From the 18th century until 1909, students at Cambridge University took a compulsory series of examinations, called the Mathematical Tripos, named after the three-legged stool that candidates originally sat on. Until the mid-18th century, these examinations were conducted orally, and only became written examinations over faculty protests. Apparently, not everyone believed that written examinations were the best or fairest way to test mathematical abilities, a view which would amaze many contemporary people – although oral examinations in mathematics are still commonly used in some countries with very strong mathematical traditions, such as Russia and the other states of the former USSR.
The Tripos became a notable annual public event in the 19th century, with The Times newspaper publishing articles and biographies before each examination on the leading candidates, and then, after each examination, the results. There was considerable public interest in the event each year, not just in Cambridge or among mathematicians, and widespread betting on the outcomes.
Continue reading ‘The Mathematical Tripos at Cambridge’
Archive for the ‘Logic’ Category
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Australian logic: a salute to Malcolm Rennie
Recently, I posted a salute to Mervyn Pragnell, a logician who was present in the early days of computer science. I was reminded of the late Malcolm Rennie, the person who introduced me to formal logic, and whom I acknowledged here. Rennie was the most enthusiastic and inspiring lecturer I ever had, despite using no multi-media wizardry, usually not even an overhead projector. Indeed, he mostly just sat and spoke, moving his body as little as possible and writing only sparingly on the blackboard, because he was in constant pain from chronic arthritis. He was responsible for part of an Introduction to Formal Logic course I took in my first year (the other part was taken by Paul Thom, for whom I wrote an essay on the notion of entailment in a system of Peter Geach). The students in this course were a mix of first-year honours pure mathematicians and later-year philosophers (the vast majority), and most of the philosophers struggled with non-linguistic representations (ie, mathematical symbols). Despite the diversity, Rennie managed to teach to all of us, providing challenging questions and discussions with and for both groups. He was also a regular entrant in the competitions which used to run in the weekly Nation Review (and a fellow-admirer of the My Sunday cartoons of Victoria Roberts), and I recall one occasion when a student mentioned seeing his name as a competition winner, and the class was then diverted into an enjoyable discussion of tactics for these competitions.
Continue reading ‘Australian logic: a salute to Malcolm Rennie’
Speech acts
Thanks to Normblog, I have seen Terry Eagleton’s recent interview on matters of religion, in which he is reported as saying:
All performatives imply propositions. There’s no point in my operating a performative like, say, promising, or cursing, unless I have certain beliefs about the nature of reality: that there is indeed such an institution as promising, that I am able to perform it, and so on. The performative and the propositional work into each other.
Before commenting on the substance here (ie, religion), some words on Eagleton’s evident mis-understanding of speech act theory and the philosophy of language, a mis-understanding that should have been clear if he tested his words against his own experiences of life. His statement concerns performatives — utterances which potentially change the state of the world by their being uttered. Examples include promises, commands, threats, entreaties, prayers, various legal declarations (eg, that a certain couple are now wed), etc. But mere propositional statements (that some description of the world is true) may also change the state of the world by the mere fact of being uttered.
Continue reading ‘Speech acts’
Great mathematical ideas
Normblog has a regular feature, Writer’s Choice, where writers give their opinions of books which have influenced them. Seeing this led me recently to think of the mathematical ideas which have influenced my own thinking. In an earlier post, I wrote about the writers whose books (and teachers whose lectures) directly influenced me. I left many pure mathematicians and statisticians off that list because most mathematics and statistics I did not receive directly from their books, but indirectly, mediated through the textbooks and lectures of others. It is time to make amends.
Here then is a list of mathematical ideas which have had great influence on my thinking, along with their progenitors. Not all of these ideas have yet proved useful in any practical sense, either to me or to the world – but there is still lots of time. Some of these theories are very beautiful, and it is their elegance and beauty and profundity to which I respond. Others are counter-intuitive and thus thought-provoking, and I recall them for this reason.
