[FOM] Certainty versus boundedness

Timothy Y. Chow tchow at alum.mit.edu
Sat May 3 17:56:37 EDT 2008

A common response to the question of what distinguishes mathematical 
knowledge from other kinds of knowledge is that mathematical knowledge 
enjoys a unique kind of certainty.  Objections to this claim of certainty 
should be familiar to most readers of this list.

An alternative answer is that the relevant evidence for a piece of 
mathematical knowledge (i.e., a theorem) is *bounded* (i.e., the proof has 
finite length).  That is, if a proof of a theorem is produced, then all 
that needs to be checked is that the proof is correct.  If the proof is 
correct, then future evidence cannot change the verdict.  In contrast, in 
any other domain of knowledge, one cannot rule out the possibility that 
future evidence may change the verdict.

One objection might be that a proof is not a proof until it is formalized 
in some formal system, and that one cannot rule out the possibility that 
the formal system will be found inconsistent in the future.  I think that 
this objection can be met.  Obviously, one may deny that a proof has to be 
formal.  Alternatively, a formalist can maintain that the real piece of 
mathematical knowledge is not the theorem, but the proof itself.  Even if 
the system is found inconsistent, it remains true that said theorem is 
provable in said system.  *That* fact at least remains impervious to 
future evidence.

On this point of view, mathematical knowledge is not necessarily certain, 
but the scope of uncertainty is carefully circumscribed: We just can't be 
absolutely certain that the alleged proof *is a proof* because it might 
contain a bug in it.

I am wondering if this point of view has been developed in detail by any 
philosophers of mathematics.


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