Church-Turing thesis is a dualism

Question

Church-Turing thesis : Every effectively calculable function is a TM-computable function.

But, hypercomputation models are strictly more powerful than TM and can solve TM-uncomputable problems on the paper.

Does this imply that, for one who believes in the Church-Turing thesis, there is only two options :

1. There is two distincts worlds : The effective one (where Church-Turing thesis applies) and the paper one (where Church-Turing thesis doesn't apply necessarily). That is the physical world and the ideas world of the classical dualism theory.
2. All the hypercomputation models are inconsistent. Using an hyper-TM in a reasoning is the same than, for instance, using an integer both even and odd in a reasoning, this can lead to false conclusions.

?

Answer 1

There's another possibility: hypercomputation is not implementable in the real world, and is only an imaginary/theoretical concept. If this is the case, then there is no contradiction with the Church-Turing thesis.

Of course, anyone can invent imaginary worlds where strange things are true. For instance, there's the computer scientist's Superman: not only can he leap tall buildings in a single bound, he can also solve undecidable problems in his head! I suppose one could imagine such a thing, but that doesn't mean it has any correspondence to the real world, and the possibility of imagining such a Superman doesn't contradict the Church-Turing thesis. The study of hypercomputation can be thought of as asking, well, ok, so Superman might not be real, but what if he were?

The Church-Turing thesis is a hypothesis about the real world, so imaginary schemes that can't be implemented in the real world don't contradict it.

Answer 2

The Church–Turing thesis only purports to describe the types of processes that qualify as “computational”. It does not assert that “hypercomputational” processes are mathematically inconsistent. Most widely accepted formalizations of mathematics do allow us to define and reason about noncomputable functions.

Some take “computation” to mean “process that can be performed on a Turing machine”. In that view, the Church-Turing thesis is just a definition of the word “computation”, and does not really claim anything.

Some take “computation” to mean “process that matches our intuitive understanding of an algorithm”. In that view, the Church-Turing thesis is just an encoding of our intuition, and its only claims are about what our intuition says.

Some take “computation” to mean “process that can be performed in the real world”. In that view, the Church–Turing thesis becomes a much stronger assertion, but it’s an assertion about physics, not about computer science. It asserts that we cannot build a device capable of hypercomputation within our physical universe. There’s still no reason to think that such a device would be inconsistent with itself mathematically.

(In fact, it’s not even obvious whether such a device would be inconsistent with known theoretical physics—see Malament–Hogarth spacetime. And there’s still a lot about physics that we don’t know. It’s possible that new physics could establish that the universe itself has a Turing-computable model, in which case the Church–Turing thesis would be true. It’s just as possible that some new phenomenon allows us perform real hypercomputation and the Church–Turing thesis would be false.)

Answer 3

No, you are confused.

Turing machines and Hypercomputation are both mathematical models, and they are both consistent because we can build mathematical models in which all functions are Turing computable, as well as models in which hypercomputable functions exist. These are of course different models. There is no mystery about having a lot of mathematical models of various theories, for instance, there are models of both euclidean and non-euclidean geometry.

The mathematical theory of computation should not be confused with "the real world". This is a bad thing to do because Turing machines are a piece of mathematics. Do you also think that infinite straight thin lines really exist in the physical world? Or maybe that they are slightly bent because of gravity?

Church's thesis is a piece of premathematical or philosophycal analysis which essentially says "Turing's mathematical definition of computation correctly models what can actually be computed in the real world". This is not a mathematical statement.

In one of the comments you made the following faulty line of reasoning: "... that imaginary worlds are from our mind and, since our brain is physical, that imaginary worlds are part of the physical world." To see why this makes no sense, consider a similar line of reasoning: "Yesterday I imagined a fairy with golden wings, and since the fairy was in my mind, and my brain is physical, the fairy is part of the physical world". The problem is that you are confusing the thoughts in your mind with their physical representation (the electro-chemical functioning of your brain).

A Belgian artist put it very eloquently: