Reciprocal Spaces and Negative Dimensionality

The logarithm function (among other things) measures (approximately) the lengths of numbers expressed in our standard place-value notation - for example (base 10 logs rounded to the nearest whole number) :

4 : length = 1 log = 1
12 : length = 2 log = 1
90 : length = 2 log = 2
100 : length = 3 log = 2
150 : length = 3 log = 2
446 : length = 3 log = 3
.
.
165462454 : length = 9 log = 8


( ....this off the reciprocal space / negative dimensionality topic but....I wonder what are the bounds on the lengths of the names that can be given to numbers, under different possible naming schemes.

There is the trivial naming scheme whereby the names of numbers are the same length as the numbers themselves, so that the length of the name of a number increases identically and linearly as the number it names :

* = 1
** = 2
*** = 3
etc

And easy to construct names whose length increases as, say, the square of a given number :

* = 1

**
** = 2

***
***
*** = 3
etc

- but what about the more useful schemes, whereby the length of the name of a number increases sub-linearly ?

Roman Numerals can be more compact

C = 100
M = 1000

....or less

XXXIV = 34

....but the Roman system is "non deterministic" - yes it is able to compress orders of magnitude to a single symbol as with C and M , but there is no automatic compact Roman symbol for 1,000,000 unless and until we intervene and assign one.

So then a question - is there a deterministic naming system for integers, under which the length of the name of a number N increases at less than log(N) ? )

(see for interest the Berry paradox relating to lengths of names of numbers - http://en.wikipedia.org/wiki/Berry_paradox )


But to return to the topic - there are some parallels between the log function and the concept of dimensionality.

• the dimensionality of a space gives the length of the "names" of points in that space, in a similar way in which the log of an integer gives the length of the name of that integer. For example points on a 2-D plain have names like (1,3) , of length 2 ; points in a 3-D space have names like (3,4,8) , of length 3 etc
  
  
• When we multiply numbers we add their logarithms ; and when we "multiply" spaces - i.e. take a cross product (aka cartesian product, direct product) - we add their dimensions. So for example, the dimension of the Cartesian plane is two , the sum of the dimensionality of the x and y axes that are "multiplied together" to create the plane.

We can use this parallel to motivate an interpretation of negative dimensionality, by considering what is the dimensional analog of a negative logarithm, via these correspondences.

Negative logarithms are obtained from positive reciprocal numbers.

Log(10) = 1
Log(1/10) = -1

This is so since the log of 1 is zero , and we must have

log(10) + log(1/10) = log ( 10 * 1/10) = log (1) = 0

This might suggest that :

• a space with negative dimensionality is in some sense perhaps a "small" space , just as a number with a negative logarithm is a smallish number
  
  
• a space S with negative dimensionality -D is in some sense a reciprocal space in that , if T is another space with dimension +D , it seems that we should have, operationally, and as with the logarithm analogy :
  
  dim ( S X T )
  = dim(S) + dim(T)
  = D - D
  = 0
  

So without further ado (there is far too much ado in this blog as it is ! ) I will take this as my working concept of a negative dimensional space : I will call it a reciprocal space , meaning that on cross multiplication with a positive dimensional space it yields (in some strange black-box way yet to be specified) a product space whose dimension is the dimension of the positive space, less the negative dimension of the reciprocal space - and if these dimensions are equal, the product space is a zero dimensional point.

The obvious fact that we cannot yet specify what actually goes on in this multiplication, and that we cannot visualise what a negative dimensional reciprocal space looks like, need not worry us for the time being. It is similarly impossible to visualise a negative length (and I do not count the metaphor of oppositely directed lengths as such a visualisation - this is just a model of a negative number (see also Roger Penrose on negative numbers, page 65 in "The Road to Reality") - yet we can still discover how negative numbers should behave, and find a use for them. (I would argue that fractional numbers are equally abstract and it is in fact impossible to visualise 1/2, but that is for another blog !)

