metaphysical algebra (2)

A previous blog suggested a metaphysical algebra for conceptually analysing emergence and the formula :

O = N x I

I = the relatively disordered input space of things, from which the emergence. A high dimensional space. Examples : 8 people around a table, a potential 8 way babel; 56 independent nucleons in a plasma ; several billion meaningles stimuli per second presenting themselves to our eyes, ears, taste, skin as we experience (say) a shower of rain.

O = the relatively ordered output space of emergent phenomena. A low dimensional space, viz : 1 polite conversation at the table rather than babel ; one iron atom condensing from 56 nucleons in the plasma ; one conscious subjective experience of a shower of rain.

N = that which transforms the input space into the output space. I suggested this be thought of as a space itself, with a negative dimension, and that some rough dimensional book-keeping can be done :

dim(O) ~ dim(N) + dim(I)

<=> dim(N) ~ dim(O) - dim(I)

So for example the book-keeping predicts that the thing that transforms the huge number of raw sense-stimuli elicited by a shower of rain into a simple subjective conscious experience of a shower of rain, will itself be a very complex space of high dimension. As indeed it is, being a brain with a dimensionality of perhaps 100 billion - the number of interconnected neurons it contains.

OK - so, the existence of an "output" space of low dimension is plausible. In each of the examples we can see the emergence of a coherent new level of reality, with fewer "things" (= lower dimension) than in the lower level, and exhibiting its own novel higher level laws of behaviour. For example iron atoms (once they have caught a few electrons from the plasma) have emergent behaviour such as chemical valence that is nowhere hinted at in the pedestrian lives of solitary protons and neutrons and the nuclear forces that govern their interactions.

And, OK - the existence of a distinct "lower level" "input" space of high dimension also seems plausible - for the iron atom example this is just the 56 independent nucleons in the plasma, a space with dimension at least 56. For the shower of rain it is all the billions of nerve signals elicited on our skin , eyes, ears, smell, taste by the physical effects on our bodies of the shower of rain.

But where does the space "N" come from ? Specifically - why call it a space and why assign it a negative dimension ? Perhaps there are other types of thing that N could be, that also transform a given space of high dimension into a resulting space of low dimension ?

Well it comes about mainly because I like the idea, and thinking about the idea, and seeing where it might go (which might be nowhere). Just maybe there is indeed some abstraction that can be done to find high level structure, in the daily scientific grind of explaining this and reducing that....explanation and scientific reduction as factorisation ? The formula implies an "ontologically opaque" view of things like iron atoms and consciousness - which is to say that, these things are to be regarded as new levels of coherent reality in their own right and they are not ontologically transparent - you can't just look in reductionist fashion through an irrelevant layer of iron atom behaviour and see nucleons, or an irrelevant layer of subjective experience of a rain shower and see neurons firing.

So far though that is at most merely proof-by-narcissism - "its so because I like it and it likes me". I suspect that's as far as I'll ever get - but maybe I can incrementally build a case and then one day do a summing up. Lets start with precedents - that's a good way to build a case in law.

My first precedent is the example of functional notation and spaces-of-functions in mathematics. Don't let that put you off - its very unlikely I know anything more about this than you do and in fact writing this little bit puts me in mind of finding out where this functional notation we all take for granted really came from. Since it is no less metaphysical and wacky in its own way than my "negative dimensional reducing spaces"

Take a few maths equations :

y = x + 2

y = x^2

y = x^2 + 4x + 4

y = sin(x) + cos(x) + 4x + 4

- these represent the daily calculational grind of multiplying this by that and adding it to the other to get some answer for some reason - and I would wildly guess that up until the beginning of the 19th century there was no higher level of abstraction than this. Though there was certainly an enormous level of skill in manipulating these equations.

But then somebody said "lets think of x^2 +4x +4 as consisting, actually , in a thing called a "function" operating on another thing called "x" - we will now represent the calculational rule x^2 + 4x +4 as being the formal product of a function, with x :

f(x) = x^2 + 4x + 4

and these "functions" will live in their own function space and we can take formal products of functions with other functions - in the above example , if

g(x) = x + 2

h(x) = x^2

f(x) = x^2 + 4x + 4

then f(x) = h(g(x)) so that f = hg

....standing back a bit - where on earth did that functional abstraction come from !? - its a huge innovation. You take an equation like

x^2 + 4x + 4

- which we can all understand as instructions to do some stuff on our calculators - and do an abstract factorisation of this into a thing called a function, "f", existing in some weird function space, acting on a thing called "x" , existing in a (less weird) space of numbers !? Its bizarre, only we are used to it so it seems bland.

The encouraging thing is that this apparently willful not to say whimsical piece of abstraction turned out to be very fruitful in mathematics in the long run.

(Which is not to say that my own wilful abstraction and factorisation of emergence and (soon) reductionism and scientific explanation and (after that ) certain physical processes, into formal products of spaces, and associated dimensional book-keeping, will be so productive).

An idea for a metaphysics of big and little emergences

Big Emergences :

• The emergence of life from an ambient "soup" of inanimate molecules.
• The emergence of consciousness from a cranial soup of neural networks and other brain parts that are not conscious
• Computational emergences - Cellular Automata :
  Examples : Life by John Horton Conway ; many examples in Stephen Wolfram's A new kind of science. The general idea is a simple algorithm, repeated application of which leads to totally unexpected "emergent" behaviours and patterns. There are well known examples from (computational) biology - along the lines of, simple postulated individual interactions from which emerge coherent behaviours of schools of fish, flocks of birds etc ; simple rules of growth and development which lead to emergent morphologies of coat patterns etc on animals.
• The emergence of a coherent regulated society from the (more or less) free participation of millions of individuals.

