One wonders, fourthly, where are recorded all the prior distributions of beliefs which this theory requires investigators to articulate before doing research.  Surely someone must be writing them down, so that we consumers of science can know that our researchers are honest, and hold them to potential account.   That there is such a disconnect between what Bayesian theorists say researchers do and what those researchers demonstrably do should trouble anyone contemplating a choice of statistical paradigms, surely.   Finally, one wonders how a theory that requires non-zero probabilities be allocated to models of which the investigators have not yet heard or even which no one has yet articulated, for those models to be tested, passes muster at the statistical methodology corral.

To my mind, Bayesianism is a theory from some other world – infinite gambles, imagined prior distributions, models that disregard time or requirements for constructability,  unrealistic abstractions from actual scientific practice – not from our own.

So, how could the Bayesians make as much headway as they have these last six decades? Perhaps it is due to an inherent pragmatism of statisticians – using whatever techniques work, without much regard as to their underlying philosophy or incoherence therein.  Or perhaps the battle between the two schools of thought has simply been asymmetric:  the Bayesians being more determined to prevail (in my personal experience, to the point of cultism and personal vitriol) than the adherents of frequentism.  Greg Wilson’s 2001 PhD thesis explored this question, although without finding definitive answers.

Now,  Andrew Gelman and the indefatigable Cosma Shalizi have written a superb paper, entitled “Philosophy and the practice of Bayesian statistics”.  Their paper presents another possible reason for the rise of Bayesian methods:  that Bayesianism, when used in actual practice, is most often a form of hypothesis-testing, and thus not as untethered to reality as the pure theory would suggest.  Their abstract:

A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypothetico-deductivism. We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science.

Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework.

References:

Andrew Gelman and Cosma Rohilla Shalizi [2010]:  Philosophy and the practice of Bayesian statistics.  Available from Arxiv.  Blog post here.

Technorati Tags: hypothesis-testing, statistics, Bayesianism, frequentism, inductive inference, hypothetico-deductivism

I have just encountered an undated essay, “The Two Cultures of Mathematics”, by Fields Medallist Timothy Gowers, currently Rouse Ball Professor of Mathematics at Cambridge.    Gowers identifies two broad types of research pure mathematicians:  problem-solvers and theory-builders.  He cites Paul Erdos as an example of the former (as I did in my earlier post), and Michael Atiyah as an example of the latter.   What I find interesting is that Gowers believes the the profession as a whole currently favours theory-builders over problem-solvers.  And domains of mathematics where theory-building is currently more important (such as Geometry and Algebraic Topology) are favoured over domains of mathematics where problem-solving is currently more important (such as Combinatorics and Graph Theory).

I agree with Gowers here, and wonder, then, why the teaching of mathematics at school still predominantly favours problem-solving over theory-building activities, despite a century of Hilbertian and Bourbakian axiomatics.   Is it because problem-solving was the predominant mode of British  mathematics in the 19th century (under the pernicious influence of the Cambridge Mathematics Tripos, which retarded pure mathematics in the Anglophone world for a century) and school educators are slow to catch-on with later trends?  Or, is it because the people designing and implementing school mathematics curricula are people out of sympathy with, and/or not competent at, theory-building?  Certainly, if your over-riding mantra for school education is instrumental relevance than the teaching of abstract mathematical theories may be hard to justify  (as indeed is the teaching of music or art or ancient Greek).   This perhaps explains how I could learn lots of tricks for elementary arithmetic in day-time classes at primary school, but only discover the rigorous beauty of Euclid’s geometry in special after-school lessons from a sympathetic fifth-grade teacher (Frank Torpie).

Technorati Tags: Timothy Gowers

My first comment is that the poster misunderstands the axiomatic method in pure mathematics.  It is not the case that “axioms are by assumption true”.  Truth is a bivariant relationship between some language or symbolic expression and the world.  Pure mathematicians using axiomatic methods make no assumptions about the relationship between their symbolic expressions of interest and the world.   Rather they deduce consequences from the axioms, as if those axioms were true, but without assuming that they are.    How do I know they do not assume their axioms to be true?  Because mathematicians often work with competing, mutually-inconsistent, sets of axioms, for example when they consider both Euclidean and non-Euclidean geometries, or when looking at systems which assume the Axiom of Choice and systems which do not.   Indeed, one could view parts of the meta-mathematical theory called Model Theory as being the formal and deductive exploration of multiple, competing sets of axioms.

