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.

Tuesday, June 10, 2008

theory of action 2

In part 1 of this theory of action it was claimed that while the valuations derived from one's theory of ones self and ones actions was a fragile thing, the value of the actions themselves can be secure - actions are substantive and incompressible, real things in time and space. If true this seems (albeit probably in a sense that would be fairly cryptic to many people !) very is legitimate, good, essential even, to be and act and do, even when one's confidence in the whole enterprise of ones' self and ones actions is lacking - when one lacks an adequate theory of action and self.

Or the argument can be put more simply like this : it is common to infer the fitness and goodness of our actions, from our estimation of the fitness and goodness of ourselves : we are good so the things we do are good ; yet even if the inference can't be made, we are not to be deterred - we can still act and our actions can still be good.

Which raises the question - how can we know our actions are fit and good when we do not currently have a theory that asserts our own goodness and fitness ? Clearly this requires some external arbiter of the moral value of our actions - we need to be able to know a priori that it is good to help an old lady across the street, for the injunctions of theory of action 1 to hold.

Yet if we do have such an external arbiter, surely this would induce a very simple theory of ourselves and our actions - which is that, we ourselves are fit and good, if our actions are fit and good.

But if this robust theory is available, why then did we get into trouble in the first place - what sort of theory did we originally have that failed, and why did it fail ?

There are two possible answers. The first is that our theory was , indeed, that we ourselves are fit and good, simply if our actions are fit and good ; but that our actions have not recently been fit and good, so that our theory has failed. In this case to continue to act we need a special act of will and steadfastness - and / or perhaps some forgiveness from the external arbiter. (Many religious theories of self and action are of this type - we are valued as the sum of our actions, and there is some process of forgiveness or other remedy available when the theory fails). I will deal with this special act of will and steadfastness, in a future part of the theory of action, since it has a very interesting structure.

The other possible answer is that our original theory was not one that judged us as the sum of our actions. Perhaps , for example , it was a theory derived from our pedigree - that we were fit and good because of high birth.....and perhaps it failed because we have just found out our birth (or that of one of our ancestors) was illegitimate ! Or perhaps it was a theory based on some other intrinsic measure - say, exceptional ability in some specific domain, such as mathematics, music, art, athletics, intelligence tests, and we have come to doubt the measure in some way.

My guess is that the theories of self and action that fail and leave us stranded and stalled are often of this second sort - based on some intrinsic metric of self.

These types of theories are not robust for several reasons. Firstly, the intrinsic metric itself is likely to be error prone - it will be difficult to be certain about the nobility of our birth, the degree of our talent in our chosen domain. Secondly the metric is relative - there is no absolute level of nobility of birth or talent in some domain : there are only relative measures - we are more or less noble or talented than somebody else. This means that the intrinsic metric can change abruptly - for example when we meet for the first time people who are vastly more nobly born or talented than we are. Thirdly, even if we could achieve some robust intrinsic metric of our own worth, the inference that if we are fit and good we will do fit and good things, is not certain.

It is tempting to speculate that theories of self and action that are based on intrinsic metrics, rather than being extensionally based - "we are what we do" - are a feature of modern times and the modern psyche and likely to become more so - for example, the progressive educational strategy of attempting to inculcate intrinsic self-esteem is a possible example ; intrinsic genetic merit will be a possible metric in the future. On the other hand, the English and other class systems are a historical counter example, which show that these theories are nothing new.

(I should confess that I have recently read Richard Reeves very enjoyable biography of John Stuart Mill so am maybe "under the influence". Not that I started these theories of action consciously thinking about Mill, but I seemed to have ended up in his vicinity. Reeves describes Mill's life-long intense hostility to "intuitionist" philosophy. I am not sure I completely understand the meaning of "intuitionist" - but I suspect that those theories of self and action that are based on intrinsic metrics, rather than extrinsic canons of deeds done, are in some sense "intuitionist" theories. And a distinctive thing about Mill was his activism - as a journalist, politician, champion of womens' suffrage and many other causes - so that I am describing a theory of action and self which might be something like the one Mill himself lived by)

Saturday, June 7, 2008

theory of action 1

When you are fit and confident and strong, you have available a theory of action. The theory is maybe just something like - that you are fit and confident and strong, loved and loving, intelligent and kind etc, and the theory is a theory of all the actions available to such a person - and the theory allows you to act and do and be, without faltering.

When as happens from time to time you are somewhat more dented and ding-ed and knocked about a bit - perhaps really quite completely crumpled by that general flux of minor or major or sometimes completely illusory rejections ,disappointments and complete failures (...that flux that appears to increase in intensity with age, though perhaps this is just a failure of recollection), you often find yourself struggling to act because you make the mistake of thinking you need that good old red-blooded theory of action before you can you stall and freeze, furiously trying to panel-beat your theory of yourself and of action back into shape so that you can set off once more.....or in other words , trying to lift yourself by your bootstraps....

But the thing to remember is that - while your theory of yourself and of action is so easily compressible down to nothing by the flux- it is made of very insubstantial stuff - your actions are substantive and incompressible. Sure - if you help an old lady across the street while you are all depressed and theoryless, it feels wrong - or at least , profoundly unfulfilling - as though there is a zombie in charge. You have no theory of your action so you can't even be sure of your motives.....but the action is real and out there in time and space, it is good and cannot be compressed away by the flux.

The thing then is to force yourself to act , even when theoryless, even if it feels profoundly unfulfilling , as though there is nobody at home, just a zombie walking.

This is probably natural to many people. But to others (well , at least one other) it is surprisingly difficult - forcing yourself to act and do and be without your theories of yourself and of action intact, can feel like walking down a path with your eyes shut, or maybe more like, landing too fast and out of control down the runway, with the machine far ahead of the pilot in command.

But this is a way of feeling that you need to get used to