- Euclid’s axiomatic treatment of (Euclidean) geometry
- The various laws of large numbers, first proven by Jacob Bernoulli (which give a rational justification for reasoning from samples to populations)
- The differential calculus of Isaac Newton and Gottfried Leibniz (the first formal treatment of change)
- The Identity of Leonhard Euler: exp ( i * \pi) + 1 = 0, which mysteriously links two transcendental numbers (\pi and e), an imaginary number i (the square root of minus one) with the identity of the addition operation (zero) and the identity of the multiplication operation (1).
- The epsilon-delta arguments for the calculus of Augustin Louis Cauchy and Karl Weierstrauss
- The non-Euclidean geometries of Janos Bolyai, Nikolai Lobachevsky and Bernhard Riemann (which showed that 2-dimensional (or plane) geometry would be different if the surface it was done on was curved rather than flat – the arrival of post-modernism in mathematics)
- The diagonalization proofof Gregor Cantor that the Real numbers are not countable (showing that there is more than one type of infinity) (a proof-method later adopted by Godel, mentioned below)
- The axioms for the natural numbers of Guiseppe Peano
- The space-filling curves of Guiseppe Peano and others (mapping the unit interval continuously to the unit square)
- The axiomatic treatments of geometry of Mario Pieri and David Hilbert (releasing pure mathematics from any necessary connection to the real-world)
- The algebraic topology of Henri Poincare and many others (associating algebraic structures to topological spaces)
- The paradox of set theory of Bertrand Russell (asking whether the set of all sets contains itself)
- The Fixed Point Theorem of Jan Brouwer (which, inter alia, has been used to prove that certain purely-artificial mathematical constructs called economies under some conditions contain equilibria)
- The theory of measure and integration of Henri Lebesgue
- The constructivism of Jan Brouwer (which taught us to think differently about mathematical knowledge)
- The statistical decision theory of Jerzy Neyman and Egon Pearson (which enabled us to bound the potential errors of statistical inference)
- The axioms for probability theory of Andrey Kolmogorov (which formalized one common method for representing uncertainty)
- The BHK axioms for intuitionistic logic, associated to the names of Jan Brouwer, Arend Heyting and Andrey Kolmogorov (which enabled the formal treatment of intuitionism)
- The incompleteness theorems of Kurt Godel (which identified some limits to mathematical knowledge)
- The theory of categories of Sam Eilenberg and Saunders Mac Lane (using pure mathematics to model what pure mathematicians do, and enabling concise, abstract and elegant presentations of mathematical knowledge)
- Possible-worlds semantics for modal logics (due to many people, but often named for Saul Kripke)
- The topos theory of Alexander Grothendieck (generalizing the category of sets)
- The proof by Paul Cohen of the logical independence of the Axiom of Choice from the Zermelo-Fraenkel axioms of Set Theory (which establishes Choice as one truly weird axiom!)
- The non-standard analysis of Abraham Robinson and the synthetic geometry of Anders Kock (which formalize infinitesimal arithmetic)
- The non-probabilistic representations of uncertainty of Arthur Dempster, Glenn Shafer and others (which provide formal representations of uncertainty without the weaknesses of probability theory)
- The information geometry of Shunichi Amari, Ole Barndorff-Nielsen, Nikolai Chentsov, Bradley Efron, and others (showing that the methods of statistical inference are not just ad hoc procedures)
- The robust statistical methods of Peter Huber and others
- The proof by Andrew Wiles of The Theorem Formerly Known as Fermat’s Last (which proof I don’t yet follow).
Some of these ideas are among the most sublime and beautiful thoughts of humankind. Not having an education which has equipped one to appreciate these ideas would be like being tone-deaf.
Myopic utilitarianism
What are the odds, eh? On the same day that the Guardian publishes an obituary of theoretical computer scientist, Peter Landin (1930-2009), pioneer of the use of Alonzo Church’s lambda calculus as a formal semantics for computer programs, they also report that the Government is planning only to fund research which has relevance to the real-world. This is GREAT NEWS for philosophers and pure mathematicians!
