[FOM] "Hidden" contradictions

[Mark Steiner]

I appreciate this response.  However, my physicist friends tell me that
the theory known as QED is thought to be inconsistent, but people use it
anyway, with great success in predictions.  I think what this means is the
claim that there is no way to formalize QED in a consistent axiomatic
system.  If this is right, then there is a sense in which formal systems
do play some kind of role in physics.

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Timothy Y. Chow
Sent: Tuesday, August 13, 2013 9:18 PM
To: fom at cs.nyu.edu
Subject: Re: [FOM] "Hidden" contradictions

Mark Steiner wrote:

> 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?

I think that there are serious problems with the way this question is
phrased.

For a start, it's not clear what you mean by "inconsistent systems
actually yielding false theorems."  For example, how is this different
from just making a mathematical error?  You make an error, producing a
provably false statement.  In effect, at that point you are starting to
"work in an inconsistent system," because you are taking your false
statement for granted.  Then before discovering the mistake, you reach
something that you decide is worth calling a theorem.  Behold, an example
of an inconsistent system actually yielding a false theorem!  This is not
just a hypothetical scenario.  The published mathematical literature
contains many provably false theorems, some of which have not been noticed
because nobody has bothered to read the papers in question.  Since it is
standard mathematical practice to take published results as axioms, the
mathematical literature as a whole is one giant inconsistent system which
has yielded false theorems, and if you have a vivid imagination then you
can easily picture some false theorem that *could* make a "bridge fall
down," especially since your scare quotes imply that no actual bridges
need to be harmed in the making of this imaginary world.

More to the point, if your response is that you're restricting your
attention to explicitly articulated formal systems that have been widely
adopted for use in science and engineering, then the problem is that
scientists and engineers never work explicitly in any formal system.
F.o.m. simply does not play that role in the real world.  If a bridge
actually falls down for some theoretical reason, it's going to be because
the mathematical *model* fails to take something important into account,
e.g.,

   http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_%281940%29

or for a less hackneyed example:

   http://www.damninteresting.com/a-potentially-disastrous-design-error/

Tim
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