Wednesday, January 08, 2020

Comparing Gisin's intuitionistic mathematics with adelic physics

Nikolina Benedikovic had an interesting posting commenting the work of physicist Nicolas Gisin. Gisin makes several strange looking statements.

  1. Gisin states that physicists apply classical physics, which is deterministic. This is of course true. They however apply also quantum physics, which involves non-deterministic state function reduction in conflict with the determinism of Schrödinger equation but is necessary for the interpretation of experiments. Statistical determinism is assumed but requires the notion of ensemble.

  2. Gisin claims and numbers with infinite number of decimals involve infinite number of information. This is not the case in general. For instance, if the decimals obey some formula the information is finite. Also rationals have infinite number of decimals but the the sequence of decimals is periodic so that the information content can be said to be finite.

  3. Gisin claims that the world is finite. Presumably he means that world is discrete and consists of finite number of points. This picture leads to insurmountable difficulties in practice.

    I believe that it is not world which is discrete but the cognitive representations about it. They are always discrete and in computer science based on the use of finite rational numbers reducing to pairs of integers. What is wrong with recent day physics is that cognition together with consciousness is kept out from consideration. Finite measurement resolution is described in ad hoc manner.

  4. Intuitionistic mathematics is the proposal of Gisin in which everything is finite. Finite number of decimals brings in indeterminism and Gisin argues that this number increases with time - kind of evolution.

It is interesting to compare this with TGD view.
  1. Finite measurement resolution instead of finite world looks to me a more realistic option. Classical realities - say space-time surfaces in TGD - are continua but our observations are always discrete because of finite observational and cognitive resolution.

  2. Gisin's approach excludes algebraic numbers. In TGD also algebraic numbers are allowed - essentially as roots of polynomials and are represented geometrically - say sqrt(2) as the length of the diagonal of square. Geometric representations complete the linear representations of numbers based on sequences of digits. This corresponds to the reductionistic-holistic dichotomy or left brain-right brain dichotomy.

  3. Cognition has as correlates p-adic numbers and their extensions induced by those of rationals- one can speak of -p-adic variants of space-time surfaces. This leads to what I call adelic physics (see also this). Cognitive representations correspond to points of space-time surface common to both real and p-adic space-time surfaces with preferred imbedding space coordinates (by symmetries) having values in extension of rationals and making sense in all number fields involved. They are essentially unique for given extension and the representation is in generic case discrete and even finite. These unique discretizations in the intersection of reality and various p-adicities are the TGD counterpart for intuitionism.

    Remark: Interestingly, the extension of rationals by powers of Neper number e or its root induces finite-D extension of p-adic numbers. So that also roots of e could be allowed by cognitive representations with finite resolution just like roots of unity. They would be very exceptional transcendentals.

  4. In TGD framework cognitive resolution is characterized by the n and the number N(p) of pinary digits and to the integer n. The mathematics of cognition is discrete and analogous to intuitionistic mathematics. n measures algebraic complexity and is kind IQ. n actually corresponds to effective Planck constant h_eff =nh_0 and measures quantum coherence scale in TGD framework so that a direct connecting with quantum physics allowing dark matter in TGD sense emerges.

  5. Gisin compares the increase of decimals as a process analogous to evolution. In TGD evolution would reduce to an unavoidable increase of n and N(p).

To sum up, the indeterminism about which Gisin talks would thus correspond to finite measurement and cognitive resolution in TGD framework. This indeterminism is in certain sense a correlate also for quantum non-determinism. For instance, geometric time order of "small" state function reductions (weak measurements) in events in zero energy ontology can vary and this variation corresponds classically to the lack of well-ordering for p-adic numbers.

Indeed, as one types text one often finds that the experienced order of digits as sequence of small state function reductions is different from that for the outcome representing corresponding sequence of moments of geometric time: you experience typing "outcome" but the result is "outocme"! Neuroscientist would of course invent other explanations.

See this and this .

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

Articles and other material related to TGD.

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