What might have seemed, for example, mere pointless musings on the correct way to undertake reasoning – by Aristotle, by Islamic and Roman Catholic medieval theologians, by numerous English, Irish and American abstract mathematicians in the 19th century, by an entire generation of Polish logicians before World War II, and by those real-world men-of-action Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein and Alonzo Church – turned out to be EXTREMELY USEFUL for the design and engineering of electronic computers. Despite Russell’s Zen-influenced personal motto – “Just do! Don’t think!” (later adopted by IBM) – his work turned out to be useful after all. I can see the British research funding agencies right now, using their sophisticated and proven prognostication procedures to calculate the society-wide economic and social benefits we should expect to see from our current research efforts over the next 2300 years – ie, the length of time that Aristotle’s research on logic took to be implemented in technology. Thank goodness our politicians have shown no myopic utilitarianism this last couple of centuries, eh what?!
All while this man apparently received no direct state or commercial research funding for his efforts as a computer pioneer, playing with “pointless” abstractions like the lambda calculus.
And Normblog also comments.
POSTSCRIPT (2014-02-16): And along comes The Cloud and ruins everything! Because the lower layers of the Cloud – the physical infrastructure, operating system, even low-level application software – are fungible and dynamically so, then the Cloud is effectively “dark” to its users, beneath some level. Specifying and designing applications that will run over it, or systems that will access it, thus requires specification and design to be undertaken at high levels of abstraction. If all you can say about your new system is that in 10 years time it will grab some data from the NYSE, and nothing (yet) about the format of that data, then you need to speak in abstract generalities, not in specifics. It turns out the lambda calculus is just right for this task and so London’s big banks have been recruiting logicians and formal methods people to spec & design their next-gen systems. You can blame those action men, Church and Russell.
Science and poetry
The Asian scholar Arthur Waley once wrote:
All argument consists in proceeding from the known to the unknown, in persuading people that the new thing you want them to think is not essentially different from or at any rate is not inconsistent with the old things they think already. This is the method of science, just as much as it is the method of rhetoric and poetry. But, as between science and forms of appeal such as poetry, there is a great difference in the nature of the link that joins the new to the old. Science shows that the new follows from the old according to the same principles that built up the old. “If you don’t accept what I now ask you to believe,” the scientist says, “you have no right to go on believing what you believe already.” The link used by science is a logical one. Poetry and rhetoric are also concerned with bridging the gap between the new and the old; but they do not need to build a formal bridge. What they fling across the intervening space is a mere filament such as no sober foot would dare to tread. But it is not with the sober that poetry and eloquence have to deal. Their te, their essential power, consists in so intoxicating us that, endowed with the recklessness of drunken men, we dance across the chasm, hardly aware how we reached the other side.” (Waley 1934, Introduction, pp. 96-97)
Reference:
Arthur Waley [1934]: The Way and its Power: A Study of the Tao Te Ching and its Place in Chinese Thought. London, UK: George Allen and Unwin.
Guerrilla logic: a salute to Mervyn Pragnell
When a detailed history of computer science in Britain comes to be written, one name that should not be forgotten is Mervyn O. Pragnell. As far as I am aware, Mervyn Pragnell never held any academic post and he published no research papers. However, he introduced several of the key players in British computer science to one another, and as importantly, to the lambda calculus of Alonzo Church (Hodges 2001). At a time (the 1950s and 1960s) when logic was not held in much favour in either philosophy or pure mathematics, and before it became to be regarded highly in computer science, he studied the discipline not as a salaried academic in a university, but in a private reading-circle of his own creation, almost as a guerrilla activity.
Pragnell recruited people for his logic reading-circle by haunting London bookshops, approaching people he saw buying logic texts (Bornat 2009). Among those he recruited to the circle were later-famous computer pioneers such as Rod Burstall, Peter Landin (1930-2009) and Christopher Strachey (1916-1975). The meetings were held after hours, usually in Birkbeck College, University of London, without the knowledge or permission of the college authorities (Burstall 2000). Some were held or continued in the neighbouring pub, The Duke of Marlborough. It seems that Pragnell was employed for a time in the 1960s as a private research assistant for Strachey, working from Strachey’s house (Burstall 2000). By the 1980s, he was apparently a regular attendee at the seminars on logic programming held at the Department of Computing in Imperial College, London, then (and still) one of the great research centres for the application of formal logic in computer science.
