Essays On Curiosity

Essays On Curiosity-17
But Turing’s argument was convincing enough that later mathematicians and scientists have for the most part been willing to accept it.

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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.


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