A handwavy model

I present an approach to how one might try to model an evolutionary system. So, suppose we have a manifold, M. Let’s say its Lorentzian (that always sounds impressive). Suppose we have a function, f : M -> R, which represents an optimal quantity. Suppose the “measured” function is g : M -> R, and we have initial data for g along a spacelike hypersurface S. The problem is to determine g, given that there is “evolutionary pressure” for it to converge to f on M, for all points p “timeward” of S.

So, without worrying too much about process, but just waving our hands and invoking “random noise”, we could say there is a forcing vector field, r : M -> TM which acts to “push” g towards f. On the other hand, g has no hope of converging to f if f is varying too much. I.e. if |grad f| > |r|, f will “repel” nearby points ie it will be an unstable fixed point or repellor of some invariant which we are trying to optimise. If, on the other hand, |grad f| < |r|, f will become a stable attractor of the same.

Hence we expect g to converge to f in regions where f is an attractor. This makes sense in the context of biological evolution on earth since local environments tend to be relatively static and not terribly variable. In the context of, say, convergence of laws in a society to some new optimum to adjust to technological change, the process fails because technological change is, at the moment, extremely variable.

Eight

Eight happens to be the number of Thurston’s model 3 geometries: $R^{3}$, $RP^{3}$, $S^{3}$; $S^{2} \times R$, $RP^{2} \times R$; $S^{1} \times S^{2}$, Nil, and Sol. The fact that there are eight different particle families in the Standard Model is, perhaps, no coincidence in lieu of this information.

Then there is the eightfold path of buddhism, which is supposed to act as guidance to find the middle path- a path that, according to the ethos, supposedly should bring peace, temperance and happiness to the practitioner.  It is perhaps no surprise either that eight is a number considered synonymous with luck and wealth in certain countries in east asia, which is where buddhism arose.

The Octonians are a number system with eight roots of unity.

In Terry Pratchet’s discworld, the eighth colour is the colour of magic, Octarine.

Not to mention that eight is the first cubic number (after one, but that doesn’t really count).

So I think you will agree that eight is a pretty cool number.