Archive for November, 2011

Digital dumbing down

Despite being all the rage, touchscreens have never impressed me.   I did not put my finger (metaphor chosen deliberately) on the reasons why until reading Edward Tufte’s criticism:  they have no hand!  They lack tactility, and of all the many possible diverse, sophisticated, subtle, and complex motions that our hands and fingers are capable of, touchscreens seem designed to accommodate just two very simple motions:  tapping and sliding.   Not something to write home about when you wake up each morning eager to digitally percuss, or have hands able  to think.    Bret Victor has a nice graphically-supported argument about the lack of embodiment of touchscreens in the world of those of us with opposable thumbs, here.  As Victor says:

Are we really going to accept an Interface Of The Future that is less expressive than a sandwich?




A brief history of mathematics

Australian category-theorist Ross Street has an elegant, one-page summary of the first 2,500 years of western mathematics, here.  This was apparently a handout given in a talk to the Macquarie University Philosophy Students Society in 1984.  I found Street’s high-level view of what (some important) mathematicians have (mostly) been doing illuminating and thought-provoking, and so I reproduce it here.

 

Space, Sets and Beyond

First Cycle:  General spaces advance the study of naive geometry

1. Naive geometry:  Zeno, Eudoxus.

2. Axiomatic geometry (unique model intended): Euclid, Apollonius (c. 300-200 BC).

3. Algebraic techniques (coordinate geometry): Descartes 1596-1650.

4. Non-Euclidean geometry (independence of the “parallels axiom”:  models without parallels axiom constructed from a model with it):  Gauss, Bolyai, Lobatchewski (early 19C).

5. Locally Euclidean spaces:  Riemann 1826-1866, Lie.

6. Relationships between spaces (continuity, linearity): Cauchy, Cayley, Weierstrass, Dedekind (1880-present).

 

Second Cycle:  Toposes can be viewed as even more general spaces

1. Naive set theory: Peano, Cantor (c. 1900).

2. Axiomatic set theory (unique model intended): Hilbert, Godel, Bernays, Zermelo, Zorn, Fraenkel.

3. Abstract algebra (mathematical logic): Boole, Poincare, Hilbert, Heyting, Brouwer, Noether, Church, Turing.

4. Non-standard set theories (independence of the “axiom of choice” and “continuum hypothesis”; Boolean-valued models; non-standard analysis): Godel, Cohen, Robinson (1920-1950).

5. Local set theory (sheaves): Leray, Serre, Grothendieck, Lawvere, Tierney (1945-1970).

6. Relationships between toposes (a “topos” is a generalized set theory): 1970-present.





Suddenly, the fog lifts . . .

Andrew Wiles, prover of Wiles’ Theorem (aka Fermat’s Last Theorem), on the doing of mathematics:

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were.

This describes my experience, over shorter time-frames, in studying pure mathematics as an undergraduate, with each new topic covered: epsilon-delta arguments in analysis; point-set topology; axiomatic set theory; functional analysis; measure theory; group theory; algebraic topology; category theory; statistical decision theory; integral geometry; etc.    A very similar process happens in learning a new language, whether a natural (human) language or a programming language.     Likewise, similar words describe the experience of entering a new organization (either as an employee or as a management consultant), and trying to understand how the organization works, who has the real power, what are the social relationships and dynamics within the organization, etc, something I have previously described here.

One encounters a new discipline or social organization, one studies it and thinks about it from as many angles and perspectives as one can, and eventually, if one is clever and diligent, or just lucky, a light goes on and all is illuminated.    Like visiting a new city and learning its layout by walking through it, frequently getting lost and finding one’s way again,  enlightenment requires work.  Over time, one learns not to be afraid in encountering a new subject, but rather to relish the state of inchoateness and confusion in the period between starting and enlightenment.  The pleasure and wonder of the enlightenment is so great, that it all the prior pain is forgotten.

Reference:

Andrew Wiles [1996],  speaking in Fermat’s Last Theorem, a BBC documentary by S. Singh and John Lynch: Horizon, BBC 1996,  cited in Frans Oort [2011 ]:  Did earlier thoughts inspire Grothendieck? (Hat tip).




Bands

Gripped as I seem to be by listmania, here is another:  of orchestras and ensembles I have seen perform live.   I omit chamber music groups, counting only ensembles large enough to need a conductor.

