What's the significance of the Church-Turing Thesis?

Question

My understanding is that the thesis is essentially a definition of the term "computable" to mean something that is computable on a Turing Machine.

Is this really all there is to it? If so, what makes this definition so important? What makes this definition so significant as to warrant having it's own name?

In most other branches of mathematics, a definition is an important part of the scaffolding, but not a result onto itself. In the case of the Church Turing Thesis, it seems like there must be more, but all I can see is the definition.

So, what is the significance of the Church-Turing Thesis?

Answer 1

The idea is that we have an informal notion of "computable" - that is, "something that can be computed". (This is explicitly not a precise definition). We also have a formal definition of "computable", that is, "computable by a Turing machine". The Church-Turing thesis is that these two notions coincide, that is, anything that "should" be computable is in fact computable by a Turing machine. (It's pretty clear that anything that is computable by a Turing machine is computable in the more informal sense).

Put another way, the Church-Turing thesis says that "computable by a Turing machine" is a correct definition of "computable".

Answer 2

The answer from user73985 explains the content of the Church-Turing thesis, but I'd like to add a few words about its value; why do we want it.

The first benefit that we get from this thesis is that it lets us connect formal mathematical theorems to real-world issues of computability. For example, the theorem that the word problem for groups is Turing-undecidable has the real-world interpretation that no algorithm can solve all instance of the word problem.

The second benefit is in mathematics itself, specifically in computability theory. Published proofs that something is Turing-computable almost never proceed by exhibiting a Turing-machine program, or indeed a program in any actual computing language. Sometimes, if the matter is simple enough, they provide some sort of pseudo-code. Most often, though, they merely give an informal description of an algorithm. It is left to the reader to see that this actually does give an algorithm (in the intuitive sense) and therefore, by the Church-Turing thesis, could be simulated by a Turing machine. The usual situation is that, although experts in Turing-machine programming would (if they existed) be able to routinely convert the intuitive algorithm into a Turing machine program, the program would be too large and complicated to be worth writing down.