On the question of economic modeling, the blogger presents the views of Gerard Debreu on why the abstract mathematicization of economics is something to be desired.   One should also point out the very great dangers of this research program, some of which we are suffering now.  The first is that people – both academic researchers and others – can become so intoxicated with the pleasures of mathematical modeling that they mistake the axioms and the models for reality itself.  Arguably the widespread adoption of financial models assuming independent and normally-distributed errors was the main cause of the Global Financial Crisis of 2008, where the errors of complex derivative trades (such as credit default swaps) were neither independent nor as thin-tailed as Normal distributions are.  The GFC led, inexorably, to the Great Recession we are all in now.

Secondly, considered only as a research program, this approach has serious flaws.  If you were planning to construct a realistic model of human economic behaviour in all its diversity and splendour, it would be very odd to start by modeling only that one very particular, and indeed pathological, type of behaviour examplified by homo economicus, so-called rational economic man.   Acting with with infinite mental processing resources and time, with perfect knowledge of the external world, with perfect knowledge of his own capabilities, his own goals, own preferences, and indeed own internal knowledge, with perfect foresight or, if not, then with perfect knowledge of a measure of uncertainty overlaid on a pre-specified sigma-algebra of events, and completely unencumbered with any concern for others, with any knowledge of history, or with any emotions, homo economicus is nowhere to be found on any omnibus to Clapham.  Starting economic theory with such a creature of fiction would be like building a general theory of human personality from a study only of convicted serial killers awaiting execution, or like articulating a general theory of evolution using only a hand-book of British birds.   Homo economicus is not where any reasonable researcher interested in modeling the real world would start from in creating a theory of economic man.

And, even if this starting point were not on its very face ridiculous, the fact that economic systems are complex adaptive systems should give economists great pause.   Such systems are, typically, not continuously dependent on their initial conditions, meaning that a small change in input parameters can result in a large change in output values.   In other words, you could have a model of economic man which was arbitrarily close to, but not identical with, homo economicus, and yet see wildly different behaviours between the two.  Simply removing the assumption of infinite mental processing resources creates a very different economic actor from the assumed one, and consequently very different properties at the level of economic systems.  Faced with such overwhelming non-continuity (and non-linearity), a naive person might expect economists to be humble about making predictions or giving advice to anyone living outside their models.   Instead, we get an entire profession labeling those human behaviours which their models cannot explain as “irrational”.

My anger at The Great Wen of mathematical economics arises because of the immorality this discipline evinces:   such significant and rare mathematical skills deployed, not to help alleviate suffering or to make the world a better place (as those outside Economics might expect the discipline to aspire to), but to explore the deductive consequences of abstract formal systems, systems neither descriptive of any reality, nor even always implementable in a virtual world.

Technorati Tags: economics, economic modelling, homo economicus, mathematical economics

Scientific American’s tribute page is here, and here is a just-posted transcript of a February 1979 conversation between Gardner and other mathematicians.   This transcript contains a wonderful statement by mathematician Stan Ulam:

In fact, you know, yesterday Ron Graham gave a marvelous, really interesting lecture about some esoteric question; and I was wondering during it, Well, the question sounds very complicated, why devote so much ingenuity? Then I remember what, I think, Fourier or Laplace wrote: That mathematics—one reason for its being—is to defend the honor of the human mind.”

Technorati Tags: Martin Gardner, Stan Ulam

Number of diatonic scale classes: 3

Number of note names (A-G); number of notes in a common scale; number of white keys per octave on a piano:  7

Number of scales one note different from the Major scale: 9

Number of notes in the most common equal-tempered scale:  12

Number of common musical keys (C + 1-6 flats/sharps):  13

Number of 7-note diatonic scales (=7 * 3):  21

Number of elementary 2-fold polychords: 23

A k-fold polychord is an n-note chord sub-divided into k non-empty subchords, for k=1,  . . ., n.  For example, the 6-note chord <C, D, E, F#, G, A> can be subdivided into the 3-note 2-fold polychords, <C, E, G> and <D, F#, A>.

