Space-time surfaces as numbers, Turing and Gödel, and mathematical consciousness

The stimulus for this work came the links to Bruno Marchal’s posts by Jayaram Bista (see this). The original comments compared the world views behind two Platonisms, the Platonism based on integers or rationals and realized by the Turing machine as a Universal Computer and the quantum Platonism of TGD (see this). Marchal also talks about Digital Mechanism and claims that it is not necessary to assume a fixed physical universe “out there”. Marschal also speaks of mathematical theology and claims that quantum theory and even consciousness reduce to Digital Mechanism.

Later these comments expanded to a vision about the geometric correlates of arithmetic and even more general mathematical consciousness based on the vision about space-time surfaces as generalized numbers and providing also a representation of the ordinary complex numbers.

This also led to a more detailed view about the TGD realization (see this) of Langlands correspondence (LC) in which geometric and function field versions naturally correspond to each other and the LC itself boils down to the condition that cobordisms for the function pairs (f₁,f₂) defining the space-time surfaces as their roots are realized as flows in the infinite-D symmetry group permuting space-time regions as roots of a function pair (f₁,f₂) acting in the “world of classical worlds” (WCW) consisting of space of space-time surfaces satisfying holography = holomorphy principle.

That space-time surfaces form an algebra with respect to multiplication and that this algebra decomposes to a union of number fields (see this) means a dramatic revision of what computation means. The standard view of computation as a construction of arithmetic functions is replaced with a physical picture in which space-times as 4-surfaces have interpretation as almost deterministic computations. Space-time surfaces allow arithmetic operations and also the counterparts of functional composition and iteration are well-defined. This would suggest a dramatic generalization of the computational paradigm and it is interesting to ponder what this might mean.

This also leads to a vision about the fundamental geometric correlates of arithmetic and even more general mathematical consciousness based on the vision about space-time surfaces as generalized numbers and providing also a representation of the ordinary complex numbers. The notion of concept, such as a set as a collection of its instances, can be realized at the level of WCW in terms of the locus of the WCW spinor field when space-time surfaces correspond to numbers in generalized sense or to ordinary complex numbers. Second realization analogous to Boolean algebra is in terms of the product of space-time surfaces as elements of the generalized number field. Also the notion of linear space can be realized in this way by realizing the ordering of the elements of the set geometrically. Also the notion of function can be realized.

Of course, my personal view of computation and metamathematics is rather limited: I am just a humble physicist thinking simple thoughts but my sincere hope is that mathematicians would realize how deep the implications of the new physics based number concept has.

See the article Space-time surfaces as numbers, Turing and Gödel, and mathematical consciousness or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.