Henley to Taieri Mouth

Happy New Year !

31/12/2009 New Years eve day was fine and warm if a little windy in Dunedin and I managed to negotiate and mount a most-of-the-family (16 y/o and over excepted) expedition, to walk from the end of Taieri Ferry Road towards Taieri mouth. They started off a bit reluctantly on the justifiable basis of many times bitten (by my various previous family trip schemes and paternalistically induced suffering) many more times shy - however in the end were very good company and we had a good day of it. There is a "millennium track" along this route - one of a number of these around the countryside, and its a very well maintained and enjoyable track to walk. Its about 9kms right through to Taieri mouth and the Pacific Ocean and me being me was keen to go all the way - groans and growls all around at this suggestion initially but once we got half way, to a fantastic picnic spot down by the river / estuary at a place called John Bull gully, my offer to walk back and drive the car around the long way to pick up the rest of them at the other end if they wanted to continue was taken up, with even moderate enthusiasm - the charm of this track on a nice day had done its job and generated a second wind in what is lets face it mostly a fairly sedentary and un-intrepid family !

(Watch the stinging nettle just before John Bull Gully ! - its an interesting looking plant which seems to appeal to children who then touch it and get a very big fright)



I'd been out that way a few years before with a couple of people from work, and two of my at that time pretty small children, on a morning's canoeing outing (aka paternalistically induced suffering) - however on that day it was too windy so we gave up, after a bit of nautical nonsense involving me in my canoe towing the two small children in theirs and everybody being slightly out of control. There is a bit of a wind-tunnel / venturi effect through this gorge I suspect. New Years day 1/1/2010 looked promising for having another go at canoeing this place, after enjoying walking it the day before (i.e. - to quickly capitalise on my no doubt very short lived credibility from that successful expedition.....and probably knowing me, thereby overdo it !) - fine and warm and although gales were forecast inland, I couldn't see much going on out the window on New Years Day out our way in Dunedin on the east coast, and as they were westerlies I figured the sea-breeze that would be trying to blow up the gorge from the Pacific as the morning wore on and the land heated up, might cancel out the situational westerlies.

It turned out a great day - we managed to fluke canoeing down the river , which is estuarine at this point, on an outgoing tide , my 11 y/o daughter Rosie in one canoe and I in the other. Our canoes are pretty basic - I've got an ancient one-seater fibreglass river (i.e. round-bottom, no keel or rudder) canoe I've had for years , and the other one is a lurid pink two-seat plastic canoe suitable for kids - i.e. quite wide and stable. But its amazing what you can stow away in even a small canoe (keys, camera, cell-phone, goggle and flippers in case we wanted to snorkel, picnic lunch, water bottles.....and still plenty of room). And how much speed you can make with these streamlined vessels. At this time of the year birds have been breeding - we spotted some ducklings and at one point cruised close by some large black-back seagull chicks - the chicks seemed almost the size of adult oyster-catchers. Forgot to time it but it can't have been much more than 1.5 hours, including calling in and disembarking for a picnic at good old John Bull gully from the day before, where we saw a family offloading huge amounts of picnic equipment from an outboard boat up from Taieri mouth.

After completing the trip, disembarking and climbing a hill to get cell-phone coverage at Taieri mouth, we (politely) requested backup from home - Andrew (9 y/o) was fairly keen to canoe back up the river with me, so his Mum Adele very kindly drove down from Dunedin (about 30 minutes by car) to pick up Rosie and deliver Andrew and a cup of tea. All the while, the tide had turned so Andrew and I were able to canoe up on a (according to somebody I was talking to ) very strong incoming tide - maybe it was a spring tide, I forgot to check. So we had a fairly easy trip back up-river, with me towing Andrew some of the way. On the return journey I noticed now the black-back gull nesting sites, which seem usually to be in the bush a metre or two above the water - marked out by one of the parents standing there and then taking off and squawking and dive-bombing if you loiter. Rosie and I had not noticed these on the way down - maybe because the tide was out and the parents were down on the mud-flats feeding or leading mostly-sedentary families on New Years Day family trips.

Its a surprisingly beautiful little stretch this - once you get into the gorge a bit, either on the track or by canoe, you could for all the world be on some river in the middle of the Brazilian rain forest.....though just up over the next ridge, its all sheep and dairy farms, commercial exotic (pinus radiata) forests, and towns. Oh and the jet skis and outboard boats that scream up and down there on a nice day give the true location away a bit.

The inland (Henley) end is a popular fishing spot, you can apparently catch ocean-going trout. I have a New Years resolution, to go down there after work the odd day while the days are still long and try to catch a fish. Easier resolved than done but I'm hoping I can make it happen.

actabolism in two Chambers

I was looking for a word to describe the sum total of the thoughts and actions of a person - over an hour or day, possibly week (but not a life or even a year) and thought of perhaps "actabolism", based on a mixed-up analogy with "metabolism" which Chambers defines (offline, 1976 - a big red hardback dictionary ) as "the sum total of chemical changes in living matter".

