80-793: Sir William Hamilton, 9th Baronet FRSE (8 March 1788 – 6 May 1856) was a Scottish metaphysician . He is often referred to as William Stirling Hamilton of Preston, in reference to his mother, Elizabeth Stirling. He was born in rooms at the University of Glasgow , He was from an academic family: his father Professor William Hamilton , had in 1781, on the recommendation of William Hunter , been appointed to succeed his own father, Dr Thomas Hamilton, as Regius Professor of Anatomy, Glasgow ; he died in 1790, aged 32. William Hamilton and his younger brother, Thomas Hamilton , were brought up by their mother. Hamilton received his early education at Glasgow Grammar School , except for two years which he spent in
160-471: A dissenter : He married into a Unitarian family, where his essentially Christian deist interpretations of scripture were welcome. Later in life he would lean more deist and join Martineau's Free Christian Union . De Morgan was on occasion accused of atheism which he dismissed as sectarianism. In his will De Morgan would write I commend my future with hope and confidence to Almighty God; to God
240-591: A Foreign Honorary Member of the American Academy of Arts and Sciences in 1855. He taught his class for the last time in the winter of 1855–1856. Shortly after the close of the session he was taken ill, and died in Edinburgh. He died on 6 May 1856 and was buried in St John's Episcopal Churchyard at the east end of Princes Street in Edinburgh. The stone is not in its original location and is used to edge
320-423: A Proposed System of Logic (1860). He showed that reasoning with syllogisms could be replaced with the composition of relations . The calculus was described as the logic of relatives by Charles Sanders Peirce , who admired De Morgan and met him shortly before his death. Historians trace several developments in modern logic directly to De Morgan's contributions to algebraic logic : "Any serious attempt to study
400-463: A confidant on personal matters. The study of logic in Britain underwent a revival following the publication of Richard Whately 's Elements of Logic in 1826. The book itself was the subject of a debate that would spur both De Morgan and George Boole to action. On the one hand, argued by William Whewell , logic, particularly syllogism as emphasized by Whately, could not arrive at "new truths" and
480-663: A consulting actuary for various life assurance firms, including the Family Endowment Assurance Office, the Albert Life Assurance Office, and the Alliance Assurance Company . He published several articles on actuarial subjects as well as the book An Essay on Probabilities and Their Application to Life Contingencies and Insurance Offices . However his most notable work as an actuary is his promotion of
560-446: A distinct system of unit-symbols, and investigating or assigning relations which define their mode of action on each other". De Morgan summarized and extended his algebraic work in his book Trigonometry and Double Algebra (1849). De Morgan was a prolific writer; an incomplete list of his works occupies 15 pages of his memoirs. While most of De Morgan's mathematical writing is educational in nature, consisting of various textbooks, it
640-415: A doctrine to the writings of forgotten thinkers; and nothing gave him greater pleasure than to draw forth such from their obscurity, and to give due acknowledgment, even if it chanced to be of the prior possession of a view or argument that he had thought out for himself. Of modern German philosophy he was a diligent, if not always a sympathetic, student. How profoundly his thinking was modified by that of Kant
720-534: A few months of his birth. His 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 was not a member of any one of the colleges. When De Morgan was ten years old, his father died. His mathematical talents went unnoticed until he
800-642: A full curriculum, from Euclid through the calculus of variations, with his classes often exceeding 100 students. His approach integrated lectures, reading, problem sets, personal instruction, and extensive course notes. He disliked rote learning and viewed mathematics education as learning to reason and core to a liberal education. Several of his students went on to become mathematicians, most notably James Joseph Sylvester , and some of them, Edward Routh and Isaac Todhunter , well known educators themselves. Many of his non-mathematician students rated him highly; William Stanley Jevons described De Morgan as "unrivalled" as
880-467: A government pension. Recruited by Elizabeth Jesser Reid , in 1849 De Morgan taught mathematics for one year at the newly founded Ladies College in Bedfored Square . In 1850 De Morgan received a book from John Elliot Drinkwater Bethune , A Treatise on Problems of Maxima and Minima , written and self-published by the self-taught Indian mathematician Ramchundra . De Morgan was so struck by
