Saturday, August 15, 2009

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).


Amanda said...

You lose me Alan :-)
I challenge you to write a post containing zero numbers, or other mathematical notations. Please!

Alan said...

Hi Mandy - thanks for your comment and loyalty ! Guilty as charged. (Of extreme solipsistic obscurity and first degree murder of the English language). I'll try and improve. Mind you, you used some advanced mathematical notation yourself but cheated by using the word "zero" ! :-)