So far this is all pretty harmless. Next blog will leave planet earth entirely and start to see reciprocal spaces everywhere - the brain as a reciprocal space ; consciousness as the lower dimensional product of this reciprocal space, with the very high dimensional space of the flux of experience and sensation - thus "explaining" certain aspects of our conscious experience. And making a few predictions though probably not testable.

(That will be pretty harmless as well , apart from the small carbon footprint made by the disk space used up in the post)

(I am dimly aware of previous characterisations of negative dimensional spaces - there is a fractal one from Mandelbrot I briefly encountered - but its good to just go Sunday driving without a map with these things sometimes - its of course impossible you will actually discover part of the countryside that hasn't been mapped already (you have to be an intellectual mountaineer for that , which I am not), but you might get to see some interesting places you would not otherwise have seen had you been better prepared ! )

(And a "reciprocal space" in crystallography is a Fourier transform - I do not mean anything like a Fourier tansform in my use of this term however)

The sea the sand the wind and your foot meet at two points (the dimensionality of paddling is -4)

How many different things can meet in one place ?

...I found myself wondering, brain no doubt starved of oxygen, three quarters of the way up the long haul by foot and bike from the city (Dunedin) to Roslyn, so I could enjoy the ride along Highgate and zoom down to home in Mornington. For some reason I was quite excited by the question and have often idled around it since then - that was about three years ago I think !

So for an example, the sea and the air meet at the ocean surface ; the earth and the air meet on the dry ground surface ; the earth and the sea meet on the sea-bed surface - with each intersection we lose a dimension, so volumes (3D) of sea, air and earth intersect at surfaces (2D).

And since surfaces (2D) intersect along lines (1D) , so then these three vast volumes of sea, earth and atmosphere all finally intersect together along a single meandering thin line - the tide-line of the sea on the sand - on an outgoing tide on a gently sloping sandy beach you can see the linear traces of this final 1 dimensional intersection of these three vast volumes contouring along the beach, marking the successively retreating limit of the pour of each wave up the sand.

So if you paddle your foot in the tide , half in the water and half out, you can add your foot to this grand intersection of earth, sea and sky and reduce the dimensionality of the final intersection to zero - two distinct points where earth, sea, sky and your foot all meet - there's one point on each side of your foot, down near the sole where the edge of the sea meets the sand meets the air in a long line to thread your foot. (Points having dimensionality zero)

Now this concept of how many things can meet in one place lacks (among other things) a clear definition , so I wanted to give it a name in the hope that a groove of clearer meaning might be worn down by usage. I decided to avoid a derivation from words like convergence, confluence, intersection etc , because these have a spatial / geometric sense, when in some cases the "one place" and the "many things" are not going to be particularly spatial. "Cardinality" is a word that can mean "how many" , but is also extensible to more abstract senses of the size of a collection, so I started with this word. And since in some cases in the animate world, the number of things that can be brought into one place is very large - they "crowd-in", I decided on the term "crowdinality" - an as yet unclaimed term, according to Google.

