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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
‘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 companyWill not allow an AxiomAn 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 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
Concept of GeometryConcept of Geometry
Euclidean Geometry
Lobachevskian, Reimannian Geometry
Tension
Pure geometry Physical Geometry
Euclidean Geometry General relativity, etc.LobachevskianReimannianSchwarzschild...
Concept of GeometryConcept of Geometry
Euclidean Geometry
Lobachevskian, Reimannian Geometry
Tension
Pure geometry Physical Geometry
Euclidean Geometry General relativity, etc.LobachevskianReimannianSchwarzschild...
Geometry after 1915
Etc.
Computers
New evidence has coming to light…
We are in period of tension
Concept of ComputabilityConcept of Computability
The Turing machine
Various new computers (mould, SADs, quantum)
tension
Pure computability Physical computability
OTM, SAD1, … Assess the physical theoriesthat house these computers
Concept of GeometryConcept of Geometry
Euclidean Geometry
Lobachevskian, Reimannian Geometry
Tension
Pure geometry Physical Geometry
Euclidean Geometry General relativity, etc.LobachevskianRiemannianSchwarzschild...
Concept of GeometryConcept of Geometry
Euclidean Geometry
Lobachevskian, Reimannian Geometry
Tension
Pure geometry Physical Geometry
Euclidean Geometry General relativity, etc.LobachevskianRiemannianSchwarzschild...
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
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).
Question: Is the SAD1 ‘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: SAD1 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 1Add 1 Reveal answerRepeat 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
Just as the New Geometry left Euclidean Geometry untouched, so the New Computability leaves Turing Computability untouched.
Note
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