Number of 7-note chords:  66

Number of distinct interval sets (partitions of 12):  77

Number of 7-note triatonic scales (=7*35):  245

Number of notationally-distinct diatonic scales (=13 *21):  273

Number of distinct chord-types (= N(12) – 1):  351

Number of 7-note musical scales (=7*66):  462

Number of scales (=Number of n-note scales, summed over all n)  (=2^(12-1) = 2^11):   2048

Number of chords without rotational isomorphism (= 2^12 – 1):  4095

Number of notationally-distinct scales (=13 * 462):  6006

Number of non-syncopated 8-bar 1/4-note rhythmic patterns:  458,330

Number of non-syncopated 8-bar 1/8-note rhythmic patterns:  210,066,388,901

Reference:

Michael Keith [1991]:  From Polychords to Polya:  Adventures in Musical Combinatorics.  (Princeton, NJ:  Vinculum Press.)

• The discipline of Computer Science has usually developed first through practice, and only later through theory.
  

  The first calculating machine was invented by Leibniz in 1671, while the first mathematical theory of computing machines was that of Turing in 1937. The first widespread programmable device was Jacquard’s Loom, invented in 1801, while the first mathematical theory of programme languages did not appear until the 1960s. The Internet has been operating (first as Arpanet) since 1969, yet we still lack a formal theory of interaction. The first online friendships between people who never met were between telegraph operators, and later between telephonists, in the 19th century. The first e-commerce network was the Florists Telegraph Delivery Association, created in the USA in 1910.

• It is therefore a profound mis-understanding to consider Computer Science to be a branch of Pure Mathematics.

The scope (the content) of the discipline of Computer Science comprises the behaviors of human artefacts of a certain sort, along with some natural phenomena. Without the artefacts, we would likely not have the theory, or at least not yet (and perhaps not for a long time). Without the theory, we would not fully understand the performance of the artefacts, so both are needed. But the artefacts came first, and practice should dominate. If the theory dominates, our discipline will shrivel and die, becoming as dessicated and as useless as mathematical economics.

• Artificial Intelligence (AI) is the study of thinking about ways of knowing and ways of acting.
  

  This statement updates a statement of Seymour Papert (1988, p.3), who considered only ways of knowing.

  Seymour Papert [1988]: One AI or Many? Daedalus, 117 (1) (Winter 1988): 1-14.

• Not all ways of thinking are equally effective in all situations.
  

  In particular, some means of representing knowledge are more effective than other representations for some purposes. For instance, non-probabilistic formalisms for representing uncertainty, such as Dempster-Shafer Theory and Possibility Theory, are more effective than Probability Theory for domains where knowledge may be inconsistent or incomplete (ie, domains where the Law of the Excluded Middle cannot be presumed to hold, such as in medical diagnosis and in criminal forensics). The statement in the previous sentence remains true even though many such non-probabilistic formalisms can be shown to be equivalent to second- or higher-order nested probabilistic formalisms. This equivalence is a quaint mathematical result; non-probabilisitic formalisms are often easier for ordinary humans to understand than nested probabilistic formalisms. This is why the statement in the second sentence in this paragraph is an instance of the statement in the first sentence.

• Corollary: It behooves no one in AI to be dogmatic about ways of thinking.
  

  This is one reason why I am not a Bayesian.

• Deductive reasoning over an abstract mathematical model will only provide information about the real world to the extent that the relationship between the model and reality is continuously dependent on the initial assumptions.
  

  In other words, the fact that the assumptions of a model are close to some real phenomenon tells us nothing about whether the outputs of the model are close to those of the real phenomenon if the relationship between model and reality is not continuous. If you derive some result from an assumption that participants in some interaction have infinite processing capabilities, for example, then it does not necessarily follow that the same or close result holds if their real processing capabilities are finite, even if very large. Economics has always suffered from forgetting this truth, but most mainstream economists only seem to have realized it following the Great Global Economic Crisis of 2007. Indeed, some of the so-called freshwater economists have still not realized it, alleging that their models are still good predictors.

• In domains with intelligent participants (such as economics and computer science), models may be performative.
  

  In other words, participants may decide their modes of behaviour based on what modelers have suggested, so that modeling becomes, in effect, a form of self-fulfilling prophecy.

