Saturday, June 28, 2008

Pythagoreans, Orbifolds and Catherine the Great

I've just decided - in future when asked to state my religion I'll enter "Pythagorean".

(reference - "The Music of Pythagoras" by Kitty Ferguson, my current reading - I'm currently a perfect fifth (1/3rd - pythagorean joke ! ) the way through it).

Not that I think Pythagoras would have admitted me as one of his disciples - I a bit too shallow and stupid and not a good enough listener. And its not as though I am confident that anybody really knows much about the pythagoreans at all - Ferguson is very up-front about how little solid historical evidence there is , and how unreliable many of the re-tellings down the ages are likely to be. Seems like Pythagoras almost made it into the classical limelight but not quite - he lived about 570-500BC - and by the second half of the fifth century BC , 450 to 400BC , we have Aeschylus, Aristophanes, Hippocrates and his oath, and the parthenon. A pythagorist Philolaus was in the limelight period and almost a direct link with Pythagoras himself - he was born only 25 years or so after the death of Pythagoras, and was educated by pythagorists, the older of whom must have known Pythagoras. Evidently Philolaus did write a detailed account of the teachings of the pythagorists - but unfortunately only fragments survive, and it is not clear which parts of those are Pythagoras and which are Philolaus. So the real Pythagoras and his teachings remain outside the limelight - in the twilight zone of history.

So why a pythagorist ? Its certainly not that I believe in numerology or have any sort of mysticism about numbers. Nor do I, as apparently did the pythagorists, believe that beans contain souls. (Their chain of reasoning was : (1) beans cause flatulence (2) flatulence is air (3) it was widely believed that souls were air (4) => beans contain souls. The belief is reputed to have led to Pythagoras's death - he was fleeing some hostile locals, whipped up by a disgruntled pillock of the community who had been refused admission to the pythagorean school : when confronted by a field of beans blocking his way, Pythagoras because of his beliefs had to run around it (can't trample all those souls !) while those in pursuit just ran across, so catching and killing him). Nor do I believe in other pythagorean (non-bean !) staples such as reincarnation or the uncomplicated purely number-based rationality of the universe . Nor do I think they sound a very attractive bunch really - rather humourless and puritanical.

Its just that, like the pythagoreans, I have this need to see - dimly, probably deludedly, and in a rather half-baked and I suppose somewhat embarrassing, dilettantish way - mathematical concepts in domains where they are not generally (currently) admitted. And I suppose, a kind of literary or ramblingly discursive or in some other sense oblique, rather than directly technical and computational, relationship with mathematics (of necessity since I do not possess any significant mathematical talent). Maybe could put it as, mathematics as a source of inspiration and metaphors, but slightly more than that - metaphors that can almost , but not quite , be used for computation and prediction in the target domain of their allusion, as well as in the source domain.

(The link between music and mathematics that was first made by Pythagoras is now a commonplace - yet still has legs for future discovery - e.g. see the recent "The Geometry of Musical Chords
Dmitri Tymoczko (7 July 2006)Science 313 (5783), 72". This uses a fascinating mathematical object known as an orbifold. Now my understanding of orbifolds is pythagoristic rather than technical - I had a concept of something like an orbifold before I came across the term - and have been assuming for awhile that what is denoted by this term corresponds to my concept.... could be wrong. I wanted - just for my own interest - something to describe the manifold represented by the combined state of a set of cyclical functions. Lets say for example - the space occupied by the expression levels of a set of genes. The expression of each gene varies continuously in some no doubt cyclical fashion (unless it is some odd one-shot developmental gene that only ever turns on once), and the high-dimensional space that the expression of N genes lives in is clearly a continuous N-dimensional manifold (my understanding of which is mostly technically OK I *think*, but also a little pythagorist in the above sense ) - yet because of the cyclical nature of each of the N expression levels, it seems to be a slightly different type of manifold, because of these orbits in each of the dimensions. Another example would be - the combined state represented by the position of the tip of every leaf of a tree blowing in the wind. Again - each leaf gyrates in its own orbit
and the combined high dimensional set of leaf-tip positions clearly lives in a continuous N-dimensional manifold (N = the number of leaves) - yet it has this structure in which each dimension lives in an orbit. (I think this is what is meant by the technical orbifold definition (e.g. see Wikipaedia ) that "Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on open subsets of Rn, an orbifold is locally modelled on quotients of open subsets of Rn by finite group actions..." - the finite group actions here being the rotations of the leaf tips of the tree, or the rotations/oscillations of the expression levels of the genes...(user beware - do not requote any of this as it may be misinformation !)....more on orbifolds in a future blog !)

(....or are the manifolds I described N-dimensional torii ? For example a cross product of two circular one-dimensional manifolds is a torus - so maybe the cross-product of N orbits of the type I describe is an N-dimensional torus. Question : is there an isomorphism of some sort between orbifolds and N-dimensional torii ?)

(Hypothesis : pythagorists (in the above sense) could play a role in bringing far-fetched mathematical objects like orbifolds down from the mount and finding them useful employment in the fields. Mathematicians themselves won't do that - they are slaving at the top of the mountain. Non mathematician specialists won't, they are too busy slaving on other mountains. Then again the non-specialist pythagorist dilettante has little credibility on either mountain so its unlikely after all but possibly worth trying, though I suspect he/she is likely to anger some technical specialist mob or other and get chased around the bean fields a bit doing this sort of thing !)

(I note in passing that Kitty Ferguson is as well as being an author on various mathematical and physics subjects also a "Juilliard trained professional musician". I can't believe the number of young people I have met in the last few years , multi-talented in this way - science / maths careers and also accomplished musicians - composers, performers, conductors.... I don't remember it being like that when I was younger and at 'varsity - there were some smart people alright (I was not one of them ) - and I know one that went on to be a well known poet and others that rose to academic success but.....we were....kind of ordinary in comparison, and really just tended to mooch around.....or at least it seems that way in retrospect. (In my case there wasn't even a "we" - I just mostly mooched my way solitarily through various crushes). Maybe there is an interaction effect, between talent and the technology and educational opportunities and expectations now that enables talented people to burn brighter sooner nowadays.....something like the famous Flynn effect maybe - an interaction specifically between natural talent and the modern world)

For something completely different - and for a much raunchier read (in between the lines ) - read the excellent "The Memoirs of Catherine the Great" , a new translation by Mark Cruse and Hilde Hoogenboom. I love a quote (from a section that was left out of a 1907 Russian edition)..."No one holds his heart in his hand and restrains or releases it by closing his hand at will." (page 200! - its hot ! ). Her diary reminds me a bit of the Pepys one, in its surprising accessibility and the way it draws you in to a story of day to day life which on the face of it would appear likely a dull read yet somehow becomes the opposite. They were both writing also at about the time of the Enlightenment when change and progress was scented on the wind - perhaps it is the background optimism shining through (and despite the various disasters and discomforts they endured - the great fire, plague, political and career problems) that make these diaries so good to read. And in Catherine's one - court life is surprisingly like office life - with courtly notes playing the role of email, and a similar caste of personalities and rivalries and manipulative behaviours.

1 comment:

Amanda said...

Hi Alan.
Great to read more of your blogging. Not pretending I understand it all - being both mathematically and historically challenged as I am. You've given me a few new words to wonder about such as orbifold! Cheers, Mandy