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.

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

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

10. 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

11. 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.

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

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

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

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

16. 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