Andrew English

Finding John Smith: A True Story is autobiographical. The Halas and Batchelor animator John Bernard Smith (1929-1967) was, I discovered around the year 2000, my real father. It is my current book project. Part I: John Smith. 1. Death by misadventure. 2. Fraser drama unfolds. 3. Accident or incident? 4. Last letters. 5. Letters to Aunt Peg. 6. Sorrow and consolation. Part II: The Long Search. 7. New beginnings. 8. Melancholia. 9. Reanimation. 10. Hell’s Gate. 11. Finding John Smith, 12. The Smiths. Afterword: Two Eclipses. As yet unfinished. Here I include the first chapter only. The structure is inspired by the travel book Alexandria: A History and a Guide by E. M. Forster (1879-1970), rendering the invisible visible.

Monsters of Truth, abandoned around 2019. The writing that sustained my philosophical thinking during most of my years teaching mathematics at Abingdon School. 1. Through the Looking Glass. 2. The Meaning of Probability. 3. Mathematical Recreations. 4. The Nature of Mathematics. 5. The Foundations of Mathematics. 6. Comparing Mathematics to a Game. 7. On Formally Undecidable Propositions. 8. Space, Time and Deity. 9. Our Knowledge of the External World. 10. The Mind and its Place in Nature. 11. Computing Machinery and Intelligence. 12. The Mysterious Universe. The original idea was for a book on philosophy as paradox, comparing philosophical paradoxes to paradoxes outside philosophy. And taking from Baltasar Gracián’s “Monsters of Truth” the idea that philosophical monsters, in a later sense, have an origin comparable to artistic monsters: centaurs, minotaurs, dragons, gryphons, and so on. But the Cambridge background to Wittgenstein’s thinking began to dominate and another idea entered, that of a “distant mirror”, borrowed from Freeman Dyson (1923-2020), in which we can see ourselves reflected, but only if we understand what Wittgenstein actually said. Hence Journeyings in Wittgenstein’s Cambridge became an alternative title. “Mirror reversal” is ultimately compared to and contrasted with the paradox of Achilles and the tortoise in its many historical manifestations. (The latter is itself a mirror, in which we see our own thinking reflected.) Included here is the final chapter “The Mysterious Universe”, research which would otherwise have been lost.

“Sitting on the foundations of Wittgenstein’s hut in Skjolden, Norway, in the late summer of 1993, looking out over the lake below, I remember thinking, ‘Now I understand what Wittgenstein meant in his letter to Moore by “quiet seriousness”’. Years later, standing by the hostel which had once been Drury’s brother’s cottage on Killary Harbour, Ireland, I could wonder ...” The book’s twelve chapters include: “Wittgenstein: The Swansea Years, 1942-1947” by Wittgenstein’s biographer Ray Monk; “The Place of Swansea on the Road to Philosophical Investigations” by Jonathan Smith, former archivist of Wittgenstein’s papers at the Wren Library, Trinity College, Cambridge; “‘It’s good to be away from Cambridge & to be here, & among friendly people’ – Wittgenstein’s letters to Ben Richards and his philosophical work in Swansea” by Alfred Schmidt, an historian and archivist at the Austrian National Library, Vienna; “The ‘Swansea School’” by Mario von der Ruhr, Honorary Senior Lecturer at Swansea; “J. R. Jones and Wittgenstein” by Huw Williams, Reader in Philosophy at Cardiff; and “‘Change and decay in all around I see’: The Wittgenstein School of English at Swansea” by M. Wynn Thomas, Professor of Welsh Writing in English at the University. The key figure is the American philosopher of remote Welsh ancestry Rush Rhees (1905-1989).

G. H. Hardy’s remarkable Indian protégé Srinivasa Ramanujan (1887-1920) is without doubt the most significant mathematician who is known to have solved one of Dudeney’s puzzles. When a student friend at Cambridge read out an arithmetical problem from Dudeney’s “Perplexities” column in the latest issue of The Strand Magazine, specifically the Grand Christmas Double Number of December 1914, Ramanujan solved it straightaway and in a generalised form, without recourse to pencil and paper. The story of this astonishing display of mathematical skill is told in some detail and its supposed implications discussed.

Henry E. Dudeney (1857-1930) was, according to pure mathematician G. H. Hardy (1877-1947), one of England’s “better makers of puzzles”. His “Perplexities” column, a regular feature in The Strand Magazine between 1910 and 1930, not only gave recreational mathematicians what they wanted – an “intellectual ‘kick’” – but also occasioned the odd sparkling display of intellect from professionals. Dudeney’s geometrical dissection puzzles are a case in point, though his ingenious solutions sometimes betray a lack of clarity about the difference between a mathematical proof and an empirical procedure. His equilateral triangle to square dissection and the equally stunning regular octagon to square dissection sent to him by Hardy’s colleague G. T. Bennett (1868-1943) are both here demonstrated visually – process and result an evident unity.