As well as the crowdinality of paddling, I also wanted to mention the crowdinality of puns (usually 2) ; the crowdinality of maps (4, by the 4 colour theorem ?); the crowdinality of stories and movies (the higher the crowdinality the better the story. The movie "O brother where art thou" has a high crowdinality on many levels (most of which I completely missed until I read the wiki page !)) ; crowdinality as an explication of consciousness (more later); crowdinality as one of the hallmarks and prerequisites of creativity (more later); the crowdinality of computers (low - around 2 ) as compared with brains (high - in the hundreds of thousands if not millions) ; the crowdinality of the nucleus of an iron atom at the center of a big old star (usually 56 , i.e. its the number of protons and neutrons crowded together and spending life as a single nuclear unit, and the most commom isotope of Fe has 26 protons and 30 neutrons); the crowdinality of a neutron ( 3 , due to its mutually intersected 3 quarks , 1 Up and 2 Down ) ; the crowdinality of a proton (also 3 quarks , 2 up and 1 down); the crowdinality of a scientific paper (the higher the better. "An Alternative Menaquinone Biosynthetic Pathway Operating in Microorganisms" , Tomoshige Hiratsuka et al. was a beauty I came across recently. I am a very lowly bioinformatics foot soldier's foot soldier by trade and I loved the intersection in one study of a bit of bioinformatics with a whole lot of other threads to yield a genuine new discovery ); the crowdinality of a sentence - rather low from the viewpoint of logic formalisms, which considers only the syntax and semantics of the symbols in the sentence - but very high according to recent alternative analyses , such as provided by "situation theory", which explicitly introduce into the analysis the context within which language is conducted (for some examples - including how this type of analysis resolves the famous and ancient Liar paradox - see "Goodby Descartes", by Keith Devlin); the high crowdinality of molecular complexes such as spliceosomes and signalling cascades in the world of molecular biology; the vast crowdinality of a richly synapsed neuron inside a brain ; crowdinality as an approach to the explanation of emergence.

Regarding creativity - A Gardiner in "The Princeton Companion to Mathematics" describes the "delight in a double-edged strategy, which points in two directions at once...[and]...has much in common with the pleasures we derive from....puns and double entendres". Gardiner goes on to describe how Koestler showed how scientific and literary creativity often flows from the identification and exploitation of "double meanings with a built-in tension". Koestler called them bisociations.

In his book "The Space Between Our Ears : How the brain represents visual space " , Michael Morgan has the picture "The Death of Marat" (http://en.wikipedia.org/wiki/The_Death_of_Marat ) with the tart caption "The only writer on consciousness that got what he deserved". (The writing referred to is Marat's "Philosophical Essay on Man (1772), in which he apparently theorises about the mind). Nice shot.

That said, point taken and duly cautioned and all that, I do have a distinct phenomenological vision of consciousness as consisting in the topmost teetering single neurological summit point , of the highest peak in a vast cerebral mountain range of intricately wavering peaks, each peak the final intersect of a huge cast of buttressing slopes of supporting neurological modules, memories , current sensations, that intersect in ascending ridges , cols and cirques of semi-thought, which in turn finally conspire in a single zero dimensional point of maximum crowdinality. (There you go I've done it , and I'll do some more , knife me pink and call me Marat !). Well actually, considering the time dimension, lets call that a one-dimensional peak of maximum crowdinality, the stream of consciousness.

(And there seemed to be some useful predictions to be made from this view, such as that other animals will lead a conscious life of some richness, differing in degree (crowdinality) but not in kind from our human kind ; that in our conscious life, which consists essentially of an intersection point, the higher the crowdinality of that point the richer will be our experience - the more and wider learning and engagement and current passing conversation and sensation we bring to that intersection point, the higher will be the peak - the prediction is that techniques like meditation and others involving the removal of stimuli actually lead to a lower rather than higher level of consciousness. Not that that lower level of consciousness is necessarily unpleasant or unworthy of pursuit for its restorative power - just that it is not in itself deeper or more meaningful than a more engaged and busy level of consciousness)

But the new term - "crowdinality"- had a few problems. Firstly it was ugly; secondly it offended ontological parsimony which should always be respected both in thought and prose style - in other words, preferably, invent no new things either deliberately, or accidentally via long winded reifications ; and finally - the new term has itself low crowdinality - there are not enough different ideas meeting in this one place to justify the creation of this new word.

In order to increase the crowdinality of the concept of crowdinality, and also to remove the need for a totally new word, I decided to attempt to intersect this concept with another idea I have idled and addled over from time to time - the idea of negative dimension. The claim will be that the arena of subjective experience is an example of a negative dimensional space, and that here is the source of the conceptual difficulties we have when trying to understand consciousness and subjective experience using analytic tools and ideas based, as they are, on positive dimensional mathematical spaces.