(And since I was in any case mooching about a way to interpret some meaning into all the slog of miscellaneous microscopic myopic thinking and doing as being maybe somewhat like that great complex of miscellaneous microscopic myopic biochemical metabolism - a rather disturbing mess up close but actually supporting a worthwhile homeostatic constancy if you stand back a bit. An intricate machine to provide constancy in the face of friction and adversity...so don't forget to dot the i's and cross the t's and put out the rubbish and file the weekly pointless management report because it all does have a point after all, its all part of the intricate metabolism of life - the worthwhile achievement of (more or less) homeostasis in the face of environmental, social, professional, psychological, familial and financial friction and adversity !)

Anyway - it turned out interesting to contrast that 1976 definition of "metabolism" - "the sum total of chemical changes in living matter" , with the slightly less poetic sounding online definition given by Chambers 33 years later in 2009 :

"sum of all the chemical reactions that occur within the cells of a living organism..."

So what has changed ? :

* "changes" has become the more technical and less agnostic "reactions"

* "sum total" , suggesting a complete enumeration of a collection, has become the more quantitative sounding "sum" - which vaguely suggests some sort of chemometric summation of reactions.

* ...and the lovely antique phrase "living matter" redolent of natural philosophy and the Victorians has become the more microscopically correct "cells of a living organism". (I wonder if the newer microscopically correct version is actually correct though...I would have thought a fair bit of metabolism occurs outside cells - e.g. in the action of salivary enzymes and gut acids on food....or maybe that is not counted as part of metabolism proper ?)

The other change in the online definition is that it goes on to add "...including both anabolism and catabolism of complex organic compounds", with these terms (anabolism and catabolism) hyperlinked as I have them. The offline version has both of these as separate dictionary entries but not of course inter-linked (and the entry for catabolism delegates to katabolism)

(I hadn't met these before but have met katabatic and anabatic winds, which are winds that blow down/up the slopes of mountains, as these cool or heat the air immediately above. Hence analogously katabolism breaks down compounds while anabolism builds them up. Nice words, I'm glad I met them).

Anyway getting back to "actabolism" - although Google finds it, its
clear it only makes it into their keyword index by virtue of being
an anagramatical typo' of "catabolism" - see for example
http://www.jipb.net/earticle_read.asp?id=4557 where both versions occur.

So I do actually have freedom to operate here - there is no such word as actabolism -but my derivation is not pretty and also wrong. I really (?) need to replace the "bolism" with something denoting a thought or action, not the "meta", as this prefix supplies (?) the "sum total" part of the concept. Or so I thought with my meagre etymological knowledge

This brings out another interesting contrast between the offline and online definitions of "metabolism". The offline version occurs within the context of an ordered list of terms appearing on the page, whereas the online definition occurs on its own without any context - true there are links to the related terms anabolism and catabolism, but links are completely different if not diametrically opposed, to context. The online definition occupies a whole web page - in fact it is really more fragmentary than that, just a web sentence.

In this case the ordering of the terms in the offline dictionary is quite interesting and informative - it is not strictly alphabetical, in fact when I first looked in the big red book for the word "metabolism" I couldn't find it because this word comes after "metacarpal" and "metacentre" which confused my tiny mind ! Well - thats because "metabolism" does not (apparently), despite appearances, derive from the "meta" root after all - and the offline Chambers collects all of the words derived from the "meta" root together, even if this upsets strict alphabetical ordering.

(This is explained in a note in the dictionary's preface section - "The Arrangement of Entries" :

"...Derivatives are not listed in crude alphabetical order but in a more logical form....etc etc")

Grumpy-old-luddite-bastard-time-out :

How much of that sort of informative and rich non-semantic context that you get from "Arranging the Entries", are we going to lose in the new strictly semantic electronic information infrastructure ?

Snap-value-judgement-time-out :

Are hyperlinks always all they are cracked up to be ? Is the current web too hyper' ? Can / will the web stay the way it is now or do we need / will there evolve some "Arrangement of Entries" ? Maybe it needs to become more of a "semantic manifold", with local contextual structure, density and continuity, and less of a porous semantic web, of tangled links between tiny fragments.

True , I was only looking at one of the "freebie" definitions - but even in a
full online entry - e.g. http://www.chambersreference.com/dict/external/site/main/quantity.htm (one of their marketing examples) - you still only get a single definition on a page - there is not the context of similar words to excite the pattern-finding and insight-forming parts of ones attention as you got in 1976. (Interestingly, the structure of that 2009 online page corresponds almost exactly to the design set out in "The Arrangement of Entries" note at the beginning of my big red dictionary of 33 years ago - even though there is no longer really any "arrangement of entries" needed, as there is just one per "page" !)

Well, after all that I still haven't settled on a word to denote , analogously with metabolism, the "sum total of the thoughts and actions of a person - over an hour or day, possibly week (but not a life or even a year)"

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.