A couple of days ago, I mentioned to CodeSlinger that one of my sons was doing research in the branch of Mathematics known as ‘Combinatorics‘. His response was not only informative, it was just as passionate as my son gets when he talks about the subject.
So, for your pleasure and elucidation, here is CodeSlinger’s commentary on Combinatorics:
Combinatorics… the art of counting. Hah. Sounds trivial. But it is slowly becoming clear that combinatorics lies at the root of everything.
The fundamental equations of physics are symmetrical in time – if we watch a movie of two particles coming in from infinity, bouncing off each other, and proceeding back towards infinity, we have no way to determine whether or not we are watching it backwards. Yet a movie in which a vase falls from the table and shatters on the floor is easily distinguishable from the time-reversed version, in which a myriad of shards come flying together, assemble themselves into a vase, and jump up onto the table!
The difference, of course, is that there are many ways for the shards to be distributed about the floor, but only one way for them to be assembled into a vase. And that difference is the essence of… counting. This leads us to the second law of thermodynamics: entropy increases with time. Or, if you prefer, systems evolve towards states of higher probability. But probability is nothing other than a relative count of possibilities. Counting again.
Without counting, there is no arrow of time.
But it gets better. The whole idea of counting presupposes the existence of things to count. Which requires us to draw distinctions. And indeed, we find that distinction is the fundamental act by which something comes out of nothing. Assuming that a distinction can spontaneously arise out of the void, it will do so – because there are more ways for the void to be cloven than for it to be whole. Counting again.
If we picture a distinction as a boundary in a space – a closed curve in the plane, a closed surface in space, and so on, then we see that the lowest number of dimensions in which a boundary can assume a configuration that cannot shrink to nothing is… three (the simplest such configuration is the trefoil knot). Thus we see hints of how a universe of 3 spatial dimensions and one time dimension can spontaneously arise out of nothing. All because of counting.
Similar considerations explain how this universe comes to contain fundamental particles, and why the have the properties they do. And ultimately, why consciousness is possible. All of human feeling can be reduced to drawing or perceiving distinctions, and all of human thought can be reduced to classifying and counting them.
Thus we have the age-old question of which is more fundamental: mathematics or logic. For centuries men have been trying to derive one from the other. Finally, a little-known genius by the name of George Spencer-Brown settled it by showing that you cannot derive mathematics from logic, and you cannot derive logic from mathematics. But there is a more fundamental system, which he called the Laws of Form, from which you can derive both.
He begins with one primal element, which can be viewed as an entity (a distinction) or an action (drawing a distinction). A boundary can be seen as a way of naming the interior (calling), or as an injunction to cross into the interior (crossing). Having drawn a distinction, we can draw another one, either beside the first (recalling), or around the first (recrossing). On this base he lays down two laws, as follows
The law of calling: recalling is the same as calling.
The law of crossing: recrossing is the same as not crossing.
If we denote a boundary as (), then recalling is ()() and recrossing is (()), and we can write these two laws very succinctly as
()() = ()
where the right hand side of the second equation is literally empty, denoting the void.
And from this basis, utterly brilliant in its irreducible simplicity, he derives all of mathematics and symbolic logic:
Spencer-Brown, G, 1969: Laws of Form, London: George Allen & Unwin.
But this is only the beginning of the story. Frederick Parker-Rhodes asked what happens when you repeatedly draw a distinction and get a multitude of identical entities. From this, he developed a calculus of distinct but indistinguishable entities:
Parker-Rhodes, A F, 1981: The Theory of Indistinguishables: A search for explanatory principles below the level of physics, Synthese Library, vol. 150, Springer.
And on that, he constructed what he called the Combinatorial Hierarchy – system whereby the spontaneous emergence of distinctions from the void leads to… the standard model of particle physics. Astounding! Even more astounding, he never published this work! It was finally published for him posthumously by John Amson (see linked pdf):
Parker-Rhodes, A F, & Amson, J C, 1998: Hierarchies of descriptive levels in physical theory. Int’l J. Gen. Syst. 27(1-3):57-80.
The construction he outlines in this paper was implemented as a computer program by H. Pierre Noyes and David McGoveran (again, see linked pdf):
Noyes, H P, & McGoveran, D O, 1989: An essay on discrete foundations for physics. SLAC-PUB-4528.
So when I say that combinatorics lies at the root of everything, I really do mean everything!
It is brilliant!