# The Constructivist Retroanalyst

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: ...

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