Paradigm, Number 9 (December, 1992)
43, St. Leonards Road, Bengeo, Herts. SG14 3JW.
Based on a paper given at the University College, London, colloquium. Copyright reserved to I. Grattan-Guinness.
We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic, the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two.
This quotation, from De Morgan in 1868, 1 shows him at his best, The point made is surprising but profoundly true, both in his time and still in ours; 2 he is one of the few to have worked in both disciplines, so that the statement is self-referential; and it is so again in a personal way, for he himself had sight in only one eye, so that his wit shines especially strongly here. In this paper a brief survey of his career and achievement is followed by somewhat more detailed remarks on his contribution, via both mathematics and logic, to mathematical education.
Augustus De Morgan was born in 1806 in Madura to a colonel in the Indian Army, who brought his family back to England within months of the childs birth. An evident talent for mathematics won Augustus a place at Trinity College Cambridge in 1823 and fourth Wranglership four years later. At that time University College London was being founded, and he was appointed founder professor in mathematics. 3 He passed his entire career there, broken only by a resignation for five years from 1831 as a protest against administrative practices. He resigned again in 1866 (five years before his death), to be succeeded by his former student T. A. Hirst (1830-1892). His other principal students were W. S. Jevons (1835-1882), best remembered now for his work in economics but then also very well known as a logician; 4 and W. K. Clifford (1845-1879), who obtained the companion chair in applied mathematics at University College in 1874. Little is known, however, about his career and general effectiveness as a teacher.
De Morgan confined his involvement in societies to two organisations: the Society for the Diffusion of Useful Knowledge (SDUK), which gave him his main venue for educational writing; and the Royal Astronomical Society, of which he was secretary for a time and for whom he wrote several historical articles. The history of mathematics and of science was one of his principal concerns, aided by the gradual building up of a magnificent library (kept in his home in Camden Town, across the fields from University College). This collection now graces the shelves of the University of London Library, where are also kept about 200 of his notebooks; otherwise, however, his manuscripts are hard to find, with a disappointing file at University College. This is a great pity, for he was also a prolific correspondent, and quantities of his letters are to be found in the Nachlässe of friends such as G. B. Airy (a former teacher at Cambridge) and J. F. W. Herschel; but we lack most replies.
De Morgans total publication list is vast. It includes research-level articles on various aspects of algebra and analysis, and especially in logic (his most important work); a pioneering book, written with his wife, on physical mediumship; two book-length articles for the Encyclopaedia Metropolitana, and hundreds of short ones for the Penny Encyclopaedia (produced by the SDUK); many book reviews in general journals; several historical books and articles; a variety of textbooks; and articles on education, especially in the Quarterly Journal of Education of the SDUK, with which we start.
The five years of resignation, 1831-1836, are particularly important for our concerns, for during this period he produced much of his most significant writing on mathematical education. The Quarterly Journal of Education contains thirty pieces in the ten volumes produced during its years 1831-1835 of publication: commentaries on the Ecole Polytechnique in Paris and the far less distinguished efforts in science coming out of Oxford; lengthy reviews of books on algebra and other areas of mathematics; and a string of articles explicitly on teaching elementary branches of mathematics, from which the following quote shows him in good form:
It is a very common notion that this subject [arithmetic] is easy; that is, a child is called stupid who does not receive his first notions of number with facility. This, we are convinced, is a mistake. Were it otherwise, savage nations would acquire a numeration and a power of using it, at least proportional to their actual wants, which is not the case. 5
In 1831 De Morgan published with the SDUK his first book on educational questions, On the study and difficulties of mathematics. It appeared again five years later together with some related writings, and a reprint was made in the USA in 1902, albeit with at least one unfortunate misprint. [See p. 11 Ed.] The book was a companion to textbooks, commenting on educational questions for teachers and others. He covered mostly arithmetic and algebra, and on the latter branch he showed difficulties perhaps of his own making, although not unique to him.
The principal worry was the epistemological status of negatives, and the manner of their accommodation into arithmetic and algebra. Two sensible interpretations had long been known as directed line segments (+3 is 3 feet in front of me, and -3 is 3 feet behind); and as debts, from which the minus sign may have its origins in Mediterranean accounting. 6 Nevertheless, De Morgan followed the naïve empiricism of many algebraists in treating negative numbers as ways of speaking that are properly expressed only by rephrasing propositions containing them into a form containing only positive numbers or differences (for example, 3-5=-2 is really, say, 3+2=5 or 5-2=3). Already under the sway of his father-in-law William Frend, a fervent doubter of the negatives, during these years he was also influenced by C. Peacock, then Dean of Ely but also a former teacher at Cambridge, and allowed himself a more abstract reading of (an) algebra. Overall, however, his position remained unclear. This is typical, for philosophical thought was not normally one of his strengths. 7
Logic arose in the closing chapters of his Study, where De Morgan considered geometrical reasoning. The best known issue was the truth and (non-)provability of the parallel axiom of Euclidean geometry, but he saw more general issues at hand. These led to notable fruit in 1839, in a pamphlet on First notions of logic (preparatory to the study of geometry, which reappeared virtually unchanged as the opening chapter of his book Formal logic (1847). A striking example of the lack of eye contact between logic and mathematics is that fact that, while Euclidean geometry had long been held to be the model of deductivism in mathematics and syllogistic as the apotheosis of Aristotelian logic, very little work had been done for 2000 years on the links between the two until De Morgans enterprise&endash;when the links turned out to be far more incomplete and complicated than might have been expected. He went on to make other notable contributions to logic (although not with educational issues in mind): he played a significant role in the early 1840s concerning the quantification of the predicate, which allowed the four types of proposition all/some Xs are all/some Ys into an extended repertoire of syllogistic forms; and, most importantly of all, he introduced a general logic of relations in a paper of 1860. Links with algebras were quite explicit; there, are even aspects of semiotics in his presentation, especially a notation using brackets and periods to characterise propositional forms. 8 His valuable contributions are rendered somewhat incoherent, however, by unclear presentation and especially by skimping on the philosophical background.
