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Obviously, this was an informal and heuristic argument!Invoking a child’s arithmetic book, or someone’s mental state is not the stuff of a rigorous, bulletproof argument.But Turing’s argument was convincing enough that later mathematicians and scientists have for the most part been willing to accept it.
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.
In elementary arithmetic the two-dimensional character of the paper is sometimes used.
But such a use is always avoidable, and I think that it will be agreed that the two-dimensional character of paper is no essential of computation. The behavior of the computer [Turing is referring to the person performing an algorithm, not the machine!
Today we’re used to the idea that computers can be programmed to do many different things.
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In Turing’s day, however, the idea of a universal programmable computer was remarkable.
Turing was arguing that a single, fixed device could imitate algorithmic process whatsoever, provided the right program was supplied.
It was an amazing leap of imagination, and the foundation of modern computing.
His argument is too long to reproduce in full here, but it’s fun and instructive to see the style of Turing’s reasoning: Computing is normally done by writing certain symbols on paper.
We may suppose this paper is divided into squares like a child’s arithmetic book.