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#1732780765646960-525: A private school at Chiswick in Kent , and in 1807 went as a Snell Exhibitioner , to Balliol College, Oxford . He obtained a first class in literis humanioribus and took his BA in 1811 (MA 1814). He had been intended for the medical profession, but soon after leaving Oxford he gave up this idea, and in 1813 became a member of the Scottish bar, as a qualified advocate . Hamilton's life continued to be that of
1040-634: A recently built townhouse at 11 Manor Place, in Edinburgh's west end. In 1829, his career of authorship began with the appearance of the well-known essay on the "Philosophy of the Unconditioned" (a critique of Victor Cousin 's Cours de philosophie )–the first of a series of articles contributed by him to the Edinburgh Review . He was elected in 1836 to the University of Edinburgh chair of logic and metaphysics, and from this time dates
1120-403: A reprint, with large additions, of his contributions to the Edinburgh Review . Soon after, his general health began to fail. Assisted by his devoted wife, he persevered in literary labour; and during 1854–1855 he brought out nine volumes of a new edition of Stewart's works. The only remaining volume was to have contained a memoir of Stewart, but this he did not live to write. Hamilton was elected
1200-528: A series of papers and the book Trigonometry and Double Algebra (1849). De Morgan's double algebra was never fully developed but remains a precursor to geometric algebra and influenced the Irish mathematician Sir William Rowan Hamilton in his development of quaternions . De Morgan and Hamilton were friends and correspondents for over 25 years, with De Morgan serving both as a colleague in mathematics, reviewing his Lectures on Quaternions (1853), and as
1280-458: A series of papers "On the foundation of algebra", describing what he called "logical" or " double " algebra, essentially an early form of geometric algebra . While these papers are perhaps most notable for their influence on Sir William Rowan Hamilton and the development of quaternions , they are also recognized to contain De Morgan's steps towards a fully abstract algebra : "Inventing
1360-469: A strict Church of England upbringing De Morgan was publicly a non-conformist , at some personal cost: His refusal to conform debarred him from further advancement at Cambridge; his marriage was without Church ceremony; and on several occasions he fought with the University College administration to maintain religious neutrality, eventually resigning over the issue. In private De Morgan was
1440-449: A student, while he was gradually forming his philosophic system. Investigation enabled him to make good his claim to represent the ancient family of Hamilton of Preston, and in 1816 he took up its baronetcy , which had been in abeyance since the death of Sir Robert Hamilton of Preston (1650–1701). Two visits to Germany in 1817 and 1820 led to William's taking up the study of German and later on that of contemporary German philosophy, which
1520-605: A teacher. Jevons, heavily influenced by De Morgan, would go on to do independent work in logic and become best known for the development of the theory of utility as part of the so-called Marginal Revolution . In 1866, the Chair of Mental Philosophy and Logic at University College fell vacant and James Martineau was recommended formally by the Senate to the Council. The Council, at the urging of George Grote , rejected Martineau on
1600-432: Is a strange compound of Kant and Reid. Its chief practical corollary is the denial of philosophy as a method of attaining absolute knowledge and its relegation to the academic sphere of mental training. The transition from philosophy to theology, i.e. to the sphere of faith, is presented by Hamilton under the analogous relation between the mind and the body. As the mind is to the body, so is the unconditioned Absolute or God to
1680-439: Is comparatively slight, but he stimulated a spirit of criticism in his pupils by insisting on the great importance of psychology as opposed to the older metaphysical method, and by his recognition of the importance of German philosophy, especially that of Immanuel Kant . By far his most important work was "Philosophy of the Unconditioned," the development of the principle that for the human finite mind, there can be no knowledge of
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#17327807656461760-469: Is evident from the tenor of his speculations; nor was this less the case because, on fundamental points, he came to widely different conclusions. Hamilton was more than a philosopher; his knowledge and interests embraced all subjects related to that of the human mind. He studied anatomy and physiology. He was also well-read in ancient and modern literature, being particularly interested in the 16th and 17th centuries. Among his literary projects were editions of
1840-399: Is for his pioneering contributions to logic for which he is best known, presented in several books and papers, notably Formal Logic (1847) and Syllabus of a Proposed System of Logic (1860). His work on algebra is also of note, in particular Trigonometry and Double Algebra (1849). De Morgan was also a well known popularizer of science and mathematics; he contributed over 600 articles to