So now the phenomenological vision of consciousness is similar but inverted by the negative dimensional space interpretation - it is the bottom-most gurgling neurological gully trap, of the lowest ravine in a vastly deeply dissected cerebral canyon of intricately carved and banded gorges, each ravine the final lowest intersecting foot of a huge cast of ascending slopes of supervening neurological modules, memories , current sensations, that intersect in descending scallops, anti-cols and anti-cirques of demi-thought, which in turn finally conspire in a precipitous tomo of maximum negative dimensionality !

I'll provide an attempt at a supporting characterisation of negative dimensional spaces in the next blog.

Its Good To Try New Things - Hormesis And The Advice Theorem

The advice theorem says the following :

"All advice is good advice, because there is some course of action , between the advised course and its complete opposite, which must with mathematical certainty lead to the best possible outcome".

An imaginary example shows how application of the advice theorem could save your life.

Suppose you are suffering from scurvy because you are eating only trace fruit and veg and are completely ignorant of the requirement for dietary vitamin C, and a well meaning but (as) ignorant friend advises you that your ill health is caused by the presence in your diet of small amounts of fruit and veg - you will be fine, your friend advises, if you switch to a diet consisting solely of corned beef from a tin.

Luckily you are in possession of the advice theorem. Applying the theorem to the advice you have received, you appreciate that whatever the merits of this advice, the fact is that the best possible outcome will be achieved with a diet somewhere between all corned beef, and the complete opposite of that - say, all fruit and veg - i.e. somewhere along the new dietary axis implied by your friend's advice.

Unfortunately the theorem is unable to help with the problem of choosing a point along that axis - but commonsense suggests that the optimum point is more likely to be an interior one, rather than at either end - there are vastly more interior points than there are boundary points (just two) - so rather than a diet of all corned beef or all fruit and veg, you decide to introduce a moderate amount of fruit and veg into your diet so as to operate at an interior point of the new advisory axis, rather than at the "all corned beef" end point suggested by your friend. Within days your health is improving - your friend's incorrect advice, moderated by the advice theorem, has saved your life.

The only prerequisites for application of the advice theorem, are that :

1. You are able to define one or more axes of action implied by the advice - so that you can identify the two extremes within which both the actual course of action you take, and the optimum outcome, must occur - i.e. you can identify a course of action which is in some sense the complete opposite of the one advised. Obviously there will generally be no unique "completely opposite" course of action - but it doesn't matter , there will be some optimum point on whatever axis you choose. Of course , some axes will be more productive than others - but any axis you choose must contain some course of action which will result in some zero or greater improvement to your current situation.

2. You are able to rank at least notionally the outcomes of actions on a numeric scale such that the concept of a maximum is meaningful.

Provided these two conditions are met, then the advice theorem may be pictured as a graph of outcomes, with the vertical axis being how good the outcome, and the horizontal axis being the course of action taken intermediate between that recommended and its opposite.

Then - if for example the graph is a horizontal straight line, it does not matter which course you take. And there will be some graphs where one of the end points *is* the best outcome. And some with a hump - the optimum in the middle , and some a wiggly line and the optimum is just somewhere along there. However - for all possible graphs , it is the case that at least one of the courses of action along the axis *must* achieve the maximum possible outcome.

I discovered the theorem late last year, while driving back from Omarama to Dunedin after a weekend away with the kids in a tent, and swimming in the Ahuriri river and having a look at the World Gliding Grand Prix. As I drove back down towards Kurow, I reflected somewhat soberly, amid that somewhat sober landscape, on a bunch of advice I had dished out to a colleague a week before, and wondered whether in fact the complete opposite of my recommendations might not be the best course.

