And back to the first part of the OP, Turing, like many of his contemporaries, was interested in procedures rather than solutions. He didn’t care what the answer was, particularly, just the various means of reaching the answer. The “satisfactory solutions” to which he referred are those means.
That’s an interesting interpretation, andros, but if you check the link, he is talking about two methods - systematic and random. The ‘solutions’ clearly refer to the equation ‘for n between 50 and 200 equal to the square of the sum of its digits’.
Even if you’re correct, and this is not a glaring error in a seminal paper missed by Turing, his typist, editor, and the mathematician in the street, his conclusion - the random method is better - also fails to pass muster.
If the solution is a square between 50 and 200, list the possible solutions (64, 81, 100, 121, 144, 169, 196). Are the sum of the digits equal to the square root of any of these seven numbers, (as opposed to the original range of 151 numbers)? Yes, one: 81.
This is a much more elegant solution than brute forcing lockstep or randomly the terrain.
I realize it is a philosophical paper (it was published in Mind, after all) and not germaine to the point he was attempting to make, but it is another example of science and philosophy not making good bedfellows, like Schrodinger’s Cat.
It seems to me that often Heisenberg’s Uncertainty Principle or Godel’s Incompleteness Theorum will be appliqued to inappropriate surfaces.
Let us call that Pinky’s Inappropriateness Theorum.
Its interesting to remember that shrodinger said ’ i wish i had never mentioned that damm cat’.
I wonder why ?
At this juncture, I feel it is my sworn duty to provide the obligatory link to Cecil Adams’s poem about Schrödinger’s Cat. There.