Pragnell’s key role in early theoretical computer science is sadly under-recognized. Donald MacKenzie’s fascinating history and sociology of automated theorem proving, for example, mentions Pragnell in the text (MacKenzie 2001, p. 273), but manages to omit his name from the index. Other than this, the only references I can find to his contributions are in the obituaries and personal recollections of other people. I welcome any other information anyone can provide.
UPDATE (2009-09-23): Today’s issue of The Guardian newspaper has an obituary for theoretical computer scientist Peter Landin (1930-2009), which mentions Mervyn Pragnell.
UPDATE (2012-01-30): MOP appears also to have been part of a production of the play The Way Out at The Little Theatre, Bristol in 1945-46, according to this web-chive of theatrical info.
UPDATE (2013-02-11): In this 2001 lecture by Peter Landin at the Science Museum, Landin mentions first meeting Mervyn Pragnell in a cafe in Sheffield, and then talks about his participation in Pragnell’s London reading group (from about minute 21:50).
UPDATE (2019-07-05): I have learnt some further information from a cousin of Mervyn Pragnell, Ms Susan Miles. From her, I understand that MOP’s mother died in the Influenza Pandemic around 1918, when he was very young, and he was subsequently raised in Cardiff in the large family of a cousin of his mother’s, the Miles family. MOP’s father’s family had a specialist paint manufacturing business in Bristol, Oliver Pragnell & Company Limited, which operated from 25-27 Broadmead. This establishment suffered serious bomb damage during WW II. MOP was married to Margaret and although they themselves had no children, they kept in close contact with their relatives. Both are remembered fondly by their family. (I am most grateful to Susan Miles, daughter of Mervyn Miles whose parents raised MOP, for sharing this information.)
References:
Richard Bornat [2009]: Peter Landin: a computer scientist who inspired a generation, 5th June 1930 – 3rd June 2009. Formal Aspects of Computing, 21 (5): 393-395.
Rod Burstall [2000]: Christopher Strachey – understanding programming languages. Higher-Order and Symbolic Computation, 13: 51-55.
Wilfrid Hodges [2001]: A history of British logic. Unpublished slide presentation. Available from his website.
Peter Landin [2002]: Rod Burstall: a personal note. Formal Aspects of Computing, 13: 195.
Donald MacKenzie [2001]: Mechanizing Proof: Computing, Risk, and Trust. Cambridge, MA, USA: MIT Press.
Computer science, love-child: Part 2
This post is a continuation of the story which began here.
Life for the teenager Computer Science was not entirely lonely, since he had several half-brothers, half-nephews, and lots of cousins, although he was the only one still living at home. In fact, his family would have required a William Faulkner or a Patrick White to do it justice.
The oldest of Mathematics’ children was Geometry, who CS did not know well because he did not visit very often. When he did visit, G would always bring a sketchpad and make drawings, while the others talked around him. What the boy had heard was that G had been very successful early in his life, with a high-powered job to do with astronomy at someplace like NASA and with lots of people working for him, and with business trips to Egypt and Greece and China and places. But then he’d had an illness or a nervous breakdown, and thought he was traveling through the fourth dimension. CS had once overheard Maths telling someone that G had an “identity crisis“, and could not see the point of life anymore, and he had become an alcoholic. He didn’t speak much to the rest of the family, except for Algebra, although all of them still seemed very fond of him, perhaps because he was the oldest brother.
Continue reading ‘Computer science, love-child: Part 2’
Thinkers of renown
The recent death of mathematician Jim Wiegold (1934-2009), whom I once knew, has led me to ponder the nature of intellectual influence. Written matter – initially, hand-copied books, then printed books, and now the Web – has been the main conduit of influence. For those of us with a formal education, lectures and tutorials are another means of influence, more direct than written materials. Yet despite these broadcast methods, we still seek out individual contact with others. Speaking for myself, it is almost never the knowledge or facts of others, per se, that I have sought or seek in making personal contact, but rather their various different ways of looking at the world. In mathematical terminology, the ideas that have influenced me have not been the solutions that certain people have for particular problems, but rather the methods and perspectives they use for approaching and tackling problems, even when these methods are not always successful.
