Gödel, Lawvere and TGD

The tweets of Curt Jaimungal (see this) inspired an attempt to understand Gödel's incompleteness theorem and related constructions from the TGD point of view.

It has remained somewhat unclear to me how the notion of conscious self is defined in theories based on pure mathematics. I however understand that the conscious system is identified as an object in a category X and the view of self about itself would be a set of morphisms of f_x of X→ X as structure-preserving descriptions, morphisms, which would give information about x to the other selves y as objects of X. One can define X^Y as an object having as objects the morphisms Y→ X. X^Y would correspond to X as seen by object Y.

This associates to every object x \in X morphism f_x\in X^X of the category X into itself. One could say that X embedded in X^X and f_x corresponds to models of x for other selves of y \in X. Under conditions formulated by Lawvere, any morphism f in X^X has a fixed point y_f. In particular, for f_x one can find y_x such that f_x(y_x)=y_x is satisfied. In some cases this might be the case. Under the assumptions of Lawvere, one can have y_x=x and this might be the case always. These kinds of objects x are very special and one can wonder what its interpretation is.

In particular, Gödel's sentence is a fixed point for a sentence f_x, which associates to a sentence y a sentence f_x(y) stating that y is not provable in the formal system considered. It turns out that f(x)=x is true. Therefore x is not provable but is true. Could this mean that this kind of object is self-conscious and has a self model?

On the other hand, self-reflection, which means that one becomes aware of the content of one's own consciousness at least partially, can be claimed to create descriptions of itself and fixed point property suggests an infinite number of levels or possibly limit cycles: for Julia sets only non-trivial limit cycles are present. Infinite regression however means a paradox. On the other hand one can argue that self-representation is trivial for a fixed point.

What is the situation in TGD? In the following the idea about physics laws, identified in the TGD frameworks as the dynamics of space-time surfaces, is discussed in detail from the perspective of the metamathematics or metaphysics.

The laws of physics as analogs for the axioms of a formal system

The basic idea is that the laws of physics, as they are formulated in the TGD framework (see this and this), can be regarded as analogs for the axioms of a formal system.

1. Space-time surface, which by holography= holomorphy vision is analogous to a Bohr orbit of particles represented as a 3-surface is analogous to a theorem. The slight classical non-determinism forces zero energy ontology (ZEO)(see this): instead of 3 surfaces the analogs of Bohr orbits for a 3-surfaces at the the passive boundary (PB) of the causal diamond (CD) are fundamental objects. By the slight classical non-determinism, there are several Bohr orbits associated with the same 3-surface X³ at the PB remaining un-affected in the sequence of "small" state function reductions (SSFRs). This is the TGD counterpart of the Zeno effect. The sequences of SSFRs defines conscious entity, self.
2. The adelization of physics means that real space-time surfaces obtained using extension of E of rationals are extended to adelic space-time surfaces. The p-adic analogs of the space-time surface would be correlates for cognition and cognitive representations correspond to the intersections of the real space-time surface and its p-adic variants with points having Hamilton-Jacobi coordinates in E (see this).  
3. Concerning Gödel, the most important question is how self reference as a metamathematical notion is realized: how space-time surfaces can represent analogs of statements about space-time surfaces. In the TGD framework holography= holomorphy vision provides an exact solution of the classical field equations in terms of purely algebraic conditions. Space-time surfaces correspond to the roots function pairs (f₁,f₂), where f_i are analytic functions of the Hamilton Jacobi coordinates of H=M⁴\times CP₂ consisting of one hypercomplex and 3 complex coordinates.

   The meta level could correspond to the maps g= (g₁,g₂): C²→ C², where g_i are also analytic functions or Hamilton-Jacobi coordinates, mapping the function pairs f=(f₁,f₂): H→ C² and giving new, number theoretically more complex, solutions. The space-time surfaces obtained in this way correspond to the roots of the composites gºf = (g₁(f₁,f₂),g₂(f₁,f₂)).

   g should act trivially at the PB of CD in order to leave X³ invariant. One can construct hierarchies of composites of maps g having an interpretation as hierarchies of metalevels. Iteration using the same g repeatedly would be a special case and give rise to the generalization of Mandelbrot fractals and Julia sets.

4. Second realization would be in terms of the hierarchy of infinite primes (see this) analogous to a repeated second quantization of a supersymmetric arithmetic quantum field theory for an extension E of rationals and starting from a theory with single particle boson and fermions states labelled by ordinary primes. Here one can replace ordinary primes with the prime of an algebraic extension E of rationals. This gives a second hierarchy. Also the Fock basis of WCW spinor fields relates to WCW like the set of statements about statements to the set of statements.

How space-time surfaces could act on space-time surfaces as morphisms

Could one, by assuming holography= holomorphy principle, construct a representation for the action of space-time X⁴ surface on other space-time surfaces Y⁴ as morphisms in the sense that at least holomorphy is respected. In what sense this kind of action could leave a system associated with X⁴ fixed? Can the entire X⁴ remain fixed or does only the 3-D end X³ of X³ at the PB remain fixed? In ZEO this is indeed true in the sequence of SSFRs made possible by the slight failure of the classical determinism.