  In Economics, for example, the Black-Scholes model of options pricing allowed traders to price options rigorously. To do so, traders adopted the assumptions made by the modelers (eg, that errors are normally distributed, that decision-makers maximize expected utility, etc).

  Philip Mirowski [2002]: Machine Dreams: Economics Becomes a Cyborg Science. Cambridge, UK: Cambridge University Press.

  For doctrines of nuclear warfare, decision-makers adopted the modes of analysis, assumptions, and decision-options suggested to them by game theorists. Some of these assumptions were questionable during the Cold War – for example, that all participants know and agree on the game they are playing. As a consequence, the US Government appears to have embarked in the late 1950s on a mission to ensure the leaders of the USSR were also using game theory (mainly by issuing high-level public statements asserting that game theory was of no use in military applications).

• We have entered an era when the prevailing paradigm for the notion of computation is computing-as-interaction.

This paradigm follows earlier paradigms of computation as calculation (c. 1600 – 1965), of computation as information-processing (1965 – 1980), and computation as cognition (1980 – 1995). The new paradigm changes everything. In particular, an abstract model of computers based on movie projectors (ie, Turing Machines) is woefully inadequate for computing where outputs may be needed before all the inputs arrive, where there are multiple threads of control, where programs may be created, composed with one another, and compiled when invoked (ie, at run-time), and where computational devices and software exist together in ever-on, dynamic ecologies. We await an adequate formal, mathematical account of computing-as-interaction, and the game semantics of Samson Abramsky et al, and the bigraphs of Robin Milner are possible candidates.

M. Luck, P. McBurney, S. Willmott and O. Shehory [2005]: The AgentLink III Agent Technology Roadmap. AgentLink III, the European Co-ordination Action for Agent-Based Computing, Southampton, UK.

• As a consequence, Computer Science has a lot to learn from disciplines that have studied interaction.
  

  Disciplines such as Argumentation Theory, the Philosophy of Language, Linguistics, Economics, Social Psychology, Sociology, Anthropology, Political Science, Marketing, Epidemiology, Ecology, and Biology.Hopefully, the learning will be in both directions. For example, it is possible (although, in my personal opinion, unwise and immoral) for economists to assume that all economic actors always act in their own self-interest, maximizing their perceived expected utility. No rational computer scientist could make this assumption, however (except pro tem), since we all know the prevalence of buggy code: such code means that software entities may act against their own-self interest, or the interests of their principals. Creating a computational theory of interacting economic actors which does not make such false and unfalsifiable assumptions will surely benefit Economics, as well as Computer Science.

  The two-way interplay of Computer Science with these other disciplines of interaction provides further evidence that Computer Science is not a branch of Mathematics.

• Conflict and disagreement is inevitable in open computer systems.
  

  It therefore seems absurd to use formal models in which conflict is not permitted. Instead, it behooves us to consider computational models which enable conflict to be identified, managed, mitigated, and possibly resolved. Argumentation theory, not classical logic, is appropriate here.

• The Killer App for multi-agent systems is Distributed Computing.
  

  I have lost count of the number of times I have been asked by people outside the agents community, particularly other computer scientists, to name the killer app for agent technologies. Look about! It is all around you!Agent methodologies and technologies enable us to model and simulate complex adaptive systems, such as distributed computer systems. They also enable us to engineer (to specify, to design, and to create) such systems. And they enable us to study the properties of such systems, and hence to manage and control them.

• Agents are not objects.
  

  Objects always execute when invoked, and execute as expected. Agents may not. Objects maintain persistent relationships between one another. Agent relationships may be dynamic.

Afterwards, my husband and I reminisced about our attempts to learn tennis when we were young. I told him that my sisters and I used to go down to the public tennis courts in Portobello. We had probably never seen a professional tennis match; we just knew that tennis was about hitting the ball to and fro across the net. We had a few lessons and became quite good at leisurely rallies, hitting the ball back and forth without any attempt at speed. Sometimes we could keep our rallies going for quite a long time, and I found this enjoyable.