Then it hit me - whatever the true situation, my advice had at least some value in that it created for my colleague a new axis of action - consisting of all courses of action between what I recommended and its complete opposite, and that somewhere along that axis there must surely be a point which would achieve the best possible outcome. On my return to Dunedin I communicated by email the exciting news to my colleague - my advice could be shown mathematically to guarantee the best possible outcome....though it may need a bit of titration to find the optimal point, between following it to the letter, and doing the complete opposite. (Shortly afterwards - on Boxing day actually - my entire family came down with Salmonella, which made for a miserable Christmas and New Year. )

I was just reading an interesting article about something called "hormesis" in the New Scientist magazine today (9 August issue). I experienced an odd sensation of anti-deja-vu....I have never seen this before ! Which is indeed odd for such an apparently basic idea. So - this *is* homeopathy , right, under a different name ? (Not that I have anything against homeopathy - I learned from Mandy's blog that she consults a homeopath, and she's a really clever bastard so it can't be complete bollocks !)

But also - I *have* seen this before - its nothing other than my advice theorem : almost anything is good for you , its just the dose that you have to get right - but that is at least in part a simple mathematical tautology, rather than being biologically meaningful.

So my advice is - always apply the advice theorem to any advice you receive. (Unfortunately this leads to a still to be resolved paradox - "the advice paradox" : should we apply the advice theorem to the advice to always apply the advice theorem ? If we choose not to apply the advice theorem to this advice, then this implies we accept without qualification the advice to always apply the advice theorem, which contradicts the assumption that we did not apply it. This suggests that it is impossible not to apply the advice theorem to this advice - yet in that case it is impossible to apply the theorem, which requires us to be able to not apply the theorem)

But....pointless paradoxes aside - the advice theorem is a wonderfully liberating thing for advice-giving busy-bodies like myself - we can go forth and dish out our hot air with promiscuous abandon - just so long as we also hand out the antidote - a pamphlet describing the advice theorem (with suitable warnings not to try applying the theorem to advice relating to the theorem itself as serious injury may result)

And we should always every minute of our lives try to find novel axes of action and titrate our way up to the optimal point along them. Since - its a simple mathematical fact that its good to try new things.

On Being Bent and Dissipated (1)

Saturday in Dunedin was cold foggy and drizzling, pretty depressing, so went out to the end of the Otago peninsula with 10 y/o daughter - who always drags me out there (I always end up glad she did) - and 16 y/o son (intellectually disabled, and an eating machine). We lease a crib.

Saturday in Otakou was also cold, foggy and drizzling - actually even worse as always is out there in any weather that's vaguely northerly or easterly. But as I mentioned - we were at the *end* of the Otago peninsula, so there was nowhere further to go so we stayed there.

Sunday (today) was a bit better - not much sun , but no wind at least and fog and drizzle lifted. We did some fishing (no bites) and mucked around and it sort of got me out of my rut as it always does , even though you don't think its going to before you set out - there's nothing all that flash out there that's going to explode you out of your rut, but still it somehow eases you out by next morning. So I'm always glad she drags me out.

And there's always some micro-interesting thing out there if you keep your eyes open. Like today I was thinking about how the motorbike zooming up and down the Aramoana beach - miles away over the other side of the harbour - sounded so close , as though up and down the road outside, and thinking how I'd noticed that effect the odd time before, and how what it probably is is : maybe it just happens when the harbour is quite glassy as today - smooth enough so that the sound waves are reflected coherently off the water surface - so the sound energy is only attenuating in our direction at half the rate that it normally does (though still as the second power , but with a halved constant of proportionality) - and how therefore the effect will not occur when the sea is rough as the complicated surface will just dissipate the energy incoherently.....but when the sea is rough there is also a wind so that would obscure both the effect and one's reasoning about it....I'll have to test this theory out some more.