1920-830: The Penny Cyclopedia and contributions to the Quarterly Journal of Education , the Gallery of Portraits , and the Companion to the British Almanac . Following his first resignation from London University, De Morgan started his work as a private tutor. One of his early students was Jacob Waley . He would tutor Ada Lovelace from 1840 through 1842, primarily via correspondence. De Morgan's great-grandfather, grandfather, and father-in-law were all actuaries ; not surprisingly, De Morgan also worked as
2000-477: The Infinite . The basis of his argument is the thesis, "To think is to condition." Deeply impressed with Kant's antithesis between subject and object, the knowing and the known, Hamilton laid down the principle that every object is known only in virtue of its relations to other objects. From this it follows that limitless time, space, power, etc., are inconceivable. The fact, however, that all thought seems to demand
2080-586: The Mathematical Tripos , earning a Bachelor of Arts degree. To obtain the higher degree of Master of Arts and become eligible for a fellowship, he was required to pass a theological test. Although he was raised in the Church of England, De Morgan strongly objected to taking this test. Unable to advance in academia due to his refusal, he entered Lincoln's Inn to pursue a career in law. The London University (now known as University College London)
2160-720: The Royal Society ) but also read and discussed. The first meeting of the London Mathematical Society was held at University College in 1865. De Morgan was the first president and his son was the first secretary. The earliest members included Benjamin Gompertz , De Morgan's personal friend and fellow actuary, William Stanley Jevons and James Joseph Sylvester , De Morgan's former students, Thomas Archer Hirst , De Morgan's colleague, and mathematicians William Kingdom Clifford and Arthur Cayley . Augustus
2240-716: The post-nominal letters FRSE, Honorary Fellows HonFRSE, and Corresponding Fellows CorrFRSE. The Fellowship is split into four broad sectors, covering the full range of physical and life sciences, arts, humanities, social sciences, education, professions, industry, business and public life. Examples of current fellows include Peter Higgs and Jocelyn Bell Burnell . Previous fellows have included Melvin Calvin , Benjamin Franklin , James Clerk Maxwell , James Watt , Thomas Reid , and Andrew Lawrence . A comprehensive biographical list of Fellows from 1783–2002 has been published by
2320-551: The Cambridge Philosophical Society in 1846. The paper describes a mathematical system that formalizes Aristotelian logic , specifically the syllogism . While the rules De Morgan defines, including the eponymous De Morgan's laws , are straightforward, the formalism is significant: it represented the first serious instance of mathematical logic, which would come to pervade the field of logic, and presaged logic programming . The subsequent dispute with
2400-514: The Father of our Lord Jesus Christ, whom I believe in my heart to be the Son of God, but whom I have not confessed with my lips, because in my time such confession has always been the way up in the world. At age 60, De Morgan's 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
2480-634: 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 , the Tower of London , or Westminster Abbey . Despite
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2560-504: The Society for 30 years. In 1836, De Morgan's replacement as Professor of Mathematics, George J. P. White, drowned; De Morgan was convinced to return and reinstated. That same year the London University was renamed University College and, together with King's College , was made an affiliate of the newly created University of London . De Morgan was a highly successful mathematics teacher. For over 30 years his courses covered
2640-559: The Society. Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician . He is best known for De Morgan's laws , relating logical conjunction, disjunction, and negation, and for coining the term " mathematical induction ", the underlying principles of which he formalized. De Morgan's contributions to logic are heavily used in many branches of mathematics, including set theory and probability theory , as well as other related fields such as computer science . Augustus De Morgan
2720-661: The Study and Difficulties of Mathematics (1831), Elementary Illustrations of the Differential and Integral Calculus (1832), The Elements of Spherical Trigonometry (1834), Examples of the Processes of Arithmetic and Algebra (1835), An Explanation of the Gnomic projection of the sphere (1836), The Differential and Integral Calculus (1842), and The Globes Celestial and Terrestrial (1845), as well as over 700 articles in
2800-539: The University of London. 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 humorously described himself using the Latin phrase ' Homo paucarum literarum ' (man of few letters), reflecting his modesty about his extensive contributions to mathematics and logic. He disliked
