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Mark Hogarth Cambridge University

In this thought-provoking exploration, Mark Hogarth discusses the implications of non-Turing computers as new non-Euclidean geometries. He highlights that understanding Turing's definition is crucial in differentiating between pure and physical models of computation. The essay delves into various geometrical concepts, including Euclidean, Lobachevskian, and Riemannian geometries, and draws parallels to computability issues, like the halting problem. Ultimately, it raises profound questions about the nature of computation and its relationship to physical reality.

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Mark Hogarth Cambridge University

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  1. Non-Turing Computers are the New Non-Euclidean Geometries(orThe Church-Turing Thesis Explained Away) Mark Hogarth Cambridge University

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

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

  4. Earman, J. and Norton, J. [1993]: ‘Forever is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes’, Philosophy of Science, 5, 22-42.

  5. Concept of Geometry Euclidean Geometry Lobachevskian, Reimannian Geometry Pure Geometry Physical Geometry Euclidean Geometry General relativity, etc. Lobachevskian Reimannian Schwarzschild ...

  6. Concept of Computability • OTM • Various new computers • Pure computabilityPhysical computability • OTM, SAD1, … • How ‘far’ are the • pure models from real computers?

  7. 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

  8. 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.

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

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

  11. The ‘true geometry’ is Euclidean geometry • For: ‘pure’, natural, intuitive, different yet equivalent axiomatizations. • Against: Reimannian geometry etc. • Neither is right

  12. The ‘Ideal Computer’ is a Turing machine For: ‘pure’, natural, intuitive, different yet equivalent axiomatizations. Against: SAD1 machine etc. Neither is right

  13. GEOMETRY • There are lots of geometries • Pure maths of a geometry derives from its logical structure • Question of physical relevance is one of physics • COMPUTABILITY • There are lots of computers • Pure maths of a computer derives from its logical structure (no CT) • Question of physical relevance is one of physics

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