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