2880-604: The Warden, Leonard Horner , a dispute arose over the handling of medical student protests calling for the removal of the Professor of Anatomy, Granville Sharp Pattison , on the grounds of incompetence. While De Morgan and others argued that students should have no influence in the matter, the University bowed to student pressure and dismissed Pattison. De Morgan resigned on 24 July 1831, followed by Professors George Long and Friedrich August Rosen . In 1826 Lord Brougham, one of
2960-514: The autumn of 1837, De Morgan married Sophia Elizabeth Frend (1809–1892), the eldest daughter of William Frend and Sarah Blackburne (1779–?), a granddaughter of Francis Blackburne (1705–1787), Archdeacon of Cleveland. De Morgan had three sons and four daughters, including fairytale author Mary De Morgan . His eldest son was the potter William De Morgan , who would marry the painter Evelyn De Morgan , nee Pickering. His second son, George, acquired distinction in mathematics at University College and
3040-504: The contemporary work of Tarski or Birkhoff should begin with a serious study of the most significant founders of their field, especially Boole , De Morgan, Pierce and Schröder ". In fact, a theorem articulated by De Morgan in 1860 was later expressed by Schrŏder in his textbook on binary relations , and is now commonly called Schröder rules . De Morgan was an early convert and supporter of Peacock's symbolical algebra but soon grew disillusioned. Starting in 1839, De Morgan authored
3120-691: The editorship of H. L. Mansel, D.D. (1862). A Memoir of Sir W. Hamilton , by Veitch, appeared in 1869. FRSE Fellowship of the Royal Society of Edinburgh ( FRSE ) is an award granted to individuals that the Royal Society of Edinburgh , Scotland's national academy of science and letters , judged to be "eminently distinguished in their subject". This society received a royal charter in 1783, allowing for its expansion. Around 50 new fellows are elected each year in March. As of 2016 there are around 1,650 Fellows, including 71 Honorary Fellows and 76 Corresponding Fellows. Fellows are entitled to use
3200-526: 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? Arthur Cowper Ranyard and George Campbell De Morgan, De Morgan's son, conceived the idea of founding a mathematical society in London, where mathematical papers would be not only received (as by
3280-514: 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 is best known for his pioneering contributions to mathematical logic , specifically algebraic logic , and, to a lesser extent, for his contributions to the beginnings of abstract algebra . De Morgan's contributions to logic are two-fold. Firstly, before De Morgan there
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3360-461: The enclosure at the east end of the church. He had married Janet, the daughter of Hubert Marshall, and was succeeded by his son Sir William Stirling-Hamilton, 10th Baronet , a general in the British Army. In 1840, the University of Leyden granted him an honorary Doctor of Divinity (DD), a rarity for persons outside the clergy. Hamilton's positive contribution to the progress of thought
3440-703: The founders of London University, founded the Society for the Diffusion of Useful Knowledge (SDUK) with the goal of promoting self-education and improving the moral character of the middle- and working- classes through cheap and accessible publications. De Morgan became involved with the SDUK in March 1827; his unpublished manuscript Elements of Statics for the society may have played a role in his appointment to London University. One of its most voluminous and effective writers, De Morgan published several books with SDUK: On
3520-406: The grounds that he was a Unitarian clergyman and instead appointed a layman, George Croom Robertson . De Morgan argued that the founding principle of religious neutrality had been abandoned and immediately resigned. De Morgan was an early proponent of symbolical algebra . First expressed by George Peacock in his Treatise on Algebra (1830) and developed by Duncan Gregory , symbolical algebra
3600-428: The idea of the infinite or absolute provides a sphere for faith, which is thus the specific faculty of theology. It is a weakness characteristic of the human mind that it cannot conceive any phenomenon without a beginning: hence the conception of the causal relation, according to which every phenomenon has its cause in preceding phenomena, and its effect in subsequent phenomena. The causal concept is, therefore, only one of
3680-464: The inclusion of logic in the Cambridge curriculum. De Morgan's paper "On the structure of the syllogism", published in 1846, mathematically defines the rules of Aristotelian logic , specifically syllogism , and including what are now known as De Morgan's laws . Historically significant as the inception of mathematical logic , at the time, De Morgan's paper initiated a dispute with Hamilton over