(So tonight I asked Google to explain this to me , and there is an article about sound carrying over lakes (http://www.mnresponsiblerec.org/previoussite/resources/sound.htm ) that suggests, rather , that the main reason is the bending of sound waves by a temperature inversion. Now, it so happens that there was indeed a temperature inversion today - you could clearly sea the smoke from chimneys collecting under a layer about 300 or so feet high. Sound travels faster in the warmer air above an inversion - and this would indeed bend sound back down - as the wave fronts angle obliquely into the inversion, the top of the waves hit the warmer layer first and are sped up - so the entire wave is steered downwards a bit - like the way the outer wheel in a turn has to go faster than the inner wheel, and turns the whole system towards the slower wheel.

Of course any time the sea is smooth, then there is no wind which is rather conducive to temperature layers developing so the possible causes confound each other.

And it seems like both could be correct since they are both variations on the same theme - refraction/reflection of sound when entering a less/more dense medium - perhaps the distinctive effect today was because both were in operation - canyoning the sound out through a layer between the water surface and the inversion, so that the rate of attenuation was even less then half, since the energy was only spreading out in two dimensions rather than three - so that in fact the rate of attenuation would have been more nearly proportional to the inverse first power of distance, rather than the inverse second power of distance.

Some experimentation is needed I guess - electronic send and receive over the surface of a pond, and the same distance apart over grass, and measure it.

But then later I thought of another couple of explanations. Firstly - there are some steep cliffs behind Aramoana so there was probably some echoing of the sound back off those and out over the harbour to Otakou.

Then secondly I got to thinking about the possibility of a wake effect - though really this should only happen (I think) if the motorbike is going faster than the speed of sound ! You drop a pebble in a pond and the waves spread out and attenuate as they go , because the energy is tansferred from a 0-D point, to the 1-D concentric growing circular wave front, and so a fixed amount of energy transferred to an ever growing circle, means the energy density - i.e. the wave - must attenuate. But say now you have a boat speeding through the pond - this is like dropping "continous" pebbles in a straight line. Now the energy is being transferred from a 1-D line (the speeding boat) to the a 1-D line - the wake on each side - a wake is a straight travelling wave. These wake waves do not need to attenuate on energy density grounds , since energy is being transferred from a 1-D line - the boat (or continously dropping pebble) , to 2 other 1-D lines of the same measure - the wakes (though will attenuate due to friction effects).

(I remember flying into Sydney once , looking down while still over the sea and being intrigued by these two long straight ribboned wakes left behind by a ship (or ferry or launch , I can't remember) - that seemed to just persist, travelling on over the sea long after the boat had passed. I say "ribboned" because, although they were straight, they were made of a series of sub-wavelets like
\
\
\
\
)

(I guess you can construct a wake mathematically by superposing the continuously infinite series of growing circular wave fronts generated by the passage of the boat - I guess interference effects result in the linear wake).

So I got to thinking - is there maybe a similar wake effect when you have sound coming from something like a motorbike speeding in a straight line along a beach like that ? - the bike is not fast enough to have a trailing wake (thats I guess what a sonic boom is !) - but maybe the energy is still transferred via a linear sound wave front that does not attenuate due to energy density considerations - as would be the case for a noise source at a single point in time.

So if you're ever out at Otakou and the harbour is glassy smooth - which is quite often - and can be all day in winter but usually only in the morning in summer - see if you notice the sound carrying ! - and see if its getting quieter as the second power of distance or as the first power of distance, or some power in between ! And how it is affected by whether the noise source is either continous and moving, or a single point in time and stationary.

Reading an interesting book by Anatole Abragam , a physicist , "Time Reversal" , at the moment - an autobiography, was born in 1914 and came of age during the second world war (I think he is still alive)....so that kind of got me in the mood for thinking about things like that. He has some very wise things to say about life as well as science. On the other hand, he's a bit of a sexist (I realised on second thoughts), and it got a bit boring towards the end and a bit too much name dropping and general bragging. But heck who am I to criticise !

