A lover of paradoxes

As I have probably mentioned more than once I served my apprenticeship as a historian of science working in a research project on the history of formal or symbolic logic. My special area within the project was British logical algebra in the 19th century and it was here that I took a long deep look at Augustus De Morgan who was born in Madurai in the Madras Presidency, an administrative sub-division of British India on the 27th June 1806. De Morgan was a brilliantly eclectic polymath with a Pythonesque sense of humour who both from his personality and from his appearance seemed to spring out of Charles Dickens’ Pickwick Papers, a mathematical second cousin to Sam Weller. De Morgan is my favourite Victorian.


Augustus De Morgan Source: Wikimedia Commons

Son of an army officer in the service of the East India Company he moved to England whilst only seven months old. At the age of sixteen he went up to Trinity College Cambridge where he quickly became part of the circle around George Peacock and William Whewell who would stimulate his life long interest in mathematics and logic. In 1826 he graduated 4th Wrangler in the mathematical tripos but already a convinced Unitarian he refused to sign the religious declaration required in Oxbridge in those days to graduate MA and so was not eligible for the fellowship for which he would normally have been destined. He went instead to London to study for the bar. However he found law boring and at the age of 21 and with no publication to his name he applied for the chair of mathematics at the newly founded University College London. This new university had been founded by a group of social reformers who felt that a university education should be open to all what ever their religious belief might be, Oxbridge only being open to confirmed Anglicans. Despite his youth and lack of experience De Morgan was appointed University College’s first professor of mathematics in 1828. He resigned the post only three years later on a mater of principle but was reappointed in 1836 and remained professor until 1866 when he again resigned on another mater of principle.

That De Morgan should be identified with an institution of social reform was not a mater of chance and social reform defined much of his life. He became professor of mathematics at the newly founded Queen’s College an institute of higher education for women founded by Frederick Denison Maurice. Most notably he was a highly active member of the Society for the Diffusion of Useful Knowledge an organisation dedicated to making scientific and other knowledge available in cheap, clear and concise printed versions written by the best authors. De Morgan was the most prolific of all the SDUK authors and wrote and published books and articles on a bewildering range of topics. Another of his social reformers contacts was the Unitarian William Frend whose daughter Sophia would become De Morgan’s wife.

De Morgan devoted part of his academic efforts to the reform and modernisation of formal logic, a subject that had been in a sort of coma in England for about three hundred years before being awakened from its slumbers by Richard Whately at the beginning of the 19th century. De Morgan who worked in the traditional syllogistic Aristotelian logic introduced the concept of quantification of the predicate enabling logically conclusion not possible in the traditional logic. This invention led to a bitter dispute with the Scottish philosopher Sir William Hamilton (not to be confused with the Irish mathematician Sir William Hamilton, a good friend of De Morgan’s) who claimed priority for this logical discovery. This dispute attracted the attention of another mathematician, George Boole, who stimulated by the discussion developed his algebraic logic. Boole and De Morgan were not only both disciples of the algebraic innovation of George Peacock and logical pioneers but shared a Unitarian religious outlook and became lifelong friends. De Morgan was especially proud of the fact that his Formal Logic and Boole’s Mathematical Analysis of Logic were published on the same day in 1847. Introducing, in his opinion, a new age in logic. In reality De Morgan was incorrect as the two books were published about a week apart. Although De Morgan’s logical work was by no means as innovative as Boole’s he was the first modern logician to work on the logic of relation an area that was later developed by Charles Sanders Peirce in America and Ernst Schröder in Germany both of whom were great admirers of De Morgan.

De Morgan made significant contributions to many areas of mathematics but his principle achievements were in trigonometry and in abstract algebras. His most lasting contribution was the formalisation of the principle of mathematical induction, an important tool in mathematical proof theory, to which he also contributed the name. Strangely he is best remembered today for De Morgan’s Laws. This is peculiar because the laws were not discovered by De Morgan but had been known both to Aristotle and the mediaeval logicians; De Morgan merely made them better known. The laws are fairly trivial “not (A or B) is equal to not A and not B” and “not (A and B) is equal to not A or not B” but very useful in deductive logical proofs.

