The Church-Turing Thesis Explained Away

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The Church-Turing Thesis is a Pseudo-proposition
Mark Hogarth Wolfson College, Cambridge
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Will you please stop talking about the
Church-Turing thesis, please
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Computability
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The current view
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It is absolutely impossible that anybody who
understands the question and knows Turings
definition should decide for a different concept
Hao Wang
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Experiment escorts us last His pungent
company Will not allow an Axiom An
Opportunity EMILY DICKINSON
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The key idea is simple
There is no natural / ideal/ given way to
compute.
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And the key to understanding this new
Computability is to think about another concept,
Geometry.
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Most important slide of this talk!
The new attitude is achieved by adopting the
kind of attitude one has to Geometry, to
Computability
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Concept change
paradigm (a theory) new evidence opposition tens
ion revolution new paradigm (a new theory)
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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
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Concept of Geometry Euclidean
Geometry Lobachevskian, Reimannian
Geometry Tension Pure geometry Physical
Geometry Euclidean Geometry General
relativity, etc. Lobachevskian Reimannian Schwa
rzschild ...
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Geometry after 1915
Etc.
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Computers
New evidence has coming to light We are in
period of tension
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Concept of Computability The Turing
machine Various new computers (mould, SADs,
quantum) tension Pure computability
Physical computability OTM, SAD1, Assess
the physical theories that house these
computers
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Concept of Geometry Euclidean
Geometry Lobachevskian, Reimannian
Geometry Tension Pure geometry Physical
Geometry Euclidean Geometry General
relativity, etc. Lobachevskian Riemannian Schwa
rzschild ...
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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
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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
(Nemetis group).
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Question Is the SAD1 less real than the OTM?
Answer Is Lobachevskian geometry less real
than Euclidean geometry?
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Pure models do not compete, e.g. no infinite vs.
finite
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The true geometry is Euclidean geometry
(Euclids thesis) For pure, natural,
intuitive, different yet equivalent
axiomatizations. Against Riemannian geometry
etc. Neither is right (pseudo statement)
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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)
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What is a computer?
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What is a geometry?
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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
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The marks seem to say to us 1,2,3,
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But this is an illusion the marks alone do
nothing
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The illusion is obvious in Geometry What
is this?
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Algorithms are like geometric figures drawn on
paper Without a background geometry, the figure
is nothing Without a background computer, an
algorithm is nothing
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Note
Just as the New Geometry left Euclidean Geometry
untouched, so the New Computability leaves Turing
Computability untouched.
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Not quite... Pure Turing model Physical Turing
model this will involve some physical theory,
T, embodying the pure model.
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According to T, this machine might be warm or wet
or rigid or expanding ...
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or possess conscious states or intelligence
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T will also give an account of how, e.g., the
machine computes 347
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Arithmetic has a physical side
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Pure mathematics has a physical side
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Think Computability, think Geometry