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But Turing’s argument was convincing enough that later mathematicians and scientists have for the most part been willing to accept it.
Might he have missed something in his informal reasoning about what an algorithm is?
In 1985, the English physicist David Deutsch suggested a deeper approach to the problem of defining what is meant by an algorithm.
Except instead of starting with two numbers, you’d start with a mathematical conjecture, and after going through the steps of the algorithm you’d know whether that conjecture was provable.
The algorithm might be too time-consuming to use in practice, but if such an algorithm existed, then there would be a sense in which mathematics was knowable, at least in principle.
Roughly speaking, Hilbert’s 1928 problem asked whether there exists a general algorithm a mathematician can follow which would let them figure out whether any given mathematical statement is provable.
Hilbert’s hoped-for algorithm would be a little like the paper-and-pencil algorithm for multiplying two numbers.
Attacking Hilbert’s problem forced Turing to make precise exactly what was meant by an algorithm.
To do this, Turing described what we now call a : a single, universal programmable computing device that Turing argued could perform any algorithm whatsoever.
I assume then that the computation is carried out on one-dimensional paper, i.e. ] at any moment is determined by the symbols which he is observing, and his “state of mind” at that moment.
We may suppose that there is a bound to the number of symbols or squares which the computer can observe at one moment.