(How do we know of a statement about life that it is a wise statement !? If we already knew it to be true it is unlikely to strike us as wise as we already knew it. On the other hand it must be something we believe to be true - for a statement to be wise , it must be true - or at least , we must believe it to be true.

So - on what evidence do we judge the truth or otherwise of a statement purporting to be a wise statement ? - since these generally do not come with any lengthy appendices offering empirical data as evidence.

I think a statement that strikes us as wise, is probably usually a statement that we have accumulated some evidence for ourselves , but which we have not yet formulated
any statement about - so that when we see the statement written down we are immediately able to corroborate it by referring to our own experience and memories.

This implies it is impossible to recognise wise statements about life as being such, until we have accumulated sufficient experience to be able to corroborate them as being true - until then , they will probably mean little to us as we will not have any rational basis on which to judge their truth value.

This does seem about how things go - and that it is perfectly rational for youth to ignore the wisdom of age. And also that , unfortunately, wise statements about life are often of little use - since we can only evaluate their wisdom when it is too late. Again - this does seem to be how things go)

Cybermen, Daleks, Terminators and other upgrades

We've had a big dumping of snow down here in the deep south - though had to drive up a few suburbs to have a go on the sled as the snow didn't quite manage to stay around on the ground around us like it sometimes has in other years. (This trip proved that alas petrol is *still* not expensive enough - we were in fear of our lives walking anywhere near the road due to boy-racers hooning up and down using their cars like motorised snow-boards ! )

My six year old is keen on Dr Who so the circle is complete - I must have been 6-ish , or perhaps a bit earlier when I was hiding behind a piece of furniture, absolutely terrified of the grey-scale (no colour TV then) Doctor and whatever he was up against - I only recall the Daleks, the Cybermen came later. I think it was the original old white-haired guy, long before the modern, post-modern and whatever-period-we-are-in-now Doctors - not to mention, long before the celebrity side-kicks (though alas Billy Piper appears to have departed into a parallel universe this week.) (I don't think it would have been a couch I was hiding behind, I don't think we had them then - and if we did it would have been called a sofa ! Like showers - didn't have those either, even in a new house that we built, I think in the late sixties, on the back of the then (alas no longer quite so) prosperous business of growing wool, lamb and mutton (we were sheep farmers) - we only had baths.....which come to think of it seems odd, given that at the end of the day we were often pretty filthy, yet often had to re-use bath water because baths use water so inefficiently and we were on rain-water supply..... - my guess is that pump, pipe, nozzle and water-heater technology and bathroom materials were not quite up to squirting warm water out over people in a pleasing fashion while yet not flooding the bathroom. (In fact I think there is still some progress to be made in this whole area - particularly that not-flooding-the-bathroom bit). This might have even been before alkathene piping - maybe getting copper pipe up and around a few bends and over our heads would have been too much of a mission ! Yikes ! Note to self - check out chemistry and history of black alkathene piping....am I really that old !?)

While my 6 year old is himself a little insecure at times with the Doctor and does sometimes disappear out of the room or come and sit on my lap when scared, he is clearly not as terrified as I was in my day, and generally far more sophisticated in his world view. He is, for example, able to observe that the Daleks can't even go up stairs, which seems odd for a race with their fearsome reputation. Its not something that ever occurred to me - and I'm not 100% sure that this is an original observation of his either - he has older siblings and also just generally lives in a much richer more sophisticated cultural environment than I did out in the back blocks of Matira. But still - its a testament to the originality of the ideas of the show and that wonderful Dr Who theme music (which 6-year old and I often sing loudly together as a duet much to the annoyance and disgust of the rest of the family- he does the high spooky weeeee-woooo bit, I do that menacing thrumming rhythmic bit in the bass) - that this little show with budget special effects is able to clear the room of 6 year olds on occasions - something that not even any of the Terminator movies was ever able to do !

