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.
Contents
Biography
Childhood
Augustus De Morgan 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 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 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.
Education
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 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 had been 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.
Career
The London University (now known as University College London) 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 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 was The Elements of Algebra (1828), a translation of a French textbook by Louis Bourdon, 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 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 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 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 the Penny Cyclopedia and contributions to the Quarterly Journal of Education, the Gallery of Portraits, and the Companion to the British Almanac.
Personal life
Augustus 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 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 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 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 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.
Retirement and death
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 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.
Mathematics
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.
Mathematical logic
De Morgan's contributions to logic are two-fold. Firstly, before De Morgan there 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 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 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 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 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 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.
Abstract algebra
De Morgan was an early convert and supporter of Peacock's symbolical algebra but soon grew disillusioned. Starting in 1839, De Morgan authored 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 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).
Works
De Morgan was a prolific writer; an incomplete list of his works occupies 15 pages of his memoirs. While most of his mathematical writing is educational in nature, consisting of various textbooks, it 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 the Penny Cyclopedia, ranging from Abacus to Young, Thomas. His most unusual work is A Budget of Paradoxes, a compilation of his writing, mostly book reviews, for The Athenæum Journal.
Algebra
While De Morgan's two early works on algebra are instructional, his translation of Bourdon's The Elements of Algebra (1828) and his own textbook The Elements of Algebra (1835), the issues he encountered while writing them would spur his later research.
De Morgan's research papers on algebra, presented in a sequence of four in the Transactions of the Cambridge Philosophical Society from 1839 to 1844 titled "On the foundation of algebra", defined what De Morgan called "logical" or "double" algebra. While the papers are most notable for their influence on Hamilton and quaternions, No. II includes the definition of what are now called fields and No. IV handles the case of "triple" algebra which eluded Hamilton.
De Morgan's book Trigonometry and Double Algebra (1849) consists of a treatise on trigonometry and a synthesis of his earlier work on algebra, tracing the development of "double" algebra, essentially geometric algebra, from arithmetic through symbolical algebra, illustrated throughout with the construction of the complex numbers.
De Morgan enumerates the laws that define an algebraic structure, in an early instance of what Whitehead would call universal algebra. While De Morgan notably omits Gregory's associative law, the selective application of laws, e.g., commutativity, is what led to Hamilton's quaternions.
Also of note is the introduction of hyperbolic functions and comparison of circular and hyperbolic trigonometry.
Logic
De Morgan's first work on logic, First Notions of Logic (1839), is pedagogical, introducing students to the necessary logic to study Euclid's Elements.
De Morgan's first research paper on logic, "On the structure of the syllogism" (1846), describing a mathematical system for Aristotelian syllogism, arguably marks the beginning of so-called mathematical logic.
Perhaps De Morgan's best known work, Formal Logic, or the Calculus of Inference, Necessary and Probable was published in 1847 in the same week (by arrangement) as George Boole's Mathematical Analysis of Logic. The book is primarily a reissue of his paper "On the structure of the syllogism" (1846) but also includes his earlier book, First Notions of Logic (1839), chapters on fallacies and probability, and the details of his dispute with the Scottish philosopher Sir William Hamilton.
De Morgan continued his research on logic in a series of papers, most notably "On the syllogism, No. IV" (1860), which introduced the logic of relations. De Morgan synthesizes much of this work in his book Syllabus of a Proposed System of Logic (1860).
A Budget of Paradoxes
Published posthumously in 1872, A Budget of Paradoxes is a compilation of De Morgan's column of the same name for the Athenæum, consisting mostly book reviews and focusing on so-called paradoxers, also referred to as pseudomaths (a De Morgan neologism) and pseudoscientists.
The pseudomaths De Morgan describes are mostly circle-squarers, such as Thomas Baxter, cube-duplicators, and angle-trisectors. One such angle-trisector was James Sabben, whose work received a one-line review from De Morgan:
"The consequence of years of intense thought": very likely, and very sad.
Another pseudomath identified by De Morgan was James Smith, a successful merchant of Liverpool, who claimed that
π
=
3
1
8
{\displaystyle \pi =3{\tfrac {1}{8}}}
. De Morgan writes:
Mr. Smith continues to write me long letters, to which he hints that I am to answer. In his last of 31 closely written sides of note paper, he informs me, with reference to my obstinate silence, that though I think myself and am thought by others to be a mathematical Goliath, I have resolved to play the mathematical snail, and keep within my shell... But he ventures to tell me that pebbles from the sling of simple truth and common sense will ultimately crack my shell...
Among the many pseudoscientific ideas De Morgan discredits are Alfred Wilks Drayson's expanding Earth theory and Samuel Rowbotham's Zetetic Astronomy, or the flat Earth theory.
