought he had laid the cornerstone of the Aristotelian arch, as he put it. Curious arch, this one, which had stood for 2000 years without a cornerstone. As a consequence, he paid no attention to De Morgan's innovations, accused him of plagiarism, and the controversy raged for years in the columns of the Athenæum magazine, and in the publications of the two writers.

Trigonometry and Double Algebra (1849)

De Morgan's paper entitled Trigonometry and Double Algebra contains two parts; the first is a treatise on trigonometry, and the second is a treatise on generalized algebra, which he called "Double Algebra".

George Peacock's theory of algebra was improved upon by D.F. Gregory, a younger member of the Cambridge School, who stressed not the permanence of equivalent forms, but the permanence of certain formal laws. This new theory of algebra as the science of symbols and their laws of combination led to De Morgan's edition based on logic; and his doctrine, on the subject, is still followed by a large number of British algebraists. Thus, George Chrystal based his algebra textbook on Morganian theory; although an attentive reader can observe that he practically abandons it when he is confronted with the subject of infinite series. Morganian theory is reaffirmed in his volume on Trigonometry and Double Algebra, where in Book II, Chapter II, dedicated to & # 34;Symbolic Algebra & # 34;, he writes:

By abandoning the meanings of symbols, we also abandon those of the words that describe them. So. "suma" It becomes, at the moment, an empty sound of meaning. It is a combination mode represented by $+$ when $+$ receives its meaning, so also receives the word "suma". It is very important that the student take into account that, with an exception, no word or sign of arithmetic or algebra has an atom of meaning throughout this chapter, whose object is symbols, and their laws of combination, generating a Symbolic algebrawhich can become the grammar of a hundred different Significant algebras. If someone were to say that $+$ and $-$ could mean reward and punishmentand $A$ , $B$ , $C$ , etc. could represent the virtues and vices, the reader can believe it, contradict it, or whatever he pleases, but not outside this chapter.
The only exception mentioned above, which has some part of meaning, is the sign $=$ placed between two symbols, as in $A=B$ . It indicates that the two symbols have the same resulting meaning, as different as the necessary steps to achieve it. Yeah. $A$ and $B$ are two quantities, it indicates that their measure is the same; if they are operators, their effect is the same.

La first stage in the development of algebra is arithmeticwhere only numbers and operators appear as $+$ , $times$ etc.

La Second stage It's the universal arithmeticwhere the letters substitute the numbers to denote them universally, and the processes are carried out without knowing the values of these symbols. You have to $a$ and $b$ repent any amount; then an expression as ${displaystyle a-b}$ cannot be calculated; although in the universal arithmetic is spoken of "prevision", that is, that the planned operation is possible.

La 3rd stage It's him. Simple algebra, where a symbol can affect an amount ahead or back, being able to properly represent itself as segments in a straight line passing through its origin. Negative quantities are then possible; they are represented with segments in the opposite direction. But an impossibility still remains in the final part of an expression as ${displaystyle a+b{sqrt {-1}}}$ which arises in the resolution of a quadratic equation.

La Fourth stage It's him. double algebra. Algebraic symbolism generally denotes a segment of a line in a given plane. In a double symbol two specifications are involved, nominally, length, and direction; ${displaystyle {sqrt {-1}}}$ is interpreted as a dimension in another quadrant. The expression ${displaystyle a+b{sqrt {-1}}}$ then represents a line on the plane with an absciss $a$ and an orderly $b$ . Argand and Warren had taken the double algebra far beyond; but they were not able to interpret expressions in their theories as ${displaystyle e^{a{sqrt {-1}}}}$ . De Morgan tried. reducing it as an expression of form ${displaystyle b+q{sqrt {-1}}}$ and showed that he had found a procedure for achieving this reduction in any case. The remarkable fact is that this double algebra satisfies all the fundamental laws listed, and any combination of seemingly impossible symbols could be interpreted as if it had the full form of algebra. Chapter 6 introduces hyperbolic functions and analyzes the connection between common and hyperbolic trigonometry.

