Are fundamental entities discrete or continous and what discretization at fundamental level could mean?

We had an interesting discussion with Tuomas Sorakivi about metamathematics yesterday. For a mainstream physicist, mathematics is "maths": a box from which you can draw calculational methods. The physical reality behind the models is real to the physicist, and mathematics is just a tool for creating models for it.

For a Platonist, on the other hand, mathematics is real. What the mathematician himself is, remains an open question in both views. A theory of consciousness is needed. TGD proposes such a theory (see for instance this and this).

In TGD, physics represents mathematics rather than vice versa. The view of the standard physicist is turned upside down. Mathematics is Platonia identified as the "world of classical worlds" (WCW) consisting of space-time surfaces in H=M⁴× CP₂. In zero energy ontology, space-time surfaces obeying holography would represent numbers and more general mathematical objects, even theorems. Theorems can be identified as space-time surfaces which obey an almost deterministic holographic time evolution. If determinism were perfect, there would be only a single theorem - not very interesting mathematics - but this is not the case, so several theorems corresponding to different time evolutions follow from the given premises. This happens naturally. In the TGD Universe, the task of the mathematician is to become aware of the proofs that Nature produces.

This solves the problem caused by the combinatorial explosion encountered in attempts to mechanize mathematical proofs. Theorems are deduced from the axioms by rules that are analogous to the dynamic equations of classical physics. One can speak of truth-preserving dynamics. Boolean logic crystallizes these traffic rules. The problem is that these traffic rules are very weak. From given premises, a huge number of theorems can be deduced in an infinite number of ways. In practice, a mathematical machine is therefore impossible.

The combinatorial explosion can be illustrated by the set-theoretic representation of Boolean algebra. The premises correspond to the set A. The logical implication A→B corresponds to the fact that B is a subset of A. The number of subsets increases exponentially as the number of elements of A increases. There are 2^N elements among the subsets if A has N elements. The result is a combinatorial explosion.

How are human mathematicians able to derive any theorems at all? Could one think that physics comes to rescue and replaces the Boolean traffic rules with holographic dynamics for proving. Could the laws of Nature define not only the axioms but also proofs of the theorems as holographic time evolutions. Given premises A would lead to a finite number of consequences B, provided that the holographic dynamics is slightly nondeterministic. The quantum-platonism of TGD would realize this dream.

See the chapter G&oum;del, Lawvere, and TGD or the article with the same title.