  • Academy of St Martin in the Fields, London
  • Aurora Orchestra, London
  • Australia Ensemble
  • Australian Brandenburg Orchestra
  • Australian Chamber Orchestra
  • Australian Youth Orchestra
  • BBC Philharmonic, Manchester
  • Boston Symphony Orchestra, Tanglewood
  • Brandenburg Baroque Soloists, London
  • Britten Sinfonia, London
  • Canberra Symphony Orchestra
  • Chicago Symphony Orchestra
  • Birmingham Contemporary Music Group
  • City of Birmingham Symphony Orchestra
  • Ensemble 10:10, Liverpool
  • Ensemble 11, Manchester
  • European Brandenburg Ensemble
  • Halle Orchestra, Manchester
  • Hamburger Symphoniker
  • Harare City Orchestra
  • Hong Kong Philharmonic Orchestra
  • Irving Symphony Orchestra
  • Isabella a Capella
  • Leipzig Gewandhaus Orchestra
  • Lismore Regional Symphony Orchestra
  • London Philharmonic
  • London Sinfonietta
  • London Symphony Orchestra
  • Mahler Chamber Orchestra
  • Manchester Camerata
  • Medici Choir, London
  • Melbourne Symphony Orchestra
  • Merseyside Youth Orchestra
  • Moscow Soloists String Chamber Ensemble
  • Northern Sinfonia, Gateshead
  • Orchestra of the Age of Enlightenment, London
  • Orchestra Mozart, Bologna
  • Orchestre Philharmonique de Nice
  • Orchestre Revolutionnaire et Romantique, London
  • Orquestra Gulbenkian, Lisbon
  • Patonga Orchestra
  • Queensland Conservatorium Chamber Orchestra
  • Queensland Symphony Orchestra
  • Royal Liverpool Philharmonic Orchestra
  • RNCM Chamber Orchestra, Manchester
  • Seoul Philharmonic Orchestra
  • Siglo de Oro, London
  • Silk Road Ensemble
  • Solistes de Musique Ancienne, London
  • Soweto Symphony Orchestra
  • Sydney Symphony Orchestra
  • Vienna Chamber Orchestra
  • University of Liverpool Symphony Orchestra

 




Conductors

This is a list, as best I can recall, of the conductors I have seen in live performance.  If memory serves, I add the ensemble I saw them direct.

  • Claudia Abbado, Orchestra Mozart Bologna
  • Thomas Ades, Britten Sinfonia, London
  • Yuri Bashmet, Moscow Soloists String Orchestra
  • Stuart Challender, Sydney Symphony Orchestra
  • Nicholas Collon, Aurora Orchestra
  • Matthew Coorey, RLPO and Halle Orchestra
  • Gustavo Dudamel, Royal Liverpool Philharmonic Orchestra
  • Mark Elder, Halle Orchestra Manchester
  • John Eliot Gardiner, Orchestre Revolutionnaire et Romantique, London
  • Edward Gardner, Orchestra of the Age of Enlightenment, London
  • John Gay, Soweto Symphony Orchestra and Maseru Singers
  • Richard Gill, Patonga Orchestra
  • Reinhard Goebel, Melbourne Symphony Orchestra
  • Hector Guzman, Irving Symphony Orchestra
  • Bernard Haitink, Sydney Symphony Orchestra
  • Nicholas Kraemer, Manchester Camerata
  • Marcelo Lehninger, Boston Symphony Orchestra
  • Charles Mackerras, Royal Liverpool Philharmonic Orchestra
  • Neville Marriner, Academy of St Martin in the Fields
  • Diego Matheuz, Orchestra Mozart/Mahler Chamber Orchestra
  • Michael Morgan, Queensland Conservatorium Chamber Orchestra
  • Joel Newsome, Solistes de Musique Ancienne
  • Gianandrea Noseda, BBC Philharmonic
  • Sakari Oramo, Birmingham Contemporary Music Group
  • Libor Pesek, Royal Liverpool Philharmonic Orchestra
  • Vasily Petrenko, Royal Liverpool Philharmonic Orchestra
  • Trevor Pinnock, European Brandenburg Ensemble
  • Simon Rattle, City of Birmingham Symphony Orchestra
  • Gerard Schwarz, Royal Liverpool Philharmonic Orchestra
  • Brian Stacey, Lismore Regional Symphony Orchestra
  • Marcus Stenz, Halle Orchestra Manchester
  • Patrick Thomas, Queensland Symphony Orchestra
  • Robin Ticciati, Royal Liverpool Philharmonic Orchestra
  • David Urquhart-Jones, Lismore Regional Symphony Orchestra.



Alan Greenspan in 2004

Alan Greenspan, then Chairman of the US Federal Reserve Bank System, speaking in January 2004, discussed the failure of traditional methods in econometrics to provide adequate guidance to monetary policy decision-makers.   His words included:

Given our inevitably incomplete knowledge about key structural aspects of an ever-changing economy and the sometimes asymmetric costs or benefits of particular outcomes, a central bank needs to consider not only the most likely future path for the economy but also the distribution of possible outcomes about that path. The decisionmakers then need to reach a judgment about the probabilities, costs, and benefits of the various possible outcomes under alternative choices for policy.”

