Monday, December 19, 2022

Goedel's theorem and TGD

The following is a response to Lawrence Crowell in the discussion group "The Road to unifying Relativistic and Quantum Theories". The topic of discussion related to Gödel's theorem and its possible connection with consciousness proposed by Penrose.

My own view is that quantum jump as state function reduction (SFR) cannot reduce to a deterministic calculation and can be seen as a moment of re-creation or discovery of a new truth not following from an existing axiomatic system summarizing the truths already discovered. My emphasis in the sequel is on how the number theoretic vision of the TGD proposed to provide a mathematical description of (also mathematical) cognition could allow us to interpret the unprovable Gödel sentence and its negation.

I decided to look more precisely at the Gödel number for polynomials with integer coefficients (no common factor coefficients) to which all rational polynomials can be scaled without changing the roots. Most of the classical physical content, if not all of it, can be coded by the coefficients [a0,...,aN] of the polynomial.

The Gödel numbering assigning to P an Gödel number G would be


where pi is i:th prime and is an injection.

The discriminant D is the determinant of an (2N-1)×(2N-1) matrix defined by P and its derivative dP/dx ([a1,2a2,...,NaN]) and is an integer decomposing to a product of ramified primes of P.

The first guess for Gödel's undecidable statement would that there exist polynomial P for which one has G=D. The number D coding a sentence, whatever it is, would be its own Gödel number. Why this guess? At least this statement is short;-). Can this statement be undecidable?

  1. The equation involves both D as a polynomial of ai and G involving transcendental functions piai (essentially exponential functions) so that one goes outside the realm of rationals and algebraic numbers.
  2. D=G is analog of Diophantine equation for a1,....,aN and both powers and exponential piai appear. If the coefficients ai are allowed to be a complex numbers, one can ask whether the complex solutions of G=D could form an N-1-D manifold. One can however assume this since piai leads outside the realm of algebraic numbers and one does not have a polynomial equation.
  3. The existence of an integer solution to D=G would mean that the primes pi for which ai are non-vanishing, correspond to ramified primes of P with multiplicity ai so that the polynomials would be very special if solutions exist.
  4. It might be possible to solve the equation for any finite field Gp, that is in modulo p approximation. Here one can use Fermat's little theorem pip= pi mod p. If integer solutions exist, they exist for every Gp.
What about the physical interpretation?
  1. The polynomials P define space-time surfaces and one possible interpretation is that the ramified primes of P define external particles for a space-time region representing particle scattering. The polynomials P which reduce to single ramified prime would represent forward scattering of a single "elementary" particle.
  2. In zero energy ontology, ordinary quantum states are replaced by superpositions of almost deterministic time evolutions so that also "elementary" particle would correspond to a scattering event. What exists would be events and TGD would predict not only scattering events but densities of particles as single particle scattering events inside a given causal diamond causal diamond representing quantization volume.
  3. What kind of scattering events would these analogues of Godel sentences correspond? Representations of new mathematical axioms as scattering events, not provable from existing axioms?
Exactly what we cannot prove to be true or not true for these special polynomials? What does the sentence labelled by G= D state?
  1. Integer D would express the sentence. D codes for the ramified primes. Their number is finite and we know them once we know P. Does the unprovable Gödel sentence say that there exists a polynomial P of some degree N, whose ramified primes are the primes p_k associated with ai? Or dös it say that there exists polynomial satisfying G=D in the set of polynomials of fixed degree N.
  2. Is it that we cannot prove the existence of integer solution ai to P=G using a finite computation. Is this due to the appearance of the functions piai or allowance of arbitrarily large coefficients ai? The p-adic solutions associated with finite field solutions have an infinite number of coefficients and can be p-adic transcendentals rather than rationals having periodic pinary expansions.
  3. Polynomials of degree N satisfying D=G are very special. The ramified primes are contained in a set of N+1 first primes pi so that D is rather small unless the coefficients ai are large. D is a determinant of 2N-1×2N-1 matrix so that its maximum value increases rapidly with N even when one poses the constraint ai< N. Rough estimates and explicit numerical calculations demonstrate that determinants involving very large primes are possible, in particular those involving single ramified prime identified as analogues of elementary particles, D can reduce to single large prime: D=P.

    What about the polynomials P in the vicinity of points of the space of polynomials of degree N satisfying D=0: they correspond to N+1 ramified primes, which are minimal (note that the number of roots is N). D is a product of the root differences and 2 or more roots coincide for D=0. D is a smooth function of real arguments restricted to the integer coefficients. The value of D in the neighborhood of D=0 can be however rather large. Note that the proposed Gödel numbering fails for D=0, and therefore makes sense only for polynomials without multiple roots.

  4. For D(P)=0 one has a problem with the equation G=D. G(P) is well-defined also now. The condition D(P)=0=G(P) does not however make sense. The first guess is that for 2 identical roots, P is replaced with dP/dx in the definition of D: D(P)-->D(dP/dx). D is nonvanishing and the ramified primes pi do exist for dP/dx. Therefore the condition D(dP/dx)=G(P) makes sense. For n identical roots one must use have D(dn-1P/dxn-1)=G(P).
  5. Interestingly, in TGD the hypothesis that the coefficients of polynomials of degree N are smaller than N, is physically very natural (see this) and would make the number of polynomials to be considered finite so that in this case one can check the existence of a G=D sentence in a finite time. It seems rather plausible that for given N, no G=D sentence, which satisfies the conditions ai< N, does exist.

    One can of course criticize the hypothesis ai< N implying a strong correlation between the degree N of P and the maximal size of ramified primes of P identified as p-adic primes characterizing elementary particles. One can argue that in absence of this correlation predictivity is lost. This hypothesis also makes also finite fields basic building bricks of number theoretic vision of TGD (see this).

  6. Could this give rise to a realization of undecidability at the level of conscious experience and cognition relying on number theoretic notions. How?

    Quantum states are superpositions of space-time surfaces determined by polynomials P and if the holography of consciousness is true, conscious experience reflects the number theoretic properties of these polynomials if associated to a localization to a given polynomial P in a "small" state function reduction (SSFR). This would be position measurement in the "world of classical worlds" (WCW)? The proof of the statement D=G would mean that a cognizing system becomes conscious of the D=G space-time surface by a localization to it.

    Suppose that for a given finite N and condition ai< N, G=D sentences do not exist. Hence one can say that G=D sentences go outside the axiomatic system realized in terms of the polynomials considered. Even the space of all allowed polynomials identified as a union of spaces with varying value for degree N would not allow this. G=D sentences would be undecidable by the condition ai< N.

    One can of course criticize the hypothesis ai< N implying a strong correlation between the degree N of P and the maximal size of ramified primes of P identified as p-adic primes characterizing elementary particles. One can argue that in absence of this correlation predictivity is lost. This hypothesis also makes also finite fields basic building bricks of number theoretic vision of TGD (see this).

See the articles Gödel's Undecidability Theorem and TGD and Finite Fields and TGD.

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

Articles related to TGD.

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