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In his famous paper, Turing briefly describes his machine and then states (p. 232) (emphasis mine):

It is my contention that these operations include all those which are used in the computation of a number. The defence of this contention will be easier when the theory of the machines is familiar to the reader.

Does this mean that all numbers that could ever be computed by a machine or a human being can be computed by a Turing Machine?

If yes, what proof has Turing, beside "It is my contention"?
Does Turing defend this contention later on, in the same paper or elsewhere?

Evariste
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A contention is something that one asserts to be true. Turing's contention here is essentially the Church–Turing thesis: that Turing machines can compute everything that can be reasonably computed. This isn't something that can be proven without a definition of "reasonably computed". But, even if one were to define that term, it would be arguable that the definition wasn't a good one and that, again, isn't something that can be proven.

Turing's paper includes a lot of material about computable sequences of numbers and also a sketch proof that Turing machines and the lambda calculus are equally powerful. That constitutes quite a lot of evidence for the original contention, though I didn't notice Turing explicitly mentioning this when I skimmed the paper just now.

In particular, one might choose to define "reasonably computable" as meaning "computable by any possible physical system." Well, Turing machines can simulate every physical process we're aware of, including quantum mechanics. But, of course, we can't prove that there are no physical processes we don't know about yet that can't be simulated by Turing machines.

David Richerby
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  • So you mean basically: we don't know for sure (in the sense of a mathematical proof) that "we" cannot compute something that a Turing Machine can't? How then do we know e.g. a quantum computer is equivalent to a Turing Machine (https://cs.stackexchange.com/a/23164/74452), or someone won't find a Silver Bullet that beats anything we know today? – Evariste Aug 13 '17 at 10:17
  • @Evariste I've added a paragraph discussing physical processes. – David Richerby Aug 13 '17 at 10:26