[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.
Tim
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