De Morgan also made important contributions to the history of science. The Scottish physicist David Brewster wrote and published the first modern English biography of Isaac Newton largely as a reaction to the English translation of the biography by the French physicist Jean – Baptiste Biot, which had been published by the SDUK. De Morgan didn’t like what he saw as Brewster’s Newton hagiography and wrote and published a series of biographical pamphlets on Newton, correcting what he saw as Brewster’s errors. This led to a literary dispute between the two men with both of them digging deeper and deeper into the original sources, Newton’s letters, papers, notebooks etc., in order to prove the correctness of their Newton picture. This development led scientific biography away from literary hagiography towards modern historiography. For the full details of these developments I recommend the very readable account by Rebekah Higgitt in her excellent Recreating Newton.

De Morgan also wrote and published his Arithmetical Books in which he discussed the work of over 1500 authors on the subject. This book is still regarded as an important source in the history of mathematics.

I said that De Morgan had a Pythonesque sense of humour and his letters, papers and notebooks are full of wonderful whimsies. His most famous book is his Budget of Paradoxes.


De Morgan collected the written products of circle squarers and other mathematical fools who he then exposed to ridicule in a series of newspaper articles. These were collected in a book and published after his death. This gem is still in print and is a secret tip amongst philosophers, mathematicians and logicians.

Unlike many of his friends and contemporaries De Morgan was not very active in the numerous scientific societies that flourished in the 19th century. He refused membership of the Royal Society on grounds of principle because he saw it as an elitist organisation. The only society of which he was a member was the Astronomical Society. However when his son George, like his father a gifted mathematician, founded the London Mathematical Society De Morgan became its first President.

De Morgan was a fascinating and stimulating polymath who certainly deserves to be better known than he is. One way you can do that is by getting hold of a copy of the very readable Memoir of Augustus De Morgan by his wife Sophia Elizabeth De Morgan.



Filed under History of Logic, History of Mathematics, History of science

10 responses to “A lover of paradoxes

  1. Pingback: A lover of paradoxes | Whewell's Ghost

  2. Rebekah Higgitt

    One of the things you didn’t mention was De Morgan’s fondness for doodling and sketching humorous pictures – they, as well as the whimsies you mention, were the great pleasure of working through his correspondence and papers. Mathematical symbols capering, Newton and Flamsteed boxing and a self-portrait that’s all overcoat, curly hair and glasses leap to mind. Senate House Library in London has De Morgan’s own library and it’s worth investigating not just for the rare Renaissance mathematical books that in the 19th century could be picked up for a song but also for De Morganiana. His copies of the journal Notes & Queries are (or were) available on the open shelves, stuffed with comments and cuttings, including lots of Punch cartoons.

    Many of DM’s sketches are pretty crude, but I think he had a certain facility, perhaps passed to his son, William, who trained at the Royal Academy Schools before making his name in ceramics and the Arts and Crafts movement. He also played the flute – clearly no end to the man’s talents!

    In terms of DM as a ‘joiner’, I would paint him as more of a society man than you. Although he eschewed the Royal Society he was active within the Astronomical, and was close friends with many of the chief figures there. With the SDUK (till the 1840s) and the Mathematical Society, within which he was active, I see him as more active than most.

    • I was comparing him with friends like Herschel or Whewell who were in the Royal Society, BAAS, Geological Society, Astronomical Society, Cambridge Philosophical Society etc etc…

  3. Great post as always! But speaking of paradoxes, obviously I would have loved to see a reference to De Morgan’s interest in spiritualism (see for example his preface to his wife’s book ‘From Matter to Spirit’).

  4. Pingback: Whewell’s Gazette: Yar 2, Vol. #47 | Whewell's Ghost

  5. Thank you so much for this post. I was fascinated by De Morgan and Hamilton when I entered graduate school the fall of 1987 and might have worked on them had my professor shown a little more guidance in his history of mathematics. He made the readings up, seemingly at random, from week to week and eventually put an end to the farce about halfway through the term. Considering that this professor had moved his research interests from the history of mathematics to the history of computing, I came very close to replicating your career. But it was not to be. I switched topics and advisors the following spring after Kathy Olesko lectured on the 1847 Group of Berlin biophysicists. This is how I ended up an historian of Germany rather than one of France or Britain.

  6. Pingback: Whewell’s Gazette: Year 3, Vol. #46 | Whewell's Ghost

  7. Ash Jogalekar

    As always, excellent post. A minor typo – “Madurai” instead of “Maduria”.

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