3760-567: The influence which, during the next 20 years, he exerted over the thought of the younger generation in Scotland. Much about the same time he began the preparation of an annotated edition of Thomas Reid's works, intending to annex to it a number of dissertations. However before this design had been carried out, he was struck, in 1844, with paralysis of the right side which seriously crippled his bodily powers, though left his mind unimpaired. The edition of Reid appeared in 1846, but with only seven of
3840-456: The intended dissertations, one unfinished. At his death he had still not completed the work; notes on the subjects to be discussed were found among his manuscripts. Considerably earlier, he had formed his theory of logic , the leading principles of which were indicated in the prospectus of "an essay on a new analytic of logical forms " prefixed to his edition of Reid. But the elaboration of the scheme in its details and applications continued during
3920-485: The intention is to attribute a quality (i.e. the predicate is used in connotation). In other words, we are not considering the question "what kind are men among the various things which must die?" (as is implied in the form "all men are some mortals") but "what is the fact about men?" We are not stating a mere identity (see further, e.g., H. W. B. Joseph , Introduction to Logic , 1906, pp. 198 foll.). The philosopher to whom above all others Hamilton professed allegiance
4000-493: The most general, and expressly asserted that it has to do, not with the objective validity, but only with the mutual relations, of judgments. He further held that induction and deduction are correlative processes of formal logic, each resting on the necessities of thought and deriving thence its several laws. The only logical laws which he recognised were the three axioms of identity, noncontradiction, and excluded middle, which he regarded as severally phases of one general condition of
4080-426: The next few years to occupy much of his leisure. Out of this arose a sharp controversy with Augustus De Morgan . The essay did not appear, but the results of the labour gone through are contained in the appendices to his Lectures on Logic . Hamilton had also drawn from the works of Wilhelm Esser in his explanation of laws in the language of agency. For instance, he cited Esser's definition of universal law, to explain
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#17327807656464160-434: The ordinary necessary forms of the cognitive consciousness limited, as we have seen, by being confined to that which is relative. As regards the problem of the nature of objectivity, Hamilton simply accepts the evidence of consciousness as to the separate existence of the object: "the root of our nature cannot be a lie." In virtue of this assumption, Hamilton's philosophy becomes a "natural realism." In fact, his whole position
4240-535: The philosopher Sir William Stirling Hamilton over the "quantification of the predicate" referred to in De Morgan's paper would lead George Boole to write the pamphlet Mathematical Analysis of Logic (1847). De Morgan elaborated upon his initial paper in the book Formal Logic, or the Calculus of Inference, Necessary and Probable (1847), published the same week as Boole's pamphlet and was immediately overshadowed by it. Nonetheless, later practitioners would recognize
4320-426: The pioneering nature of his work; C. I. Lewis wrote, "His originality in the invention of new logical forms, his ready wit, his pat illustrations, and clarity and liveliness of his writing did yeoman service in breaking down the prejudice against the introduction of 'mathematical' methods into logic". De Morgan developed the calculus of relations in his paper "On the syllogism, No. IV" and in his book Syllabus of
4400-447: The position. Ultimately the search committee, steered by founder Lord Brougham , Olinthus Gregory , and Henry Warburton , selected De Morgan from a field of at least 31 candidates including Dionysius Lardner , Peter Nicholson , John Radford Young , Henry Moseley , John Herapath , Thomas Hewitt Key , William Ritchie , and John Walker . De Morgan's work during this period focused on mathematical instruction: His first publication
4480-420: The possibility of existence and, therefore, of thought. The law of reason and consequent he considered not as different, but merely as expressing metaphysically what these express logically. He added as a postulate—which in his theory was of importance--"that logic be allowed to state explicitly what is thought implicitly." in logic, Hamilton is known chiefly as the inventor of the doctrine of the "quantification of
4560-417: The predicate," i.e. that the judgment "All A is B " should really mean "All A is all B," whereas the ordinary universal proposition should be stated "All A is some B." This view, which was supported by Stanley Jevons , is fundamentally at fault since it implies that the predicate is thought of in its extension; in point of fact when a judgment is made, e.g. about men, that they are mortal ("All men are mortal"),
4640-469: 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 he was from happiness. He never sought to become a Fellow of