Then our tennis teacher explained that we should now learn to play “properly”. It was only then that I realised we were meant to hit the ball in such a way that the other person could not hit it back. This came as an unpleasant surprise. As soon as we started “playing properly”, our points became extremely short. One person served, the other could not hit it back, and that was the end of the point. It seemed to me that there was skill in hitting the ball so that the other person could hit it back. If they could, the ball would flow, one got to move about and there was not much interruption to the rhythm of play. It struck me that hitting the ball deliberately out of the other person’s reach was unsportsmanlike. When I tell my husband all this, he laughs and says: “There speaks a true chamber musician.”

This story resonated strongly with me.  Earlier this year, I had a brief correspondence with mathematician Alexandre Borovik, who has been collecting accounts of childhood experiences of learning mathematics, both from mathematicians and from non-mathematicians.  After seeing a discussion on his blog about the roles of puzzles and games in teaching mathematics to children, I had written to him:

Part of my anger & frustration at school was that so much of this subject that I loved, mathematics, was wasted on what I thought was frivolous or immoral applications:   frivolous because of all those unrealistic puzzles, and immoral because of the emphasis on competition (Olympiads, chess, card games, gambling, etc).   I had (and retain) a profound dislike of competition, and I don’t see why one always had to demonstrate one’s abilities by beating other people, rather than by collaborating with them.  I believed that “playing music together”, rather than “playing sport against one another”, was a better metaphor for what I wanted to do in life, and as a mathematician.

Indeed, the macho competitiveness of much of pure mathematics struck me very strongly when I was an undergraduate student:  I switched then to mathematical statistics because the teachers and students in that discipline were much less competitive towards one another.  For a long time, I thought I was alone in this view, but I have since heard the same story from other people, including some prominent mathematicians.  I know one famous category theorist who switched from analysis as a graduate student because the people there were too competitive, while the category theory people were more co-operative. 

Perhaps the emphasis on puzzles & tricks is fine for some mathematicians – eg, Paul Erdos seems to have been motivated by puzzles and eager to solve particular problems.  However, it is not fine for others – Alexander Grothendieck comes to mind as someone interested in abstract frameworks rather than puzzle-solving.  Perhaps the research discipline of pure mathematics needs people of both types.  If so, this is even more reason not to eliminate all the top-down thinkers by teaching only using puzzles at school.” 

 

Technorati Tags: Paul Erdos, Alexander Grothendieck

Proof is supposed to be what sets mathematics apart from the other sciences. Traditionally, the subject has not been an evolutionary one in which the fittest theory survives. New insights don’t suddenly overturn the theorems of the previous generation. The subject is like a huge pyramid, with each generation building on the secure foundations of the past. The nature of proof means that mathematicians, to use Newton’s words, really do stand on the shoulders of giants.

In the past, those shoulders have been extremely steady. After all, in no other science are the discoveries of the Ancient Greeks still as valid today as they were at the time. Euclid’s 2300-year-old proof that there are infinitely many primes is perhaps the first great example of a watertight proof.

The reason for this widespread view is that mathematics uses deduction to reach its conclusions.  At least, that is true of pure mathematics, or was so until computers began to be used in proofs (a topic which du Sautoy discusses in that article).  But all deduction does is to show that, given some assumptions and given some rules of inference, a certain conclusion follows from those assumptions by applying those rules of inference.  If either the assumptions are false or the rules of inference not acceptable, then the stated conclusions will not, in fact, follow.

Du Sautoy is quite wrong to claim that new insights do not overturn the theorems of the previous generation.  The history of pure mathematics is replete with examples where proven conclusions were later revealed to depend on assumptions not made explicit, or on assumptions previously thought to be obvious but which were later shown to be false, or on rules of inference later considered invalid.   For over a century, mathematicians thought that everywhere-continuous functions were also everywhere-differentiable, until shown a counter-example.  For a similar period, they thought that the convergent limit of an infinite sequence of continuous functions was itself also continuous, until shown a counter-example.  They thought that there could not exist a one-to-one and onto mapping between the real unit interval and the real unit square, until shown such a mapping (a so-called space-filling curve).  In fact, there are infinitely-many such mappings; indeed, an uncountable infinity of them.  In all these case, “proofs” of the erroneous conclusions existed, which is why the earlier mathematicians believed those conclusions.  The proofs were later shown to be flawed, because they depended on (usually-implicit) assumptions which were false.   For the differential calculus, the fixing effort was begun by Cauchy and Weierstrauss, using epsilon-delta arguments which were more rigorous than the proofs of the earlier generation of analysts.  