Soon after writing Study, De Morgan turned his attention to the calculus, the dominant branch of mathematics of the time. A booklet of Elementary illustrations of its procedures appeared from the SDUK in 1832, to be followed by a textbook on The elements of algebra, preliminary to the differential calculus and fit for the higher classes of schools (1835). Here he defined the notion of the continuity of a mathematical function f(x) in the modern way: given that x changes in absolute value by a small amount, then another small amount can be found by which f(x) changes in absolute value.
This definition uses limits, and its significance needs explanation. The calculus had received various formulations since its rise in the late 17th century. Newtons fluxional version using limits (in a notoriously mysterious manner) had not gained favour on the Continent. Most of the major work was done there, after 1750, in the differential and integral version (which explicitly avoids limits) of Leibniz refined by Euler. It was challenged late in the century by an algebraic symbol-swinging approach proposed by Lagrange, in which both limits and infinitesimals were (allegedly) avoided. Now these two latter approaches were the choices made at Cambridge by Peacock, Herschel and Charles Babbage to replace fluxions in the mid-1810s, shortly, before De Morgans arrival there.
However, in this book of 1835 De Morgan followed recent French work (especially with A. L. Cauchy at the Ecole Polytechnique) and his slightly senior Cambridge contemporary William Whewell from a more closely Newtonian point of view in reaffirming limits as the basic tool by articulating their basic definitions and properties in a far more clear form. The formulation of continuity lay within this approach.
De Morgan continued his account in the following year, 1836 when he began to publish with the SDUK his most substantial educational volume, on The differential and integral calculus. The Society followed its practice of publishing the book in parts, a (non-consecutive) numbers of its Library of Useful Knowledge series, in order that the poorer subscribers were able to buy it in instalments. Eventually the book was completed in 1842, in twenty-five numbers.9 The print-run was probably quite large, and the books can be found quite frequently today in libraries and in booksellers catalogues; but it is not easy to determine the original clientele as De Morgan went far beyond an elementary introduction to the calculus and presented a comprehensive survey of several aspects of mathematical analysis, including many recent results.
During these years Cauchys approach gradually gained ascendancy, to reach a position of dominance which (for better or worse) it has enjoyed ever since. De Morgans book is a quite important source for this change in Britain, in what amounted to a second reform of the calculus; but it proceeded much more gradually than had the putsch of the Analytical Society. In particular, he showed himself to be a transitional figure by treating limits as a branch of algebra rather than the autonomy of limits from both algebra and geometry that Cauchy affirmed; 10 and he showed his algebraic side again by including a chapter on the manipulation of divergent infinite series of terms, which Cauchy had proclaimed to be illegitimate. De Morgan could play both sides, although again the philosophical price was rather high; rather one-eyed, as it were.
1. A. De Morgan, Review of a book on geometry, The Athenaeum, (1868), vol. 2, pp. 71-73.
2. On the period 1800-1914 see my Living together and living apart: on the interactions between mathematics and logic from the French Revolution to the First World War. South African Journal of Philosophy, 7 (1988), no. 2, pp. 73-62.
3. The appointing of an evidently able young man who however had produced nothing comprises an interesting commentary on the status of the new university. His position as Christian unattached would have helped, for Cambridge (and Oxford) were still very closely associated with the Church of England.
4. However, Jevons focused upon the ideas of De Morgans contemporary Boole rather than those of his former teacher (which are summarised in the next section). Boole and De Morgan corresponded at length on logic (and also a little on mathematical education), though with little commensurability; see G. Smith (ed.), The Boole-De Morgan correspondence 1842-1864 (Oxford: Clarendon Press, 1982), which contains a chronological list of De Morgans papers and books.
5. A. De Morgan, On teaching arithmetic, Quarterly Journal of Education, (1833), pp. 1-16 (p. 2). All contributions to this journal were anonymous; his pieces are identified, along with his other writings, in his widows biography, S. E. De Morgan, Memoir of Augustus De Morgan (London: Longmans, Green, 1882), pp. 401-402.
6. De Morgan himself wrote upon this piece of history in On Leonardo de Vincis use of + and &emdash;", Philosophical Magazine, (3) 20 (1842), pp. 135-13 and On the early history of + and 2, (4) 30 (1865), p. 376.
7. See H. Pyclor, Augustus De Morgans algebraic work: the three stages, Isis, 74 (1983), pp. 211-226.
8. On the connections between De Morgans logic and algebras, especially functional equations and the logic of relations, see M. Panteki Relationships between algebra, differential equations and logic in England: 1800-1860, C.N.A.A. (London) doctoral dissertation, 1992, chs. 3 and 6.
9. The details of publication of the book are given on p. iv of the covering Preface.
10. In his Advertisement of 1836 De Morgan claimed priority for writing a textbook solely based upon limits. This seems to be an incredible claim by an author who had written recently on the Ecole Polytechnique; perhaps was restricting his remark to English-language works, when it would be justified. His failure to mention Whewell, however, is surprising.