[FOM] "Hidden" contradictions

Mark Steiner mark.steiner at mail.huji.ac.il
Mon Aug 12 07:39:07 EDT 2013

In 1939, Turing debated Wittgenstein in a class the latter was giving at
Cambridge.  Turing argued that working with an inconsistent system could
result in “bridges falling down.”

To Wittgenstein’s argument that we don’t need to reason from a
contradiction, Turing replied that we might not ever see the contradiction,
and yet the inconsistency could still result in wildly incorrect

It would seem that in principle that Turing was correct (this is an example
of Wittgenstein himself): suppose we give the axioms for multiplication
without limiting the cancellation law  ab = ac à b=c to nonzero a.  One
could imagine that many students would not notice the problem.  Indeed, in
a recent survey at the City University of New York, 93% of precalculus
students asked to solve the equation x^2 = x, divided by x and got x = 1
(as the only solution).  One could dress this problem up quite a bit.  In
these cases, one gets correct solutions, but misses others, and this could
indeed make “bridges fall down.”

Are there any historical examples in which inconsistent systems actually
yielded false theorems that could have made “bridges fall down” without
anybody noticing the inconsistency?


Mark Steiner
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