Not only does Du Sautoy have his history wrong, but there is shurely shome mishtake in his mentioning Euclid here.  The 19th century was consumed by a controversy over the truth-status of Euclidean geometry, and the discovery of apparently-logical alternatives to it.   As clever a man as the logician and philosopher Gottlob Frege (an intellectual hero of Wittgenstein) could not get his head around the idea that these different versions of geometry could all simultaneously be true.   Yet that is the conclusion mathematicians came to: that, depending on the assumptions you made about the surface on which you doing geometry, there were in fact valid alternatives to the discoveries of the Greeks:  draw your triangles on the surface of a sphere, instead of on a flat plane, for example, and you could readily draw triangles whose three angles did not sum to 180 degrees.  You choose your assumptions, you gets your geometry!  This is not a secure pyramid of knowledge, but many pyramids, post-modernist style.

And in the first part of the 20th century, pure mathematics was consumed with a bitter argument over whether a particular rule of inference – reductio ad absurdem (RAA), or reasoning from an assumption thought to be false – was valid in deductive proofs of the existence of mathematical objects.   The dissidents created their own school of pure mathematics, constructivism, which is still being studied.  Indeed, it turns out that a closely-related logic, intuitionistic logic, appears naturally elsewhere in mathematics (as part of the internal structure of a topos). Once again, you choose your rules of inference, you gets your mathematical theorems.  

There is no single, massive pyramid of knowledge here, as du Sautoy claims, but lots of smaller pyramids.  Every so often, a great mathematician is able to devise a new conceptual framework which allows some or all of these baby pyramids to appear to be part of some larger pyramid, as Pieri and Hilbert did with geometry in the 1890s, or as Lawvere and others did with category theory as a foundation for mathematics in the 1960s.   But, based on past experience, new baby pyramids will continue to be created by mathematicians arguing about the assumptions or rules of inference used in earlier proofs.    To consider this process of contestation, splitting, and attempted re-unification to be somehow different to what happens in other domains of human knowledge may be comforting to mathematicians, but is myth nonetheless.

Technorati Tags: Marcus du Sautoy, mathematics, deduction, Cauchy, Weierstrauss, epsilon-delta, Euclid, Frege, reductio ad absurdem, Pieri, Hilbert, Lawvere

Godel was famous for having kept every piece of paper he’d ever encountered, and the set design (pictured here) included many file storage boxes.  Some of these were arranged in a checkerboard pattern on the floor, with gaps between them.  As the Godel character (Phillips) tried to prove something, he took successive steps along diagonal and zigzag paths through this pattern, sometimes retracing his steps when potential chains of reasoning did not succeed.   This was the best artistic representation I have seen of the process of attempting to do mathematical proof:  Imre Lakatos’ philosophy of mathematics made theatrical flesh.

The photograph of the La Mama billboard is from Paola’s site.

Technorati Tags: Kurt Godel, Steven Schiller, Steven Phillips

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.

The human toll of the examination on its candidates in terms of nervous breakdowns, stress and suicides was considerable. This effect is in addition to the disastrous distortion which the exam had on pure mathematics in Britain in the 19th century. The British lost a century of pure mathematics because of the focus of the Tripos on mathematics applied to physics at the expense of other branches of the discipline. The great British mathematicians of the 19th century – Babbage, Boole, Cayley, de Morgan – worked outside the mainstream of continental pure mathematics. Even 70 years after its ending in 1909, and 10,000 miles away, the pernicious influence of the Tripos could still be felt: when I commenced undergraduate study in mathematics in Australia, I could find only one university which would let me study pure mathematics without also having to do so-called Applied (mechanics, hydrodynamics, mathematical physics, etc).  And, arguably, mathematical economics, whose founders were Cambridge Tripos men, is still to throw off its heritage in the Tripos.

Andrew Warwick wrote a superb history of the exam (details below), and I quote some interesting passages from his book.  I am struck by how closely the skills required for successful performance in the oral examination (apart from knowledge of Latin) are those displayed today by successful management consultants.