To express my gratitude, I thought I would list some of the people whose ideas have influenced me, either directly through their lectures, or indirectly through their books and other writings. In the second category, I have not included those whose ideas have come to me mediated through the books or lectures of others, which therefore excludes many mathematicians whose work has influenced me (in particular: Newton, Leibniz, Cauchy, Weierstrauss, Cantor, Frege, Poincare, Pieri, Hilbert, Lebesque, Kolmogorov, and Godel). I have also not included the many writers of poetry, fiction, history and biography whose work has had great impact on me. These two categories also exclude people whose intellectual influence has been manifest in non-verbal forms, such as through visual arts or music, or via working together, since those categories need posts of their own.
Teachers & lecturers I have had who have influenced my thinking include: Leo Birsen (1902-1992), Sr. Claver Butler RSM (ca. 1930-2009), Burgess Cameron (1922-2020), Sr. Clare Castle RSM (ca. 1920- ca. 2000), John Coates (1945-2022), Dot Crowe, James Cutt, Bro. Clive Davis FMS, Tom Donaldson (1945-2006), Gary Dunbier, Sol Encel (1925-2010), Felix Fabryczny de Leiris, Claudio Forcada, Richard Gill (1941-2018), Myrtle Hanley (1909-1984), Sr. Jennifer Hartley RSM, Chip Heathcote (1931-2016), Hope Hewitt (1915-2011), Alec Hope (1907-2000), John Hutchinson, Marg Keetles, Joe Lynch, Robert Marks, John McBurney (1932-1998), David Midgley, Lindsay Morley, Leopoldo Mugnai, Terry O’Neill, Jim Penberthy* (1917-1999), Malcolm Rennie (1940-1980), John Roberts, Gisela Soares, Brian Stacey (1946-1996), James Taylor, Frank Torpie (1934-1989), Neil Trudinger, David Urquhart-Jones, Frederick Wedd (1890-1972), Gary Whale (1943-2019), Ted Wheelwright (1921-2007), John Woods and Alkiviadis Zalavras.
People whose writings have influenced my thinking include: John Baez, Ole Barndorff-Nielsen (1935-2022), Charlotte Joko Beck (1917-2011), Johan van Bentham, Mark Evan Bonds, John Cage (1912-1992), Albert Camus (1913-1960), Nikolai Chentsov (1930-1992), John Miller Chernoff, Stewart Copeland, Sam Eilenberg (1913-1998), Paul Feyerabend (1924-1994), George Fowler (1929-2000), Kyle Gann, Alfred Gell (1945-1997), Herb Gintis, Jurgen Habermas, Charles Hamblin (1922-1985), Vaclav Havel (1936-2011), Lafcadio Hearn (1850-1904), Jaakko Hintikka (1929-2015), Eric von Hippel, Wilfrid Hodges, Christmas Humphreys (1901-1983), Jon Kabat-Zinn, Herman Kahn (1922-1983), John Maynard Keynes (1883-1946), Andrey Kolmogorov (1903-1987), Paul Krugman, Imre Lakatos (1922-1974), Trevor Leggett (1914-2000), George Leonard (1923-2010), Brad de Long, Donald MacKenzie, Saunders Mac Lane (1909-2005), Karl Marx (1818-1883), Grant McCracken, Henry Mintzberg, Philip Mirowski, Michel de Montaigne (1533-1592), Michael Porter, Charles Reich (1928-2019), Jean-Francois Revel (1924-2006), Daniel Rose, Bertrand Russell (1872-1970), Pierre Ryckmans (aka Simon Leys) (1935-2014), Oliver Sacks (1933-2015), Gunther Schuller (1925-2015), George Shackle (1903-1992), Cosma Shalizi, Rupert Sheldrake, Raymond Smullyan (1919-2017), Rory Stewart, Anne Sweeney (d. 2007), Nassim Taleb, Henry David Thoreau (1817-1862), Stephen Toulmin (1922-2009), Scott Turner, Roy Weintraub, Geoffrey Vickers VC (1894-1982), and Richard Wilson.
FOOTNOTES:
* Which makes me a grand-pupil of Nadia Boulanger (1887-1979).
** Of course, this being the World-Wide-Web, I need to explicitly say that nothing in what I have written here should be taken to mean that I agree with anything in particular which any of the people mentioned here have said or written.
A more complete list of teachers is here.