[FOM] "Hidden" contradictions

[elliott mendelson]

Dear Professor Steiner:                                          In the example you gave that involved dividing by x, the result would have been 2 = 1, a contradiction.So, people would have seen the mistake right away.Best wishes,Elliott Mendelsonemenqc at msn.com
From: mark.steiner at mail.huji.ac.il
Date: Mon, 12 Aug 2013 14:39:07 +0300
To: FOM at cs.nyu.edu
Subject: [FOM] "Hidden" contradictions


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 calculations.
 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?
 Thanks,Mark Steiner  

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