If the previous theory is true, the Next stage of development should be the "Triple algebra" And yes. ${displaystyle a+b{sqrt {-1}}}$ really represents a line in a given plane, it should be possible to find a third element that added to the previous ones was assimilable to a straight in space. Argand and many others assumed it was ${displaystyle a+b{sqrt {-1}}+c{sqrt {-1}},^{sqrt {-1}}}$ although this contradicted the postulates of Euler, in which ${displaystyle {sqrt {-1}},^{sqrt {-1}}=e^{-{frac {1}{2}}pi }}$ . From Morgan and many others worked hard on this unsuccessful problem until Hamilton intervened. Now you can see clearly why: the symbolism of the double algebra denotes not a length and an address; if not One module and angle. The angles are confined to a plane. Then the next stage will be a "Quadruple algebra", in which the axis of the plane becomes variable. This gives the answer to the first question; the Algebra double analytically represents the trigonometry of the plane, and for this it became the natural tool of the analysis of the alternating electric current. But De Morgan never went that far. He died with the conviction that "the double algebra will allow to complete the development of the conceptions of the arithmetic earrings, as far as these symbols are involved, as the arithmetic itself suggests immediately."

In chapter 2 of the second book, continuing his theoretical approaches to symbolic algebra, De Morgan proceeds to invent both the fundamental symbols of algebra and its laws. The symbols are ${displaystyle 0}$ , $1$ , $+$ , $-$ , $times$ , $div$ , ${displaystyle ()}$ ⁽⁾, and letters; only these, all the others are derived from the above. As De Morgan explains, the last of these symbols allows to write an exponential, placing it above and then a given expression. Their inventory of fundamental laws is reduced to fourteen points, although some are mere definitions. The previous list of symbols appears under the first of these fourteen points. The laws themselves can be reduced to the following, which he himself admits, are not totally independent of each other, "but the asymmetric nature of the exponential operation, and the desire to connect the processes of $+$ and $times$ ... they necessarily lend themselves to keeping them separately":

Identity Laws: ${displaystyle a=0+a}$ ${displaystyle =+a}$ ${displaystyle =a+0}$ ${displaystyle =a-0}$ ${displaystyle =1times a}$ ${displaystyle =times a}$ ${displaystyle =atimes 1}$ ${displaystyle =adiv 1}$ ${displaystyle =0+1times a}$
Sign Acts: ${displaystyle +(+a)=+a,}$ ${displaystyle +(-a)=-a,}$ ${displaystyle -(+a)=-a,}$ ${displaystyle -(-a)=+a,}$ ${displaystyle times (times a)=times a,}$ ${displaystyle times (div a)=div a,}$ ${displaystyle div (times a)=div a,}$ ${displaystyle div (div a)=times a}$
Commutative Law: ${displaystyle a+b=b+a,}$ ${displaystyle atimes b=btimes a}$
Distributive Act: ${displaystyle a(b+c)=ab+ac,}$ ${displaystyle a(b-c)=ab-ac,}$ ${displaystyle (b+c)div a=(bdiv a)+(cdiv a),}$ ${displaystyle (b-c)div a=(bdiv a)-(cdiv a)}$
Laws on Exponentiation: ${displaystyle a^{0}=1,}$ ${displaystyle a^{1}=a,}$ ${displaystyle (atimes b)^{c}=a^{c}times b^{c},}$ ${displaystyle a^{b}times a^{c}=a^{b+c},}$ ${displaystyle (a^{b})^{c}=a^{btimes c}}$

From Morgan proceeds to give a complete inventory of the laws to which the algebra symbols obey, stating that "Any system of symbols that obey these rules and not others; except that they are formed by combinations of these same rules; it is then a Symbolic algebra." From this point of view, none of the above principles are rules; they are formally laws, that is, arbitrarily choose relations to which algebraic symbols are subject. De Morgan does not mention the law, which had previously been pointed out by Gregory, nominally: ${displaystyle (a+b)+c=a+(b+c),(ab)c=a(bc)}$ later called Associative Law. If the Commutative Law fails, the Associative Law should be better established; but not the other way around. It is a unfortunate circumstance for symbolists and formalists who in universal arithmetic ${displaystyle m^{n}}$ It's not like ${displaystyle n^{m}}$ ; then the Commutative Law would have full scope.Why was it not given full scope? Because the foundations of algebra are, after all, real and nonformal, material and non symbolic. For the formalists, exponential operations are too unmanageable, therefore they do not consider them, bringing them to applied mathematics.

