the Church-Turing thesis
/ Church -> CHURCH, Turing -> TYOOR-ing /
Long before computers existed, mathematicians had an intuitive idea of an 'effective procedure' or 'algorithm': a finite list of completely mechanical, unambiguous steps that a clerk with paper and pencil, plenty of time, and no cleverness could carry out. The puzzle was that this intuitive idea had no precise definition. The Church-Turing thesis is the bold claim that we finally captured it: anything that can be computed by any mechanical procedure whatsoever can be computed by a Turing machine.
The thesis equates an informal notion ('effectively computable by some algorithm') with a precise mathematical one ('computable by a Turing machine'). It is named for Alonzo Church, who defined computability via the lambda calculus, and Alan Turing, who defined it via his machines; both, working in the 1930s, arrived at the very same class of functions. Crucially, this is a thesis, not a theorem. It cannot be proved, because one side of the equation, 'effectively computable', is an informal, intuitive idea, and you cannot prove an informal notion equals a formal one. It could, in principle, be refuted by exhibiting a clearly mechanical procedure that no Turing machine can carry out, but in roughly a century no one has.
What the thesis does assert: the boundary of what Turing machines can do is the boundary of mechanical computation, so a problem proved undecidable for Turing machines (like the halting problem) is undecidable for any algorithm, in any programming language, on any conceivable digital computer. What it does NOT assert: it says nothing about speed or efficiency, nothing about what humans can intuit or prove, and (in its basic form) nothing about exotic physical processes. Stronger, more speculative versions, the physical thesis (nature itself computes nothing uncomputable) and the extended thesis (a Turing machine can simulate any model with only polynomial slowdown), go beyond this and are far more contested.
When a textbook says 'there is an algorithm to multiply two numbers', it gives a high-level recipe in plain English and does not bother encoding a Turing machine. The Church-Turing thesis is the unspoken license for this: any such clearly mechanical recipe could be turned into a Turing machine, so describing the algorithm informally is enough.
The thesis is why we may describe algorithms in prose and still call them Turing-computable.
It is a thesis, not a theorem: it cannot be proved because 'effectively computable' is informal. It also says nothing about efficiency, and it is not a claim about what the human mind can grasp.