Cellular automaton dynamics is a good model for high level self-organized systems: the laws of physics at this level are "traffic rules" selected by a convention. Moral in society is an abstract example. Whether they are obeyed is not always obvious. The dynamics at fundamental level is to my opinion more naturally given by a variational principle. In TGD variational principle would determine spacetime as 4-surface in M4xCP2 or equivalently by algebraic equations in 4-surfaces in M8 with associative tangent space.
In the recent proposal Wolfram starts from graphs with relationships between points and makes graphs and thus dynamical and able to change. He tries to get continuum, curved space-time of general relativity as a limit of graph with very large number of nodes. There would be some rules to produce new graphs - kind of space-time dynamics. I think here one ends up with problems or at least challenges: there is no limit for possible rules. There are objections.
- How to obtain 3-space and 4-D space-time? Restrictions on homology could give analog of n-D manifold which is approximated by collection of simplexes. Why just these dimensions? This is very tough problem.
- Difficulties begin as one tries to get the notion of distance and metric to get Einstein's theory. One must assign a unit distance between nearest neighbours. Distance d between to points would be minimal number of steps between them. This looks fine at first.
But what if one adds one point x between neighbours a and b? Is the distance between a x d/2 halved? Or is the distance between a and b now 2d. Both interpretations would mean dramatic change of the discretized metric in an addition of single point.
One might intuitively argue that the addition of new points improves resolution and one adds new points by some rule everywhere so that distances by nearby points would be naturally scaled down by same factor. But talking about resolution means that one already talks about conscious entities. And now there is danger that one thinks the structure as imbedded in continuous space in order to make the rule determining the distances realistic. In other words, one is discretizing continuous surface!
In any case, the idea is that adding more and more points to the graph one obtains at the limit of infinitely many points standard physics and the graphs looks like 3-manifold. I am skeptic: the arguments involving infrared limit as it is called are hand weaving.
- A further basic argument against discretization at fundamental level is that one loses the nice symmetries of continuum space-time. Even more, geometrization of these symmetries leads to a unique choice of imbedding space in TGD framework forced both by both standard model physics and by the existence of twistor lift of TGD.
Not surprisingly therefore, Wolfram has really hard time in trying to convince the reader that energy and momenta emerge from his theory. Some kind of flows in the graphs must be introduced. The reason is obvious. Noether's theorem giving extremely deep connection between symmetries and conservation laws is lost and one can only try to invent purely ad hoc arguments, which are doomed to be wrong.
One could of course consider discretization of the fundamental symmetries for the graphs fixing also the distances between the points of graphs but this would lead to the identification of them as surfaces in some space - say M4xCP2 or equivalently M8 - to get the distances between the points of discretization correctly.
- As a conscious theorist I have been talking a lot about adelic physics as number theoretical generalization of TGD in which space-times are 4-D continuous surfaces in certain 8-D imbedding space H=M4xCP2 or complexified M8 having interpretation in terms of octonions - the choices are equivalent by M8-duality - having symmetries of special relativity and standard model. One introduces besides reals also p-adic number fields as correlates of physics of cognition. p-Adic variants of space-time surfaces obey same field equations as their real counterparts and one can say that they mimic the real physics.
At M8 level one can say that space-time surfaces are roots of octonionic continuations of polynomials having rational coefficients thus define extensions of rationals via their roots. Physics at M8 level would be purely algebraic. At M8 H-level space-time surfaces would be minimal surfaces with string world sheets as singularities. Minimal surface is a geometric analog ofn massless field.
- Extensions of rationals assignable to polynomials with rational coefficients play a key role in the theory and the space-time surfaces allow a unique discretization in in highly unique preferred coordinates made possible by the symmetries of imbedding space. At the level of M8 the coordinates are determined apart from time translation.
The points of discretization have coordinates in the extension of rationals considered. I call these discretizations cognitive representations. What is remarkable that cognitive representation makes sense both as points of real space-time surfaces and its p-adic counterparts. It is in the intersection of reality and various p-adicities (or their extensions induced by the extension of rationals defining the adele).
- The space-time would not be discrete at fundamental level. The cognitive representations about it would be however discrete. At the limit of algebraic numbers the cognitive representation would be a dense set of algebraic points of space-time surfaces. The hierarchy of extensions of rationals would define evolutionary hierarchy since in quantum jumps the dimension of extension must increase in statistical sense. Dimension d of extension would actually correspond to effective Planck constant h_eff=nh_0 assignable to dark matter as phases of ordinary matter. This would give direct connection with quantum physics.
See for instance The philosophy of adelic physics .
For a summary of earlier postings see Latest progress in TGD.
Of course is spacetime discrete. Space is a tetrahedron based network with Axion-Higgses oscillating between common points.
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