1 / 16

Mark Hogarth Cambridge University

Non-Turing Computers are the New Non-Euclidean Geometries (or The Church-Turing Thesis Explained Away). Mark Hogarth Cambridge University. ‘It is absolutely impossible that anybody who understands the question and knows Turing’s definition should decide for a different concept’ Hao Wang.

roary-spence
Télécharger la présentation

Mark Hogarth Cambridge University

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related