Both the big and computational emergences have a kind of implicit payload of rabbit-out-of-the-hat magic and rare privilege - events that you are unlikely to see very often and only in special staged situations, such as at the primordial birth of life, or of consciousness, or on running a computer program like Conway's Life or on occasional witness to the impressive synchrony of a flock of birds or school of fish. But they are (in my view) no different to everyday emergences...

Little Emergences :

• 8 people sitting at a table at morning tea discussing a single topic, one speaker is speaking. One topic and thread emerging from a complex of 8 different people.
• an iron atom : 56 protons and neutrons sitting at a table at morning tea discussing a single topic - how to be an iron atom. One coherent and very useful metal atom emerging from 56 unruly nuclear personalities.
• a well designed computer program interface, marshaling millions of pixels and bits of information into a simple coherent metaphorical world with which the user interacts.
  

Unifying Small, Large (and Computational) Emergences : Dimensional Reduction

My idea for a metaphysics of emergence, is a factorisation into

(1) an input factor space (I) of relatively high dimension

(2) an output product space (O) of relatively low dimension

(3) mediation of the transform from input space to output space by a complex (i.e. complicated) intermediate factor space (N) of negative dimension.

So that we may write formally as :

I x N = O

with

dim(I) + dim(N) = dim(O)

In words - an input space of very high dimension is formally composed with a negative dimensional space to yield an output space of low dimension - the dimension of the product space is the sum of the dimensions of the two factor spaces.

For a visual metaphor - we might think of the intermediate space as being like a lens, which transmits and transforms a coherent input scene into a transformed coherent output image. All three of the elements of this metaphor have themselves internally a coherent space-like structure, and with a formal composition of the lens space with the input-scene-space generating a coherent but transformed output product space.

Taking as an example a small emergence - conversation at morning tea - the input space has dimension 8, which is the dimension of the conversation space if all 8 individuals talked independently and simultaneously ; the output space has dimension 1 - the single coherent conversation that occurs. The dimensional reduction is mediated by a relatively intricate set of social mores, hierarchies, and behaviours - but which in this metaphysics we will notate and size as a negative dimensional space of a certain dimension - in this case -7.

A living creature enjoys a coherent high-level existence within a space that is quite distinct from the melee of chemical reactions that it is supported by. Furthermore by comparison with the lower substrate - it is a much simpler, lower dimensional space. High level laws of behaviour, nutrition, reproduction operate at this level. Just as in conscious life, our brains build for us an experience of a seamless space of streets, trees, flowers, colours, selfhood....that is both quite distinct from the substrate of our brain and the actual physical activity of the external world, and also vastly simpler.

But what is the point of introducing the mathematical fictions - the negative dimensional space , and the abstract composition of spaces ?

The main idea is that this analysis is philosophically useful - but also with a hope for empirical potential in that it suggests a way to predict or measure the dimensional size of these internal engines of emergence. As a philosophical guide, the analysis recommends that, where we see simple ("low dimensional") behaviour or structures emerging from a complex ("high dimensional") input substrate, then we should expect that this emergence will always be mediated by a complex intermediate structure,with a (negative) dimensional "size" almost as high as that of the input substrate.

And as an engineering guide, the analysis recommends that (for example), where we wish to engineer software interfaces or societies that are simple and coherent, we should in general expect that the internal engines of organisation that generate these interfaces and societies will be complex and messy. And vice versa we should perhaps be wary of simple and elegant internal engineering data and object models and political ideologies - according to the metaphysics-of-emergence formula, this internal elegance will be achieved at the cost of emergent societies and software interfaces that are less coherent and more complex.

So what does the complicated intermediate negative dimensional space N look like ? It is no more possible to visualise a negative dimensional space than it is to visualise other mathematical inventions such as negative numbers and potential energies : but we can say that it encapsulates a forest of dimension-reduction-engine-room internals that actually get the job of dimensional reduction done...constraints, relationships, surfaces, volumes, intersections, knottings and braiding of dimensions, dynamic censoring and suppression of dimensions...that our new metaphysics parlays into a space-like structure.

Philosophically this analysis is in contradiction to the cellular automata view - it encourages us to look for complex machinery underlying simple emergent phenomena, rather than simple machinery. (Obviously this particular antagonism will require some explication in view of the long history and many examples and proponents of cellular automata explanations of emergent phenomena).

Freeman Dyson wrote an essay entitled "Why is Life So Complicated". He meant, why is the machinery of life so complicated. He writes "It seems to be true, both in the world of cellular chemistry and in the world of ecology, that homeostatic mechanisms have a tendency to become complicated rather than simple"

My answer would be - the machinery of life is complicated because

(1) the non-living world (I) with which it interacts is complicated : dim(I) is large

(2) life itself (O) is - almost tautologically - simple : dim(O) is low

(3) the machinery of life (N) is therefore mathematically required by the metaphysical emergence formula to be complex since

dim(N) = dim(O) - dim (I)

Life is complicated because there is a higher level metaphysics governing emergence, and for the same reasons that societies are complicated, the laws of nuclear physics that make iron atoms possible are complicated etc.

The simplicity of life itself - as opposed to the machinery that supports life - is perhaps the key part of this equation. One of the striking things about a living creature is its coherence and unity of purpose - how so many intricate parts mesh into a coherent whole that....swims, runs, searches for food, mates....has a being. But this is another way of saying that life itself - the end product of all of the machinery - exists in a rather low dimensional space. Indeed one way of explicating "being" is that it is a space of dimension 1 - this is the dimension of the thread of existence that we envisage for ourselves, stretching linearly back into the past and into the future.

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 !)