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The Church-Turing Thesis is a Pseudoproposition Mark Hogarth Wolfson College, Cambridge
Will you please stop talking about the Church. Turing thesis, please
The current view
‘It is absolutely impossible that anybody who understands the question and knows Turing’s definition should decide for a different concept’ Hao Wang
Experiment escorts us last — His pungent company Will not allow an Axiom An Opportunity EMILY DICKINSON
The key idea is simple There is no ‘natural’ / ‘ideal’/ ‘given’ way to compute.
And the key to understanding this new Computability is to think about another concept, Geometry.
Most important slide of this talk! The new attitude is achieved by adopting the kind of attitude one has to Geometry, to Computability
Concept change paradigm (a theory) new ‘evidence’ opposition tension revolution new paradigm (a new theory)
Tension (roughly late 18 the 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
Concept of Geometry Euclidean Geometry Lobachevskian, Reimannian Geometry Tension Pure geometry Euclidean Geometry Lobachevskian Reimannian Schwarzschild. . . Physical Geometry General relativity, etc.
Geometry after 1915 Etc.
Computers New evidence has coming to light… We are in period of tension
Concept of Computability The Turing machine Various new computers (mould, SADs, quantum) tension Pure computability Physical computability OTM, SAD 1, … Assess the physical theories that house these computers
Concept of Geometry Euclidean Geometry Lobachevskian, Reimannian Geometry Tension Pure geometry Euclidean Geometry Lobachevskian Riemannian Schwarzschild. . . Physical Geometry General relativity, etc.
Typical geometrical question: Do the angles of a triangle sum to 180 ?
Typical computability question: Is the halting problem decidable? Pure: No by OTM, Yes by SAD 1, etc. Physical: problem connected with as yet unsolved cosmic censorship hypothesis (Nemeti’s group).
Question: Is the SAD 1 ‘less real’ than the OTM? Answer: Is Lobachevskian geometry ‘less real’ than Euclidean geometry?
Pure models do not compete, e. g. no infinite vs. finite
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)
The ‘Ideal Computer’ is a Turing machine (CT thesis) For: ‘pure’, natural, intuitive, different yet equivalent axiomatizations. Against: SAD 1 machine etc. Neither is right (this is no ideal computer, just as there is no true geometry)
What is a computer?
What is a geometry?
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
The marks seem to say to us 1, 2, 3, …
But this is an illusion: the marks alone do nothing
The ‘illusion’ is obvious in Geometry What is this?
Algorithms are like geometric figures drawn on paper Without a background geometry, the figure is nothing Without a background computer, an algorithm is nothing
Note Just as the New Geometry left Euclidean Geometry untouched, so the New Computability leaves Turing Computability untouched.
Not quite. . . Pure Turing model Physical Turing model – this will involve some physical theory, T, embodying the pure model.
According to T, this machine might be warm or wet or rigid or expanding. . .
or possess conscious states or intelligence
T will also give an account of how, e. g. , the machine computes 3+4=7
Arithmetic has a physical side
‘Pure mathematics’ has a physical side
Think Computability, think Geometry