Augustus De Morgan
Augustus De Morgan 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.
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, who held various appointments in the service of the East India Company, and his mother, Elizabeth, was the granddaughter of James Dodson, who computed a table of anti-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
London University, 1827–1831
The London University 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, a translation of a French textbook by, followed by Elements of Arithmetic, a widely used
and long-lived textbook, and The Study and Difficulties of Mathematics, 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.
The Society for the Diffusion of Useful Knowledge
In 1826 Lord Brougham, one of the founders of London University, founded the Society for the Diffusion of Useful Knowledge 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, Elementary Illustrations of the Differential and Integral Calculus, The Elements of Spherical Trigonometry, Examples of the Processes of Arithmetic and Algebra, An Explanation of the Gnomic projection of the sphere, The Differential and Integral Calculus, and The Globes Celestial and Terrestrial, 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.Private tutor
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.Actuary
De Morgan's great-grandfather, grandfather, and father-in-law were all actuaries; not surprisingly, De Morgan also worked as 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 the work of Benjamin Gompertz, whose "law of mortality" was both under-appreciated and plagiarized.Royal Astronomical Society
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 the Society for 30 years.London University, 1836–1866
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 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 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 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.
Abstract algebra and Sir William Rowan Hamilton
De Morgan was an early proponent of symbolical algebra. First expressed by George Peacock in his Treatise on Algebra and developed by Duncan Gregory, symbolical algebra 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, 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 a series of papers and the book Trigonometry and Double Algebra. 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, and as a confidant on personal matters.
Mathematical logic and George Boole
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 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 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 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 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 a government pension.