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. . . The problem of finding
different ways of reaching
the same result in mathematics may seem analogous to finding different
ways of looking into the next room. It might be said that any way of solving
the problem will do, so long as it is in accordance with the rules of arithmetic.
But the ways will differ according to the system of arithmetic. What one
calls mathematical problems may be utterly different. There are the problems
one gives a child, e.g., for which it gets an answer according to the rules
it has been taught. But there are also those to which the mathematician
tries to find an answer which are stated without a method of solution.
They are like the problem set by the king in the fairy tale who told the
princess to come neither naked nor dressed, and she came wearing fish net.
That might have been called not naked and yet not dressed either. He did
not really know what he wanted her to do, but when she came thus he was
forced to accept it. The problem was of the form. Do something which I
shall be inclined to call neither naked nor dressed. It is the same with
a mathematical problem. Do something which I shall be inclined to accept
as a solution, though I do not know now what it will be like. (Wittgenstein,
Lectures
1932–1935, pp. 185–186)
John Cage quoting Wittgenstein
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