There's actually an interesting contrast between the Dr Who dystopians - OK, scary dudes - and more recent ones like the terminators - which is that, the ones on Dr Who tend to be meat wrapped in metal, whereas the modern ones - like the terminators - are metal wrapped in meat. I can still remember the shock when one of the previous Doctors somehow opened up a Dalek and we got to see the pathetic little mutant that lived inside it. And when you got to see the first tantalising (and I think at that point unexpected, though I forget now) bits of robot under Arnie's skin, and realised he was metal wrapped in meat, in the first terminator movie - it was a similar kind of mild shock.

Well firstly - I guess there is something about "peeling back the layers" , that is extremely suggestive and metaphorical and all that, and that really works well as a dramatic and storytelling device. Consider for example how bland , by comparison , were the metal-wrapped-in-metal robots in something like "I Robot" ,
or even "AI" - unless there are layers hinted at, there is little dramatic potential - in any drama, not just sci-fi and cyborgs and all that, you need characters that only reveal part of themselves to start with, but hint at (and deliver) more later, as layers are peeled down to. Though the completely meat-less robots in the "Aliens" series of movies are a slight counter-example - they were pretty cool ! - but once again, still worked by appearing to be one thing -then shocking you when you discover, is something else. And I think I'd class those as metal-wrapped-in-meat anyway - even though technically I guess they were plastic-and-white-goo-wrapped-in-latex ! Dramatically and conceptually, they were metal-wrapped-in-meat - robots that appear human and organic.

Its interesting to speculate about the inside-outness of the modern cyborgs as compared with the older ones - why it is that metal-wrapped-in-meat has taken over from meat-wrapped-in-metal. Maybe material for a future blog ! It may just be that our technology - both "meat related" (genetics , cell cultures - flesh-in-a-dish...) , and "metal related" (micro-devices, quite advanced embedded smarts - though we are still miles from anything approaching AI - at least, AFAIK !) - has advanced to the point where the metal-wrapped-in-meat is more conceivable and credible.

The other meat-in-metal cyborgs on Dr Who I am familiar with are the "cybermen" - though it is only in this latest series that I saw their genesis and process of manufacture - which is , that you take a stock-standard human and, in a lurid blood-spattering process called "upgrading" , extract his or her brain and stick inside a metal suit ! (I am sure that the pun on software "upgrades" , and the similar level of unpleasantness and buggering of one's brain when you do one of these as compared with being upgraded to being a cyberman , is deliberate !).

One last thing though is.....we are in fact being cyber-ishly upgraded even as we speak, but in a more subtle way. The English language is being "upgraded" at a rather frighteningly fast rate , by cell-phone technology. Consider this extract of written English from a young 20-something :

"Sup? im laxin chch for a bit b4 a little travel n amped for the snow. Listenin 2 da top40"

Which translates roughly as ....

"Hows it going - what are you up to ? I am relaxing for awhile before doing some travelling - keen to do some skiing when the snow arrives. Listening to the top 40 music show at the moment"

....and you get the feeling when you see the way groups of people are half in conversation with those present and half with other people via text - that there is maybe a change to a somewhat different, more collective consciousness going on , different to the isolated awareness we have known.....its possibly not too far from the truth to describe a gaggle of 14 y/o teenagers + cell phones as a single cybernetic organism !

....and of course the incredible access to knowledge that we now have - this futuristic network we take for granted that seems like it arrived out of nowhere, that we can address plain language queries to on almost any subject and get answers - this also is part of the "upgrade" each of us is receiving.

(....though the extent to which we were ever isolated awareness's, as opposed to brains always talking to other brains across time and space in a very complex and intrinsically social way, has I think been grossly over-stated - we got led astray by the French school (Descartes, JP Sartre) , not to mention a number of Germans, Dutch....actually it may be more of a historical time period, a fad we went through - though does seem to be a particularly continental Europe development. It was a possibly a very big, perhaps fatal mistake....more thought needed !)

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.

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 liberating...it 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)

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 act.......so 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