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’
Author Archive for peter
Page 64 of 83
Two kinds of people
K. Kram in Glee and Disaffection (translated by Mark Kaplan):
When I was an adolescent it struck me, rather narcissistically, that there were two kinds of people, politically speaking. On the one hand, there were those who had realised, at first dimly and intuitively, that there was something profoundly wrong with the social and political order in which they lived. It was wasteful, unjust, amoral and much more besides. Its language seemed formulaic and false, a screen of clichés and convenient fictions. Following up these dim intuitions, turning them into genuine understanding, would be no easy task. One had been thrown into this world, grown up with its assumptions and habits of thought, and these had (to use a phrase I would learn later) deposited a kind of inventory, and this inventory had to be painstakingly scrutinised and thought through. This thinking through would involve dragging into visibility and naming the whole social order. It would be a long game. One would have to relearn how to think and speak. But only fidelity to this project was worthwhile. And this type of person pledged that they would never succumb to the easy rewards of this social order, they would do everything they could to maintain their critical distance. Otherwise, they could not live with themselves. From this social order which they had not chosen they would at least win for themselves insight into its workings, and would attempt to prepare and imagine alternatives.And the other type? These consisted of those scandalised by the very presence of the first type. For these people, the mere fact that a form of life existed seemed to be sufficient proof that it should. And for them, the first type of person could only be motivated by resentment or fashion.”
Poem: The Workman's Friend
One of my favourite poems, by Irish comic novelist and journalist Flann O’Brien (aka Brian O’Nolan aka Myles na gCopaleen) (1911-1966):
The Workman’s Friend
When things go wrong and will not come right,
Though you do the best you can,
When life looks black as the hour of night –
A pint of plain is your only man.
When money’s tight and hard to get
And your horse has also ran,
When all you have is a heap of debt –
A pint of plain is your only man.
When health is bad and your heart feels strange,
And your face is pale and wan,
When doctors say you need a change,
A pint of plain is your only man.
When food is scarce and your larder bare
And no rashers grease your pan,
When hunger grows as your meals are rare –
A pint of plain is your only man.
In time of trouble and lousey strife,
You have still got a darlint plan
You still can turn to a brighter life –
A pint of plain is your only man.
Evil intentions
A commentator on Andrew Sullivan’s blog asks: Where is the Darwinian theory of evil? Because modern biologists this last century or so have been very concerned to avoid teleological arguments, modern biology has still only an impoverished theory of intentionality. Living organisms are focused, in the standard evolutionary account, on surviving themselves in the here-and-now, apparently going through these daily motions unwittingly to ensure those diaphonous creatures, genes, can achieve THEIR memetic goals. Without a rich and subtle theory of intentionality, I don’t believe one can explain complex, abstract human phenomena such as evil or altruism or art or religion very compellingly.
Asking for a theory of intentions and intentionality does not a creationist one make, despite the vitriol often deployed by supporters of evolution. One non-creationist evolutionary biologist who has long been a critic of this absence of a subtle theory of intentionality in biology is J. Scott Turner, whose theories are derived from homeostasis he has observed in natural ecologies. I previously discussed some of his ideas here.
References:
Alfred Gell [1998]: Art and Agency: An Anthropological Theory. Oxford, UK: Clarendon Press.
J. Scott Turner [2007]: The Tinkerer’s Accomplice: How Design Emerges from Life Itself. Cambridge, MA, USA: Harvard University Press.
Vale William Safire
William Safire, a speech-writer for Richard Nixon and later an op-ed columnist with The New York Times, has just died. To his memory, I retrieve a statement from his novel Full Disclosure, which nicely expresses a different model of decision-making to that taught in Decision Theory classes:
The truth about big decisions, Ericson mused, was that they never marched through logical processes, staff systems, option papers, and yellow pads to a conclusion. No dramatic bottom lines, no Thurberian captains with their voices like thin ice breaking, announcing, “We’re going through!” The big ones were a matter of mental sets, predispositions, tendencies – taking a lifetime to determine – followed by the battering of circumstance, the search for a feeling of what was right – never concluded at some finite moment of conclusion, but in the recollection of having “known” what the decision would be some indeterminate time before. For weeks now, Ericson knew he had known he was ready to do what he had to do, if only Andy or somebody could be induced to come up with a solution that the President could then put through his Decision-Making Process. That made his decision a willingness not to obstruct, rather than a decision to go ahead, much like Truman’s unwillingness to stop the train of events that led to the dropping of the A-bomb – not on the same level of magnitude, but the same type of reluctant going-along.” (pp. 491-492)
Reference:
William Safire [1977]: Full Disclosure. (Garden City, NY, USA: Doubleday and Company).
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’
Poem: Tu ne quaesieris
Ode I:XI of Horace, Tu ne quaesieris (translated by David West), ending with the advice, carpe diem.
Don’t you ask, Leuconoe – the gods do not wish it to be known –
what end they have given me or to you, and don’t meddle with
Babylonian horoscopes. How much better to accept whatever comes,
whether Jupiter gives us other winters or whether this is our last
now wearying out the Tyrrhenian sea on the pumice stones
opposing it. Be wise, strain the wine and cut back long hope
into a small space. Even as we speak, envious time
flies past. Harvest the day and leave as little as possible for tomorrow.
Reference:
Horace [1997 AD/23 BCE]: The Complete Odes and Epodes. Translation by David West. Oxford, UK: Oxford University Press.
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’
Neighbours
I had mentioned previously the unusually close political relationship between geographic neighbours Australia and New Zealand. But would you let your neighbours use your bathroom when their’s was broken? I guess you would if they were getting dressed to meet the US President for lunch:
Speaking at a lunch in New York, [Australian Prime Minister] Kevin Rudd revealed that he had woken on Wednesday morning, New York time, to find [New Zealand Prime Minister John] Key and his Foreign Affairs Minster, Murray McCully, lining up in their dressing gowns to use his bathroom at the residence of the Australian ambassador to the United Nations. It seems that, in a very Brian-like moment, the plumbing in the Kiwis’ hotel next door had failed.”
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.