[FOM] "Hidden" contradictions
Prof. Ranjit Nair
director at cpfs.res.in
Thu Aug 15 11:52:58 EDT 2013
Elementary error. The example has x raised to the power 2. Not
multiplication by 2 as Prof. Mendelson supposes
Ranjit Nair
________________________________________________
From:"elliott mendelson" <emenqc at msn.com>
To:"Foundations of Mathematics" <fom at cs.nyu.edu>
Date:Tue, August 13, 2013 3:57 am
Subject:Re: [FOM]
"Hidden" contradictions
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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 Mendelson
emenqc at msn.com
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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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Professor Ranjit Nair
Director
Centre for Philosophy & Foundations of Science
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