What the action of X⁴ on Y⁴ could be?

1. The action of X⁴ on Y⁴ would be a morphism respecting holomorphy if X⁴ on Y⁴ have a common Hamilton-Jacobi structure (see this). This requirement is extremely strong and cannot be satisfied for a generic pair of disjoint surfaces X⁴ and Y⁴. The interpretation would be that this morphism defines a kind of perception of Y⁴ about X⁴, a representation of X⁴ by Y⁴. Ψ

   A naive proposal for the action of X⁴ on Y⁴ assumes a fixed point action for Y⁴=X⁴. The self-perception of X⁴ would be trivial. Non-triviality of self-representation since is in conflict with the fixed point property: this can be seen as the basic weakness of the proposal that conscious experience could be described using a formal system involving only the symbolic description but no semantics level.

2. The classical non-determinism of TGD comes to rescue here. It makes possible conscious memory and memory recall (see this and this) and the slightly non-deterministic space-time surface X⁴ as an analog of Bohr orbit can represent geometrically the data making possible conscious memories about the sequence of SSFRs. The memory seats correspond to loci of non-determinism analogous to the frames spanning 2-D soap films. In the approach based on algebraic geometry, the non-determinism might be forced by the condition that space-time surfaces have non self-intersections. Second possibility is that space-time surfaces consist of regions, which correspond to different choices of (f₁,f₂) glued together along 3-D surfaces.
3. Purely classical self-representation would be replaced at the quantum level by a quantum superposition of the Bohr orbits for a given X³. A sequence of "small" state function reductions (SSFRs) in which the superposition of Bohr orbits having the same end at the PB is replaced with a new one. SSFRs leave the 3-surfaces X³ appearing as ends of the space-time surface at the PB invariant. The sequence of SSFRs giving rise to conscious entity self, would give rise to conscious self-representation.
4. The fixed point property for X⁴ making the self-representation trivial would be weakened to a fixed point property for X³, and more generally of 3-D holographic data.

How zero energy states identified as selves could act on each other as morphisms?

How the superposition Ψ(X³) of Bohr orbits associated with X³ can act as a morphism on Ψ(Y³)? The physical interpretation would be that Ψ(X³) and Ψ(Y³) interact: Ψ(X³) "perceives" Ψ(Y³) and vice versa and sensory representations are formed. This sensory representation is also analogous with the quantum counterpart of the learning process of language models producing associations and association sequences as analogs of sensory perceptions (see this).

1. These "sensory" representations must originate from a self-representation. This requires a geometric and topological interaction X⁴ and Y⁴ as a temporary fusion of X⁴ and Y⁴ to form a connected 4-surface Z⁴. This would serve as a universal model for sensory perception. In the TGD inspired quantum biology, a temporary connection by monopole flux tubes serves as a model for this interaction. If the flux tubes serve as prerequisites and correlates for entanglement, entanglement could also be generated.
2. The holomorphy for Z⁴ requires that X⁴ on Y⁴ have a common Hamilton-Jacobi structure during the fusion but not necessarily before and after the fusion. Therefore the defining analytic function pairs (f₁,f₂) (see this) can be different before and after the fusion and during the fusion and also for X⁴ and Y⁴ after and before the fusion. This might be an essential element of classical non-determinism. Continuity requirement poses very strong conditions on the function pairs involved. The representations produced in the interaction would be highly unique. As already mentioned, also the absence of self-intersections could force classical non-determinism.

   The outcome of the temporary fusion would give rise to a representation of the action of X⁴ on Y⁴ and vice versa. The representation would be a morphism in the sense that outcomes are holomorphic surfaces and the ends of X⁴ and Y⁴ at the PB of CD remain unaffected.

3. The fixed point property for Z⁴ making the self-representation trivial would be replaced with the fixed point property for Z³ and therefore also X³ and Y³.
4. The time reversed variant of sensory perception has an interpretation as motor action between them and would involve a pair of BSFRs induced by a subsystem of Z⁴. Now the end of Z⁴ at the PB of CD would be changed. X⁴ would affect Y⁴ in a non-deterministic way. The construction of the representation of X⁴ on Y⁴ would reduce to a construction of a self-representation for Z⁴.

This view is inspired by the TGD view in which self is identified as a sequence of non-deterministic SSFRs and is thus not "provable" and has also free will. The holographic data would be in the role of the assumptions of a theorem, which need not to be proved and reduce to axioms, and the Bohr orbits would correspond to theorems deducible from these assumptions. In the interaction of X³ and Y³ a larger self Z³ would be created and would involve quantum entanglement. In this view, the infinite self reflection hierarchy is replaced with a finite sequence of SSFRs providing new reflective levels and self is a dynamical object.

See the article Gödel, Lawvere and TGD.

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.