Sunday, November 30, 2008

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" ( ) 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.