In his discussion of calculations of
Spiritualism
Later in his life, De Morgan developed an interest in spiritualism. Initially intrigued by clairvoyance, he conducted paranormal investigations with the American medium Maria Hayden. The results of these investigations are documented in the book From Matter to Spirit: The Result of Ten Years Experience in Spirit Manifestations (1863), written by Sophia De Morgan and published anonymously to avoid repercussions.
Sophia was likely a convinced spiritualist, but De Morgan himself was neither a firm believer nor a skeptic. He maintained that the methodology of the physical sciences does not automatically exclude psychic phenomena, suggesting that such phenomena might eventually be explained by natural forces not yet identified by physicists. In the preface to From Matter to Spirit (1863), De Morgan writes:
Thinking it very likely that the universe may contain a few agencies – say half a million – about which no man knows anything, I can not but suspect that a small proportion of these agencies – say five thousand – may be severally competent to the production of all the [spiritualist] phenomena, or may be quite up to the task among them. The physical explanations which I have seen are easy, but miserably insufficient: the spiritualist hypothesis is sufficient, but ponderously difficult. Time and thought will decide, the second asking the first for more results of trial.
De Morgan was one of the first notable scientists in Britain to take an interest in the study of spiritualism, influencing William Crookes to also study spiritualism.
Legacy
The headquarters of the London Mathematical Society are called De Morgan House, and the top prize awarded by the Society is the De Morgan Medal.
The student society of the Mathematics Department of University College London is called the Augustus De Morgan Society.
The lunar crater De Morgan is named after him.
Collections
De Morgan's extensive library of mathematical and scientific works, many historical, was acquired by Samuel Jones-Loyd for the University of London and is now part of the Senate House Libraries collection. The London Mathematical Society's archive (up until 1994) is held in University College London (UCL) Special Collections. The collection primarily contains letters to Thomas Archer Hirst, but also includes correspondence to other European mathematicians. UCL holds around 150 rare books from the Society which focus on the history of mathematics and were published between 1701 - 1850. UCL holds letters from James Smith to Hepworth Dixon, editor of The Athenaeum, arguing against De Morgan's theory on the quadrature of the circle.
Publications
Books
Bourdon, Pierre Louis Marie (1828). The Elements of Algebra. Translated by De Morgan, Augustus. London: John Taylor.
De Morgan, Augustus (1831). On the Study and Difficulties of Mathematics.
An Explanation of the Gnomonic Projection of the Sphere. London: Baldwin. 1836.
Elements of Trigonometry, and Trigonometrical Analysis. London: Taylor & Walton. 1837a.
Elements of Algebra. London: Taylor & Walton. 1837b.
An Essay on Probabilities, and Their Application to Life Contingencies and Insurance Offices. London: Longman, Orme, Brown, Green & Longmans. 1838.
De Morgan, Augustus (1839). First Notions of Logic, Preparatory to the Study of Geometry. London: Taylor & Walton.
The Elements of Arithmetic (4th ed.). London: Taylor & Walton. 1840a.
The Differential and Integral Calculus. London: Baldwin. 1842.
The Globes, Celestial and Terrestrial. London: Malby & Co. 1845.
De Morgan, Augustus (1847). Formal Logic or The Calculus of Inference, Necessary and Probable. London: Taylor & Walton.
Trigonometry and Double Algebra. London: Taylor, Walton & Malbery. 1849.
De Morgan, Augustus (1860). Syllabus of a Proposed System of Logic. London: Walton & Malbery.
Journal articles
De Morgan, Augustus (1839). "On the Foundation of Algebra". Transactions of the Cambridge Philosophical Society. 7: 173–187.
De Morgan, Augustus (1841). "On the Foundation of Algebra, No. II". Transactions of the Cambridge Philosophical Society. 7: 287–300.
De Morgan, Augustus (1846). "On the structure of the syllogism, and on the application of the theory of probabilities to questions of argument and authority". Transactions of the Cambridge Philosophical Society. 8 XXIX: 379–408.
De Morgan, Augustus (1843). "On the Foundation of Algebra, No. III". Transactions of the Cambridge Philosophical Society. 8: 139–142.
De Morgan, Augustus (1844). "On the Foundation of Algebra, No. IV, On triple algebra". Transactions of the Cambridge Philosophical Society. 8: 241–253.
De Morgan, Augustus (1850). "On the Symbols of Logic, the Theory of the Syllogism, and in particular of the Copula, and the application of the Theory of Probabilities to some questions of evidence". Transactions of the Cambridge Philosophical Society. 9: 79–127.
De Morgan, Augustus (1858). "On the syllogism, No. III, and on logic in general". Transactions of the Cambridge Philosophical Society. 10 (1): 173–230.
De Morgan, Augustus (1860). "On the syllogism, No. IV, and on the logic of relations". Transactions of the Cambridge Philosophical Society. 10 (2): 331–358.
De Morgan, Augustus (1863). "On the syllogism, No. V, and on various points of the onymatic system". Transactions of the Cambridge Philosophical Society. 10 (2): 428–487.