Tegmark and TGD

Max Tegmark has proposed that all that exists at the fundamental level is mathematics. Tegmark could be called a Platonist. What exists at the fundamental level would be Platonia, the world of mathematical objects. In some way, Tegmark wants to add consciousness to Platonia (see this) and of course faces the same problem as the materialists.

There is no hint of what qualia, the contents of consciousness, could be attached to mathematical objects. What would happen when 2^1/2 gets depressed or falls in love with 3^1/2. What happens when 5^1/2 suffers jealousy towards roots of higher order polynomials because they are more complex algebraic numbers. What emotions could the roots of a polynomial feel and what sensory qualia could they experience? What about more complex structures like sets and Hilbert spaces?

Something is needed and it is a geometric representation of numbers and a quantum jump and its representation: it brings in awareness and free will.

1. A representation of numbers as physical quantum objects is needed. Frenkel wondered in his marvellous AfterMath podcast (see this) how the numbers are physically represented. Frenkel emphasized that numbers cannot not presented in spacetime. TGD offers a solution to the puzzle: numbers are represented as spacetime surfaces in H= M⁴×CP₂. Holography=holomorphy vision (see this and this) makes this representation possible. The details of the emerging view are discussed in the article describing the TGD view of Langlands correspondence (see this) and in the article describing the two ways to interpret space-time surfaces as numbers (see this).
   1. The spacetime surface can correspond to numbers in a functional algebra with product (f₁,f₂)*(g₁,g₂)= (f₁f₂,g₁g₂) or, more restrictively, as elements of a function field with product (f₁,g)*(g₁,g)= (f₁f₂,g) with g fixed so that one obtains a family of function fields.
   2. Space-time surfaces also correspond to complex numbers: M⁸-H duality (see this). The discriminant determined by the product of the differences roots of the function for 2-D parton surfaces determines the discriminant, which is a complex number in the general case and defined also for general analytic function. In school days we encountered the discriminant while solving roots of a second order polynomial.
   Space-time surfaces form an entire evolutionary hierarchy (or rather hierarchies) depending on how algebraically complex they are. Also p-adic number fields and their extensions as correlates of cognition emerge naturally through the hierarchy of algebraic extensions of rationals.

Now we have the numbers represented as space-time surfaces and one can ask how abstractions so central for mathematical thinking emerge. For instance, how could sets and Hilbert spaces and operators in Hilbert spaces could?

1. Since we are in the quantum world, we start to build wave functions in WCW, the Platonia. More specifically we construct WCW spinor field, Ψ in space and therefore also in the space the complex numbers represented as spacetime surfaces. We get quantum Platonia. The WCW spinors correspond to many-fermion Fock states in quantum field theory for a given 4-surafce and Ψ assigns to the space-time surfaced such a spinor. The spinor fields of WCW are induced from the second quantized free spinor fields of H and the 4-dimensionality of space-time surfaces and associated exotic smooth structures make possible fermion-pair creation but only in 4-D space-time.
2. WCW spinor field Ψ can be restricted, for example, to the set of positive and odd numbers, i.e. corresponding spacetime surfaces. The subset of WCW in which Ψ is non-vanishing, defines a subset of numbers and the concept in the classical sense as the set of its instances. For example, one obtains the concept of an odd number as a set of space-time surfaces representing odd integers. Ψ can be also restricted to the roots of polynomials of a certain degree (corresponding space-time surfaces): this gives the notion of the root of a polynomial of a given degree. Quantum concept is not a mere set but an infinite number of different WCW spinor fields that give different perspectives on the concept.
3. There is also a second way to define the notion of set: Boolean algebraic definition is possible using function algebra consisting of pairs (f₁,f₂) of a some function field obtained by keeping f₂==g fixed. The product in the function field induces the product of surfaces and this product is just the union of space-time surfaces. A given set of ordinary complex numbers represented in terms of discriminants defined by the roots of analytic functions defined at partonic 2-surfaces corresponds to the product of the spacetime surfaces corresponding to the numbers in the sense of functional algebra.

   What is of fundamental importance is that these space-time surfaces in general have a discrete set of intersection points so that there is an interaction in fermionic degrees of freedom and one obtains n-point scattering amplitudes. Fermionic Fock states restricted to the intersection indeed define naturally a Boolean algebra.

See for instance 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.