The gradual shift from oral to written examination described above should not be understood simply as a change in the method by which a student’s knowledge of certain subjects was assessed. The shift also represented a major change both in what was assessed and the skills necessary to succeed in the examination. Consider first the major characteristics of the oral disputation. This was a public event in which a knowledge of Latin, rhetorical style, confidence in front of one’s peers and seniors, mental agility, a good memory, and the ability to recover from errors and turn the tables on a clever opponent were all necessary to success. The course of the examination was regulated throughout by its formal structure and the flow of debate between opponent, respondent, and moderator. The respondent began by reading an essay of his own composition on an agreed topic and then defended its main propositions against three opponents in turn according to a prescribed schedule (fig. 3.2). The public and oral nature of the event meant that the processes of examination and adjudication were coextensive. The moderator formed an opinion of the participants’ abilities as the debate unfolded and the nature and fairness of his adjudication were witnessed by everyone present. The examination was also open-ended. The moderator could prolong the debate until he was satisfied that he had correctly assessed the abilities of opponents and respondents, and would generally quiz the respondent himself. Finally, once the disputation was completed, the only record of the examination was the recollections of those involved.

The economy of a written examination was quite different in several important respects. First, the oratorical skills mentioned above become irrelevant, as the examination focused solely on the reproduction of technical knowledge on paper and the ability to marshal that knowledge in the solution of problems. Second, the process of examination was separated from that of adjudication, a change that destroyed the spontaneity of the public Act. Without the formal procedure of a disputation and the rhythm of debate between opponent and respondent, each candidate was left to work at his own pace with little sense of how well or how badly he was doing. And, once the written examination was completed, the student had no opportunity either to wrangle over the correctness of an answer to to recover errors. The advent of written examinations also brought substantial change to the role of moderators. Instead of overseeing and participating in a public debate, they became responsible for setting questions, marking scripts, and policing the disciplined silence of the examination room (fig. 3.3). This brings us to a third important difference between oral and written assessment. As I noted above, setting identical questions to a group of students made it much easier directly to compare and rank the students, especially if each question was marked according to an agreed scheme. Furthermore, the written examination scripts provided a permanent record of each student’s performance which could be scrutinized by more than one examiner and reexamined if disagreements emerged. These differences between written and oral assessment altered the skills and competencies required of undergraduates and completely transformed their experiences of the examination process. (pages 122-124).

The ideal of a liberal education prevalent in Georgian Cambridge was, as we have seen, one in which students were supposed to learn to reason properly through the study of mathematics and to acquire appropriate moral and spiritual values through the subservient emulation of their tutors. This kind of education, which had long been seen as an appropriate preparation for public life, valued the qualities of good character, civility, and gentility above those of introspection, assertiveness, and technical expertise. The oral disputation formed an integral part of this system of education as it tested a student’s knowledge through a public display of civil and gentlemanly debate. Timidity and diffidence were indeed qualities that handicapped a student in this kind of examination, though not because they prevented him from answering technical problems in mathematics, but because they were seen as undesirable qualities in a future bishop, judge, or statesman. (page 127)

We have seen that when students were examined by oral disputation on, say, Newton’s Principia, they were generally required to defend qualitative propositions against carefully contrived objections of several opponents. These verbal encounters implied that there were seemingly plausible objections to Newton’s celestial mechanics and required the student to locate the fallacies in such objections from a Newtonian perspective. In providing written answers to questions, by contrast, students were required either to reproduce as bookwork the laws and propositional theorems found in such books as the Principia, or to assume their truth in tackling questions on the problem papers. The move from oral disputation to written examination was therefore accompanied by a far more dogmatic approach to the physical foundations of mixed mathematics. (pages 139-140)

This example highlights the fact that the kind of originality required of mathematics undergraduates was largely defined by the form of the problems they were expected to solve. They were not required to invent and deploy novel or unfamiliar principles or mathematical methods, nor even to analyze novel or unfamiliar phenomena. They were required, rather, to show that they could understand the enunciation of a well-formulated problem, analyze the physical system described using the principles and techniques they had been taught, and use that analysis to generate specific mathematical expressions and relationships. (page 166)

Reference:

Andrew Warwick [2003]:  Masters of Theory: Cambridge and the Rise of Mathematical Physics (Chicago, IL, USA: University of Chicago Press).

Technorati Tags: Cambridge University, Mathematical Tripos, examinations, Andrew Warwick, management consultants