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The Church-Turing Thesis is a Pseudo-proposition

The Church-Turing Thesis is a Pseudo-proposition. Mark Hogarth Wolfson College, Cambridge. Will you please stop talking about the Church-Turing thesis, please. Computability. The current view.

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The Church-Turing Thesis is a Pseudo-proposition

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  1. The Church-Turing Thesis is a Pseudo-proposition Mark Hogarth Wolfson College, Cambridge

  2. Will you please stop talking about the Church-Turing thesis, please

  3. Computability

  4. The current view

  5. ‘It is absolutely impossible that anybody who understands the question and knows Turing’s definition should decide for a different concept’ Hao Wang

  6. Experiment escorts us last — His pungent company Will not allow an Axiom An Opportunity EMILY DICKINSON

  7. The key idea is simple There is no ‘natural’ / ‘ideal’/ ‘given’ way to compute.

  8. And the key to understanding this new Computability is to think about another concept, Geometry.

  9. Most important slide of this talk! The new attitude is achieved by adopting the kind of attitude one has to Geometry, to Computability

  10. Concept change paradigm (a theory) new ‘evidence’ opposition tension revolution new paradigm (a new theory)

  11. Tension (roughly late 18the century-1915) ‘I fear the uproar of the Boeotians’ (Gauss) Kant (EG synthetic a priori) EG is ‘natural’, ‘perfect’, ‘intuitive’, ‘ideal’ Poincaré: EG is conventionally true Russell (1897): only non-Euclidean geometries with constant curvature are bona fide

  12. Concept of Geometry Euclidean Geometry Lobachevskian, Reimannian Geometry Tension Pure geometry Physical Geometry Euclidean Geometry General relativity, etc. Lobachevskian Reimannian Schwarzschild ...

  13. Geometry after 1915 Etc.

  14. Computers New evidence has coming to light… We are in period of tension

  15. Concept of Computability The Turing machine Various new computers (mould, SADs, quantum) tension Pure computabilityPhysical computability OTM, SAD1, …Assess the physical theories that house these computers

  16. Concept of Geometry Euclidean Geometry Lobachevskian, Reimannian Geometry Tension Pure geometry Physical Geometry Euclidean Geometry General relativity, etc. Lobachevskian Riemannian Schwarzschild ...

  17. Typical geometrical question: Do the angles of a triangle sum to 180? Pure: Yes in Euclidean geometry, No in Lobachevskian, No in Reimannian, etc. Physical: Actually No

  18. Typical computability question: Is the halting problem decidable? Pure: No by OTM, Yes by SAD1, etc. Physical: problem connected with as yet unsolved cosmic censorship hypothesis (Nemeti’s group).

  19. Question: Is the SAD1 ‘less real’ than the OTM? Answer: Is Lobachevskian geometry ‘less real’ than Euclidean geometry?

  20. Pure models do not compete, e.g. no infinite vs. finite

  21. The ‘true geometry’ is Euclidean geometry (‘Euclid’s thesis’) For: ‘pure’, natural, intuitive, different yet equivalent axiomatizations. Against: Riemannian geometry etc. Neither is right (pseudo statement)

  22. The ‘Ideal Computer’ is a Turing machine (CT thesis) For: ‘pure’, natural, intuitive, different yet equivalent axiomatizations. Against: SAD1 machine etc. Neither is right (this is no ideal computer, just as there is no true geometry)

  23. What is a computer?

  24. What is a geometry?

  25. Another question: what is pure mathematics? Partly symbols on bits of paper We might write, e.g. Start with 1 Add 1 Reveal answer Repeat previous 2 steps

  26. The marks seem to say to us 1,2,3,…

  27. But this is an illusion: the marks alone do nothing

  28. The ‘illusion’ is obvious in Geometry What is this?

  29. Algorithms are like geometric figures drawn on paper Without a background geometry, the figure is nothing Without a background computer, an algorithm is nothing

  30. Note Just as the New Geometry left Euclidean Geometry untouched, so the New Computability leaves Turing Computability untouched.

  31. Not quite... Pure Turing model Physical Turing model – this will involve some physical theory, T, embodying the pure model.

  32. According to T, this machine might be warm or wet or rigid or expanding ...

  33. or possess conscious states or intelligence

  34. T will also give an account of how, e.g., the machine computes 3+4=7

  35. Arithmetic has a physical side

  36. ‘Pure mathematics’ has a physical side

  37. Think Computability, think Geometry

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