4720-509: The role of mathematics in logic; "mathematics can not conduce to logical habits at all," Hamilton would write. The dispute would focus on the so-called quantification of the predicate , which Hamilton claimed, but as the dispute wore on in the pages of the Athenæum and in the publications of the two writers, it became apparent that Hamilton and his supporters were wrong and that De Morgan's mathematically precise description of Aristotle's logic
4800-568: The sense or "quality" of "necessary". Hamilton also prepared extensive materials for a publication which he designed on the personal history, influence and opinions of Martin Luther . Here he advanced so far as to have planned and partly carried out the arrangement of the work; but it did not go further, and still remains in manuscript. In 1852–1853 appeared the first and second editions of his Discussions in Philosophy, Literature and Education ,
4880-575: The study of mathematics as a mental gymnastic, which excited much opposition, but which he never saw reason to alter. As a teacher, he was zealous and successful, and his writings on university organisation and reform had, at the time of their appearance, a decisive practical effect, and contain much that is of permanent value. His posthumous works are his Lectures on Metaphysics and Logic (1860), 4 vols., edited by H. L. Mansel , Oxford, and John Veitch ( Metaphysics ; Logic ); and Additional Notes to Reid's Works, from Sir W. Hamilton's Manuscripts. , under
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#17327807656464960-440: The very principles of Hamilton's philosophy are apparently violated in his theological argument. Hamilton regarded logic as a purely formal science; it seemed to him an unscientific mixing together of heterogeneous elements to treat as parts of the same science the formal and the material conditions of knowledge. He was quite ready to allow that on this view logic cannot be used as a means of discovering or guaranteeing facts, even
5040-552: The work of Benjamin Gompertz , whose " law of mortality " was both under-appreciated and plagiarized. De Morgan became involved with the Astronomical Society of London in 1828. He would be appointed honorary secretary in 1831, the year in which it received its Royal Charter and became the Royal Astronomical Society . He would continue as secretary for 18 years and remain actively involved in
5120-438: The work that he entered into correspondence with Ramchundra and arranged for the book's re-publication in London in 1859, targeting a European audience; De Morgan's preface surveyed classical Indian mathematical thought and urged a contemporary return of Indian mathematics: 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,
5200-443: The works of George Buchanan and Julius Caesar Scaliger . His general scholarship found expression in his library, which became part of the library of the University of Glasgow. He also may have influenced subsequent philosophy as the inspiration for a critique by John Stuart Mill which resulted in perhaps the clearest statements ever of the idea of matter as the permanent possibility of sensation . His chief practical interest
5280-514: The world of the conditioned. Consciousness , itself a conditioned phenomenon, must derive from or depend on some different thing prior to or behind material phenomena. Curiously enough, however, Hamilton does not explain how it comes about that God, who in the terms of the analogy bears to the conditioned mind the relation which the conditioned mind bears to its objects, can be unconditioned. God can be regarded only as related to consciousness, and insofar is, therefore, not absolute or unconditioned. Thus
5360-411: Was The Elements of Algebra (1828), a translation of a French textbook by Louis Bourdon [ fr ] , followed by Elements of Arithmetic (1830), a widely used and long-lived textbook, and The Study and Difficulties of Mathematics (1831), a discourse on mathematical education. Following a series of squabbles between the faculty, including De Morgan, and the administration, in particular
5440-480: Was founded in 1826 as a secular alternative to Oxford and Cambridge; Catholics, Jews, and dissenters could enter as students and hold positions. Prior to opening in 1828, the University advertised 24 vacancies for professorship, two in mathematics, to which De Morgan applied. De Morgan was appointed Professor of Mathematics on 23 February 1828 at the age of twenty-one. The Council of the London University had failed to recruit Charles Babbage and John Herschel to
5520-500: Was Aristotle. His works were the object of his profound and constant study, and supplied in fact the mould in which his whole philosophy was cast. With the commentators on the Aristotelian writings, ancient, medieval and modern, he was also familiar; and the scholastic philosophy he studied with care and appreciation at a time when it had hardly yet begun to attract attention in his country. His wide reading enabled him to trace many
