abstraction and structure April 5 2011

portuguese gaita again. today i’ll be continuing the subject of structure and quantification. this time, i will use our concept of thing directly in the calculation of structure.

we first saw that a thing of layer l $t_l = \{ t_{1_{l-1}}, \dots, t_{i_{l-1}}, \dots t_{N_{l-1}} \}$ , i.e., a thing is made of things. for each thing, its information (quantification of structure), is a function of the arrangement of its constituents, $I(t_l) = f(\{ t_{1_{l-1}}, \dots, t_{i_{l-1}}, \dots t_{N_{l-1}} \}) = -log_2(\frac{1}{N^N}) = N log_2(N)$ . note this is only true if there is no mutual information between the constituents. if there is, we can just conjure up a higher level “middle layer” that hides this mutual information as a single entity. this makes the structure of a thing of any layer quantifiable and independent of its constituents, only their quantity and arrangement. this means we do not need to know the information of constituents to know the information of a higher level system. this is a bit mind boggling, for sure, but this means our layers are irrelevant for the calculation of information, what matters is the quantity of sub-elements.

how is this relevant? this means that to properly quantify any structure, we can either quantify it in a single layer, just like the real world, or quantify it including explicitly all the information of the parts (as we, sentient beings, define them). if we choose to quantify and include the parts, we get $I(t_t) = I(t_l) + N I(t_{l-1})$ . infinite recursion, just as we would expect from a seemingly fractal definition. what does this lead to? let’s calculate it using induction (philosophers start panicking!).

$I(t_t) = I(t_l) + N_l I(t_{l-1}) = N_l log_2 (N_l) + N_l ( N_{l-1} log_2 ( N_{l-1} ) + I(t_{l-2}) )$

grouping,

$N_l ( log_2 (N_l) + ( N_{l-1} log_2 ( N_{l-1} ) + I(t_{l-2}) ) = N_l log_2(N_l) + N_l N_{l-1} log_2(N_{l-1}) + N_l N_{l-1} N_{l-2} log_2(N_{l-2}) + \dots$

we can now group this beast using sum and product operators. this will only disguise the big beast this is.

$I(t_t) = \sum_{i=0}^N \prod_{j=0}^i N_{l-j} log_2(N_{l-i})$

let’s hope nobody notices how big these numbers are. it’s obvious it is a divergent series and grows with N.

what this means is that we can expand or contract our “zoom” to define structure at any level and accounting for any layer whatsoever. we’ll need this in the future. for now, let’s let these concepts settle. when structure is quantified over a hierarchy as ours, the quantity always depends on the zoom chosen. this paradoxical result is well known from fractal mathematics. i guessed our structure was fractal, this is the proof. fractals are a consequence of abstraction. for nature, since she is abstraction-less, there is only one value for the structure of the universe: the value for the structure of the universe. nothing like ending a philosophical text with a tautology