## de Morgan, Augustus (1806–1871)

## de Morgan, Augustus (1806–1871)

# DE MORGAN, AUGUSTUS

*(1806–1871)*

A British mathematician and logician, Augustus De Morgan was born at Madura, India, where his father was an army officer. After early education in the west of England, he entered Trinity College, Cambridge, in 1823 and graduated fourth wrangler in 1827. His refusal to subscribe to the religious tests then in force precluded him from further advancement at Cambridge, but he was fortunate enough to be appointed first professor of mathematics at the newly opened University of London. Because of his habit of resigning on matters of principle, he twice vacated this chair, once at the beginning and once at the end of his career; but he enjoyed, in the interval, the highest repute and affection as a teacher and had many pupils who later achieved distinction.

In addition to numerous important papers on the foundations of algebra and the philosophy of mathematical method, De Morgan was the author of several excellent elementary textbooks; a standard bibliography, *Arithmetical Books* (London, 1847); a large treatise on the calculus (London, 1842); and an enormous quantity of learned journalism, mostly in the shape of review articles in the London *Athenaeum* and contributions on mathematical and astronomical subjects to the *Companion to the Almanac* (1831–1857) and to the *Penny* (later *English* ) *Cyclopaedia.* His best-known work in this line is the posthumously assembled *Budget of Paradoxes* (London, 1872), a still-diverting miscellany from the lunatic fringes of science and mathematics, originally serialized in the *Athenaeum.* Despite many years' service as secretary of the Royal Astronomical Society, De Morgan was in general suspicious of official bodies and distinctions, never sought membership in the Royal Society, and declined an Edinburgh LL.D. Indifferent to politics and society—and professedly hostile to the animal and vegetable kingdoms as well—he nonetheless maintained an extensive scientific correspondence with such friends as William Whewell, George Boole, Sir John Herschel, Sir William Rowan Hamilton (the mathematician), and John Stuart Mill. His crotchets did little to disguise his exceptional benevolence and firmness of character or to inhibit his talents as a humorist and a wit.

De Morgan's outlook was that of a philosophical mathematician and historian of science; he did not claim to be a philosopher in any narrow sense of the term. He admired Berkeley and followed him to the extent of holding the existence of minds to be more certain, as a fact of experience, than that of a material world. But his general attitude to such questions may be gathered from his remark that, while he would not dissuade a student from metaphysics, he would warn him, "when he tries to look down his own throat with a candle in his hand, to take care that he does not set his head on fire."

In common with other mathematicians of his time, De Morgan realized that algebra could be conceived as a system of symbols whose laws could be codified independently of any arithmetical or other interpretation that might be given to them. His logic had a similar aim. Deeply versed in the history of logic, he was able to freshen and illuminate the subject by generalizing its traditional principles along mathematical lines. In this respect he ranks as the chief precursor of Boole; but his views attained notice chiefly through the controversy that arose when Sir William Hamilton (of Edinburgh) accused him of plagiarizing the doctrine of a quantified predicate.

De Morgan's *Formal Logic* (London, 1847) represents the best-known, though by no means the most mature, statement of his logical views. Among its many excellences, the chapter on fallacies is worthy of mention. De Morgan's later work is dispersed in pamphlets and periodicals, most notably in five memoirs contributed to the *Cambridge Philosophical Transactions* (Vols. 8–10, 1847–1863) and in his *Syllabus of a Proposed System of Logic* (London, 1860, reprinted in *On the Syllogism* (London, 1964). Though too largely concerned with polemics against Hamilton, and hampered by a notation that found no acceptance, these writings display much originality in the handling of negative terms, compound propositions, and numerous unorthodox varieties of syllogistic reasoning. Apart from the well-known "De Morgan laws" for the negation of conjunctions and disjunctions (or logical sums and products), the most important development was the recognition that the copula performs its function in the syllogism solely by virtue of its character as a transitive and convertible relation. De Morgan was led by this to examine the logic of relations in general and so paved the way not only for Peirce's "logic of relatives" but for all that has since been done in this branch of the subject.

As a skilled actuary, who was often in demand as a consultant to insurance companies, De Morgan was not unnaturally interested in the mathematical theory of probability and the problems of applying it to the hazards of mortality and other types of experience. His treatise "Theory of Probabilities," in the *Encyclopaedia Metropolitana* (London, 1837) and the more popular *Essay on Probabilities* (London, 1838) were among the earlier discussions of this topic in English (see further relevant chapters of *Formal Logic* and the papers on the evaluation of argument and testimony attached to the first two Cambridge memoirs above). De Morgan's conception of probability was largely derived from Pierre Simon de Laplace, whose ideas (and errors) he was thus instrumental in propagating among his nineteenth-century successors. His method of approach was to construe the theory as an extension of formal logic, that is, as an investigation of the rules whereby propositions not absolutely certain affect the certainty of other propositions with which they are connected. He also employed the "inverse" procedures founded on Bayes's theorem, whereby, from known factual premises, it is sought to conjecture the probabilities of their likely or possible antecedents. In attempting to quantify the degree of uncertainty involved, De Morgan identified it with the amount of belief that is, or rather, that ought to be attached to it by a rational person, and proceeded on this basis to discuss the compounding and derivation of partial beliefs in accordance with the mathematical rules of the calculus of chances. His view of the matter was thus both a priori and subjective, though not in the objectionably psychological sense that has sometimes been ascribed to him. There are better reasons for censuring the technical errors he fell into through uncritical reliance on the Laplacean "rule of succession" and "principle of indifference"; even here, however, his confidence in the mathematical apparatus was often less blindly trusting than that of the writers who preceded him.

De Morgan's conception of scientific method may be gathered primarily from a review of Francis Bacon's works inserted in the *Budget of Paradoxes.* He there embraced what is essentially the modern "hypothetico-deductive" view of the subject; but one has to go to William Whewell before him or to W. S. Jevons after him to see it worked out in full.

** See also ** Bacon, Francis; Bayes, Bayes' Theorem, Bayesian Approach to Philosophy of Science; Berkeley, George; Boole, George; Hamilton, William; Herschel, John; Jevons, William Stanley; Laplace, Pierre Simon de; Logic, History of; Logic, Traditional; Logical Terms, Glossary of; Mill, John Stuart; Scientific Method; Whewell, William.

## Bibliography

Apart from the works mentioned under the History of Logic entry there is not much literature on De Morgan. The best general accounts are in A. Macfarlane, *Ten British Mathematicians* (New York, 1916) and J. A. Passmore, *A Hundred Years of Philosophy* (London: Duckworth, 1957). The *Memoir* by his wife, Sophia Elizabeth De Morgan (London: Longmans, Green, 1882), contains an excellent account of him. The details of the quarrel with Hamilton may be found in Hamilton's *Letter to Augustus De Morgan* (Edinburgh, 1847) and De Morgan's *Statement in Answer* (London, 1847) or more readily in the appendices to De Morgan's *Formal Logic.*

*P. L. Heath (1967)*