Quantum Platonia and space-time surfaces as numbers

Yesterday evening we had a very interesting Zoom discussion with Marko and Ville related to Large Language Models (LLMs) and computationalism in general. We also discussed Platonism, propagated by Max Tegmark. The idea that only mathematical objects exist and that they can exist as long as they do not lead to internal logical contradictions is extremely fascinating and economical.

The basic problem of Tegmark’s theory is that conscious experiences are absent from Platonia. Mathematical objects are most naturally zombies. It is really hard to imagine, for example, how the square root two could have intentions and free will.

In the zero energy ontology (ZEO) of TGD, the habitants of the Quantum Platonia would be quantum states defined as superpositions of space-time surfaces of H= M⁴×CP₂, satisfying holography = holomorphy principle, which guarantees that one gets rid of path integral full of divergences. The choice of H is unique by the mathematical existence of TGD and so called M⁸-H duality leading in turn to a 4-D variant of Langlands correspondence is essential and brings the entire number theory part of TGD. Geometric and number theoretic views are dual.

The quantum jumps between zero energy states makes the Quantum Platonia conscious. In quantum jump, the final state contains classical information from previous states and quantum jumps. Therefore the Universe learns and is able to remember since the classical physics for the space-time surfaces implementing holography is not completely deterministic and makes possible memory recall (see this). The number-theoretic vision forces the increase of the algebraic complexity of space-time surfaces during the sequence of quantum jumps, which means evolution.

The most beautiful thing is that the space-time surfaces, represented as roots for pairs of polynomials with rational coefficients, correspond to integers (also more general analytic functions of generalized complex coordinates of H are possible). The integer corresponds to the products of discriminants D of polynomials associated with the regions (partonic 2-surfaces) of the space-time surface. The classical universe as a space-time surface is a number! The basic arithmetics could not emerge at a more fundamental level.

On the geometric side, the vacuum functional is identified as the exponent of the Kähler function (geometry) having a space-time surface as its argument . One the number theory side it corresponds to a power of D by M⁸-H duality (see this). Therefore the natural number theoretic and geometric invariants of the space-time surface correspond to each other: this is in agreement with the Langlands correspondence.

The discriminant D splits into the product of ramified primes, which have a direct physical interpretation that I arrived at about 30 years ago from p-adic mass calculations. The spacetime surface decomposes into particles corresponding to these primes.

Some mathematician, maybe it was Kronecker, said that God created the integers and the rest is man-made. In the TGD Universe, God also created algebraic integers, in fact an infinite hierarchy of these because he wanted evolution, and also these can be realized as space-time surfaces. There is no reason why God could not have also created transcendentals. Or maybe they have not yet emerged in the number theoretic evolution. The re-creation of the Universe continues forever. And one must not forget p-adic number fields and adeles since cognition is needed.

For the TGD as its is now see for instance the articles this and this.

For the most recent results related to the number theoretic aspects of TGD see this and this .

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.