Algebra of Relations (1860)

De Morgan discovered the algebra of relations in his Syllabus of a Proposed System of Logic, published in 1860. This algebra it was extended by Charles Sanders Peirce (who admired De Morgan and came to meet him), and again expounded and extended in vol. 3 of Vorlesungen über die Algebra der Logik by Ernst Schröder. The algebra of relations was a critical test of Bertrand Russell and Alfred North Whitehead's Principia Mathematica. In turn, this algebra became the subject of much more work, begun in the 1940s by Alfred Tarski and his colleagues and students at the University of California.

Writings for the Cambridge Philosophical Society (1864)

Mainly through the efforts of Peacock and Whewell, a Philosophical Society had been created at Cambridge; and to his Annals ((Mathematical Proceedings of the Cambridge Philosophical Society)) De Morgan contributed four memoirs on the foundations of algebra, and an equal number of writings on formal logic.

These memoirs on logic that De Morgan contributed to the Transactions of the Cambridge Philosophical Society after the publication of his book on "Formal Logic" are, with By far the most important contributions he made to science, especially his fourth memoir, in which he begins work in the broad field of "Logic of Relatives". This field has acquired great relevance later, oriented to a better knowledge of the language and thought processes. Identity and difference are the two main relationships considered by logicians; but there are many others equally deserving of study, such as equality, equivalence, consanguinity, affinity, etc.

De Morgan Proposed Laws

In modern mathematical logic, the following fundamental principles of the algebra of logic are known as Laws proposed by De Morgan:

«The denial of the conjunction equivalent to disjunction of denials»
«The denial of the disjunction equivalent to conjunction of denials»

In formal symbolic writing:

${displaystyle lnot (Aland B)Leftrightarrow (lnot A)lor (lnot B)}$
${displaystyle lnot (Alor B)Leftrightarrow (lnot A)land (lnot B)}$

Main publications

De Morgan's writings, published as complete works, would form a small library, as is the case, for example, with his writings for the Useful Knowledge Society. His view of algebra is best presented in the volume entitled Trigonometry and Double Algebra, published in 1849. His most peculiar work was his study of what he called & #34;paradoxers" (A Budget of Paradoxes); which originally appeared published in the form of letters in the columns of the Athenæum review; it was revised and expanded by De Morgan in the last years of his life, and published posthumously by his widow. From a strictly academic point of view, his most recognized work is Formal logic or the calculation of necessary and probable inferences (1847).

Chronologically, his main works are ordered as follows (list of original titles in English):

1836. An Explanation of the Gnomonic Projection of the Sphere. London: Baldwin.
1837. Elements of Trigonometry, and Trigonometrical Analysis. London: Taylor & Walton.
1837. The Elements of Algebra. London: Taylor & Walton.
1838. An Essay on Probabilities. London: Longman, Orme, Brown, Green & Longmans.
1840. The Elements of Arithmetic. London: Taylor & Walton.
1840. First Notions of Logic, Preparatory to the Study of Geometry. London: Taylor & Walton.
1842. The Differential and Integral Calculus. London: Baldwin.
1845. The Globes, Celestial and Terrestrial. London: Malby & Co.
1847. Formal Logic or The Calculus of Inference. London: Taylor & Walton.
1849. Trigonometry and Double Algebra. London: Taylor, Walton & Malbery.
1860. Syllabus of a Proposed System of Logic. London: Walton & Malbery.
1872. A Budget of Paradoxes. London: Longmans, Green.

Acknowledgments and Honors

Since 1884, the London Mathematical Society delivers the Morgan Medal every three years in recognition of the achievements of leading mathematicians.
The headquarters of the London Mathematical Society is called "The De Morgan House".
The student society of the University College of London Department of Mathematics is called "The August De Morgan Society".
The lunar crater De Morgan carries his name.

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