The product of a low-probability event and a potentially severe outcome was judged a more serious threat to economic performance than the higher inflation that might ensue in the more probable scenario.”




Limits of Bayesianism

Many proponents of Bayesianism point to Cox’s theorem as the justification for arguing that there is only one coherent method for representing uncertainty. Cox’s theorem states that any representation of uncertainty satisfying certain assumptions is isomorphic to classical probability theory. As I have long argued, this claim depends upon the law of the excluded middle (LEM).

Mark Colyvan, an Australian philosopher of mathematics, published a paper in 2004 which examined the philosophical and logical assumptions of Cox’s theorem (assumptions usually left implicit by its proponents), and argued that these are inappropriate for many (perhaps even most) domains with uncertainty.

M. Colyvan [2004]: The philosophical significance of Cox’s theorem. International Journal of Approximate Reasoning, 37: 71-85.

Colyvan’s work complements Glenn Shafer’s attack on the theorem, which noted that it assumes that belief should be represented by a real-valued function.

G. A. Shafer [2004]: Comments on “Constructing a logic of plausible inference: a guide to Cox’s theorem” by Kevin S. Van Horn. International Journal of Approximate Reasoning, 35: 97-105.

Although these papers are several years old, I mention them here for the record –  and because I still encounter invocations of Cox’s Theorem.

IME, most statisticians, like most economists, have little historical sense. This absence means they will not appreciate a nice irony: the person responsible for axiomatizing classical probability theory – Andrei Kolmogorov – is also one of the people responsible for axiomatizing intuitionistic logic, a version of classical logic which dispenses with the law of the excluded middle. One such axiomatization is called BHK Logic (for Brouwer, Heyting and Kolmogorov) in recognition.




Underground Languages

Conversations overheard on the London Underground in:

Afrikaans, Amharic, Arabic, Cantonese, Catalan, Czech, Dutch, English*, Farsi, Finnish, French, German, Gujarati, Hausa, Hebrew, Italian, Japanese, Korean, Norwegian, Polish, Portuguese, Russian, Spanish, Swahili, Swedish, Tagalog, Thai, Urdu, isiZulu.

* Overheard regional variants of English from:  Australia, Britain (Brummie, Estuary, Geordie, Glasgow-Scottish, Mancunian, Edinburgh-Scottish, RP, Sarf Lonon, Scouse, Ulster, West Country), Canada, Eire, New Zealand, South Africa, USA (Barst’n, Bronx, Brooklyn, ‘Gisland, Midwest, Northeastern, Southern).




Automating prayer

I have recently re-read Michael Frayn’s The Tin Men, a superb satire of AI.  Among the many wonderful passages is this, on the semantic verification problem of agent communications:

“Ah,” said Rowe, “there’s a difference between a man and a machine when it comes to praying.”   “Aye. The machine would do it better. It wouldn’t pray for things it oughtn’t pray for, and its thoughts wouldn’t wander.”

“Y-e-e-s. But the computer saying the words wouldn’t be the same . . .”

“Oh, I don’t know. If the words ‘O Lord, bless the Queen and her Ministers‘ are going to produce any tangible effects on the Government, it can’t matter who or what says them, can it?”

“Y-e-e-s, I see that. But if a man says the words he means them.”

“So does the computer. Or at any rate, it would take a damned complicated computer to say the words without meaning them. I mean, what do we mean by ‘mean’? If we want to know whether a man or a computer means ‘O Lord, bless the Queen and her Ministers,’ we look to see whether it’s grinning insincerely or ironically as it says the words. We try to find out whether it belongs to the Communist Party. We observe whether it simultaneously passes notes about lunch or fornication. If it passes all the tests of this sort, what other tests are there for telling if it means what it says? All the computers in my department, at any rate, would pray with great sincerity and single-mindedness. They’re devout wee things, computers.” (pages 109-110).

Reference:

Michael Frayn [1995/1965]: The Tin Men (London, UK: Penguin, originally published by William Collins, 1965)




The sociology of cosmology

Physicist Per Bak:

“I once raised this issue among a group of cosmologists at a high table dinner at the Churchill College at Cambridge. “Why is that you guys are so conservative in your views, in the face of the almost complete lack of understanding of what is going on in your field?” I asked. The answer was as simple as it was surprising. “If we don’t accept some common picture of the universe, however unsupported by facts, there would be nothing to bind us together as a scientific community. Since it is unlikely that any picture that we use will be falsified in our lifetime, one theory is as good as any other.” The explanation was social, not scientific.” (Bak, page 86)

Reference:

Per Bak [1999]: How Nature Works: The Science of Self-Organized Criticality. (New York, USA: Copernicus)