[FOM] "Hidden" contradictions

Timothy Y. Chow tchow at alum.mit.edu
Tue Aug 13 14:18:08 EDT 2013

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 

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, 


or for a less hackneyed example:



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