[FOM] 461: Reflections on Vienna Meeting
Timothy Y. Chow
tchow at alum.mit.edu
Tue Jun 14 21:35:32 EDT 2011
Alex Simpson wrote:
> AM: How is he confused? It *might* be inconsistent. I mean, it would
> be a shock. That would be a genuine crisis. I think the inconsistency
> of ZFC would not be. ... None of us know that PA is consistent.
>
> I don't read Angus here as saying that he believes the consistency of PA
> to be a "legitimate mathematical problem".
Does this just come down to a linguistic quibble about what the term
"legitimate mathematical problem" means? Compare the following
statements:
1. The Riemann Hypothesis *might* be false. I mean, it would be a shock.
But none of us *knows* that the Riemann Hypothesis is true.
2. The square root of 2 *might* be rational. I mean, it would be a shock.
But none of us *knows* that the square root of 2 is irrational.
#1 is a completely normal assertion. #2 sounds like someone is trying to
pick a philosophical fight. The main reason for the distinction is that
it is considered a proven fact that the square root of 2 is irrational,
while the Riemann Hypothesis is certainly not a proven fact. Thus,
implicit in Angus's statement is a claim that the consistency of PA is not
a proven fact in the sense that other mathematical theorems are proven
facts. I think that's enough to shock Harvey, regardless of how one
chooses to define the term "legitimate mathematical problem."
Tim
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