The mathematical prowess of pure mathematician J. E. Littlewood (1885-1977), and of his elder cousin the mathematical educator Philippa Fawcett (1868-1948), is illustrated in the context of the Mathematical Tripos examination at Cambridge. Littlewood’s brilliant though highly condensed treatment in his splendid Miscellany (1953) of a perplexing problem from an old Tripos paper – familiar to some as “The Paradox of the Second Ace” – is then expanded with reference to Coxeter’s treatment of it in his revision of Rouse Ball’s Mathematical Recreations (1939) and ultimately explained in the light of Wittgenstein’s later philosophy. The discussion touches on the nature of mathematical supposition.

Wittgenstein said in the Investigations, ‘A philosophical problem has the form: “I don't know my way about”’ (§ 123). The problem of mirror reversal—specifically the twentieth-century transatlantic controversy between the psychologist Richard Gregory, the mathematical columnist Martin Gardner, the physicist Richard Feynman and various analytic philosophers, including David Pears, Ned Block and Don Locke—is presented here as an instructive case of our not knowing our way about. ‘Why do mirrors reverse left and right but not up and down?’ We discover that our perplexity disappears as we get clearer about the employment of the familiar oppositions left and right, up and down, back and front, and others. Finding our way about in these logico-grammatical fells, we see connections with our troubles in more vertiginous landscapes. Our disquietude may not be deep (§ 111) but the ‘battle against the bewitchment of our intelligence’ (§ 109) is surprisingly severe.

Sand drawings are introduced in relation to the fieldwork of British anthropologists John Layard and Bernard Deacon early in the twentieth century, and the status of sand drawings as mathematics is discussed in the light of Wittgenstein’s idea that “in mathematics process and result are equivalent”. Included are photographs of the illustrations in Layard’s own copy of Deacon’s “Geometrical Drawings from Malekula and other Islands of the New Hebrides” (1934). This is a brief companion to my article “Wittgenstein on string figures as mathematics” (2022), published in the journal Philosophical Investigations.

Wittgenstein's farcical clash with literary critic F. R. Leavis over the analysis of Empson's poem "Legal Fiction" is well known to devotees of Wittgenstein's life (Ludwig Wittgenstein: Personal Recollections (1981), edited by Rush Rhees, Oxford: Basil Blackwell, 80). Less well known is the value of studying Empson's artistic and intellectual achievement as part of the wider cultural background for the appreciation of Wittgenstein's views and influence, early and late. This talk sketches some diverting byways awaiting further exploration. A recurrent theme is contradiction.

In Part I, an attempt is made to survey the original source material on which any detailed assessment of Wittgenstein's remarks on the foundations of mathematics from his middle and later periods ought to be based. This survey is presented within the context of a sketch of Wittgenstein's biography, which also mentions some of the major developments in his thinking. In addition, certain main themes are emphasized; these have to do primarily with the Kantian aspects of Wittgenstein's thought and with his mysticism or the 'religious point of view'. In Part II, Kreisel's critique of Wittgenstein's remarks on the foundations of mathematics, which has been developed since 1958 in a series of published articles, receives close examination, and, in connection with this, different approaches to the philosophical investigation of mathematics are considered which represent genuine alternatives to Wittgenstein's approach. There are separate sections on Lakatos's Proofs and Refutations and Bourbaki's 'L'Architecture des Mathématiques'. Finally, besides a bibliography which surveys the reception of Wittgenstein's views on the foundations of mathematics, there are two substantial appendices, which are supplemental to Part I. The first of these gives the manuscript sources for typescripts 221 and 222-4, and the correspondences in both directions between these typescripts. The second appendix is part of a chronological version of von Wright's catalogue of Wittgenstein's papers, beginning in 1929.

Wittgenstein’s ‘ethnological approach’ to the philosophy of mathematics, in particular his discussion of calculation as an experiment and the limits of empiricism in mathematics, is presented against three interrelated backdrops: (1) James’ critique of Spencer’s evolutionary empiricism, specifically regarding necessary truths; (2) the Cambridge Anthropological Expedition to Torres Straits, led by Haddon and Rivers, whose Reports implicitly confuted Spencer; and (3) the subsequent work of Malinowski, especially his supplement to Ogden and Richards’ The Meaning of Meaning, a book sent to Wittgenstein upon its publication in 1923. String figures as mathematics is a main illustrative example.