by Alexander George, 2002

Constructivism in logic or mathematics issues from the perspective that refuses to allow a general appeal to the determinate nature of mathematical reality in the course of proving assertions. Rather, a particular proof, or construction, must accompany each claim, the precise nature of which will vary with the logical structure of the claim being established.

For instance, in order to prove a disjunction “X or Y,” the constructivist demands nothing less than a proof of X or a proof of Y. The non-constructive logician – sometimes called the “classical” logician – makes no such demand; he might be willing to accept an assertion of that form even though he is neither in a position to prove X nor prepared to prove Y.

Consider the particular disjunction “X or not-X.” A classical logician will accept every such disjunction, whatever X might be; indeed, this is often known as the Law of the Excluded Middle. This is because such a logician imagines that, for any meaningful statement X, the mathematical universe is such that either X is true or X is false. The constructivist, by contrast, makes no such assumption: if one wishes to assert “X or not-X” then one must either present a proof of X or present a proof of not-X. It might well be that one is neither in a position to prove X nor in a position to prove not-X; for the constructivist, there would then be nothing for it but to refrain from asserting the disjunction “X or not-X.”

These distinctions can be brought out nicely by considering an example from retrograde analysis, a particular kind of chess problem in which the solver must use her powers of deduction, and of course her knowledge of the rules of chess, to determine facts about the history of the position with which she is presented.

Consider the following, for example: ...