The Church-Turing Thesis is a Pseudoproposition
Mark Hogarth
Wolfson College, Cambridge
Will you please stop talking about the ChurchTuring 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 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 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 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, SAD1, … 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?
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 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