5600-637: Was a first step towards abstract algebra , separating the manipulation of symbols from their arithmetic meaning. While symbolical algebra could mechanically construct negative and imaginary numbers, as in the work of Adrien-Quentin Buée [ fr ] , Jean-Robert Argand , and John Warren , it could not provide their interpretation; De Morgan observed that a similar problem troubled the classical Indian mathematician Bhāskara II in his work Bijaganita . De Morgan would move on from symbolical algebra to develop what he called "logical" or " double " algebra in
5680-468: Was almost entirely neglected in British universities. In 1820 he was a candidate for the chair of moral philosophy in the University of Edinburgh , which had fallen vacant on the death of Thomas Brown , colleague of Dugald Stewart , and Stewart's consequent resignation, however he was defeated on political grounds by John Wilson (1785–1854), the "Christopher North" of Blackwood's Magazine . In 1821 he
5760-498: Was appointed professor of civil history and delivered several courses of lectures on the history of modern Europe and the history of literature. The salary was £100 a year, derived from a local beer tax, and was discontinued after a time. No pupils were compelled to attend, the class dwindled, and Hamilton gave it up when the salary ceased. In January 1827 his mother, to whom he had been devoted, died. In March 1828 he married his cousin, Janet Marshall. Around this time he moved to live in
5840-590: Was born in Madurai , in the Carnatic region of India , in 1806. His father was Lieutenant-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 the granddaughter of James Dodson , who computed a table of anti-logarithms (inverse logarithms ). Augustus De Morgan became blind in one eye within
5920-595: Was correct. On realizing this, Hamilton would claim that De Morgan had committed plagiarism. Boole, a friend of De Morgan's since 1842, motivated in part by the disputes between Whewell and Hamilton and De Morgan and Hamilton, would write The Mathematical Analysis of Logic , published in 1847 on the same day as De Morgan's Formal Logic . Boole's work would eclipse De Morgan's and come to define early mathematical logic. De Morgan continued to support Boole's efforts, proofreading and advocating for Boole's work. Upon Boole's death, De Morgan worked to ensure Boole's family received
6000-713: 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 . He received his secondary education from Mr. Parsons, a fellow of Oriel College, Oxford , who preferred classics to mathematics. In 1823, at the age of sixteen, De Morgan enrolled in Trinity College, Cambridge , where his teachers and tutors included George Peacock , William Whewell , George Biddell Airy , H. Parr Hamilton , and John Philips Higman . Both Peacock and Whewell would influence De Morgan's selection of algebra and logic for further research. De Morgan placed fourth in
6080-408: Was in education, an interest which he manifested alike as a teacher and as a writer, and which had led him long before he was either to study of the subject both theoretical and historical. He thence adopted views as to the ends and methods of education that, when afterwards carried out or advocated by him, met with general recognition; but he also expressed in one of his articles an unfavourable view of
6160-411: Was likely to promote native effort towards the restoration of the native mind in India. The influence of classical Indian logic on De Morgan's own work on logic has been speculated upon. Mary Boole , claimed a profound influence—via her uncle George Everest —of Indian thought in general and Indian logic, in particular, on her husband George Boole , as well as on De Morgan: Think what must have been
6240-473: Was no mathematical logic— logic , including formal logic , was the domain of philosophers; De Morgan was the first to make formal logic a mathematical subject. Secondly, De Morgan would develop the calculus of relations, essentially abstracting logic via the application of algebraic principles. De Morgan's first original paper on logic, "On the structure of the syllogism", appeared in the Transactions of
6320-670: Was one of seven children, only four of whom survived to adulthood. These siblings were Eliza (1801–1836), who married Lewis Hensley, a surgeon living in Bath; George (1808–1890), a barrister-at-law who married Josephine, daughter of Vice Admiral Josiah Coghill, 3rd Baronet Coghill ; and Campbell Greig (1811–1876), a surgeon at the Middlesex Hospital. When De Morgan moved to London, he befriended William Frend (1757–1841). Both had studied mathematics at Cambridge and subsequently left for religious reasons, and both were actuaries . In
6400-505: Was therefore inferior to and distinct from scientific reasoning; on the other hand, argued by the Scottish philosopher Sir William Hamilton , Whately's effort to equate logic to a "grammar for reasoning" was wrong and reductive. De Morgan, perhaps influenced by the writings of Sylvestre François Lacroix , saw the utility of Whately's logic in mathematics, both in its emphasis on the syllogism and in its grammar-like abstraction, as evidenced in his own writings on education and in his demand for
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