[FOM] Fwd: invitation to comment
Kevin Watkins
kevin.watkins at gmail.com
Sat May 21 17:57:23 EDT 2011
2011/5/21 <Andre.Rodin at ens.fr>:
> Here in my view is the core problem. From
> a mathematical viewpoint formal propositions and formal theories are
> mathematical *objects* like points, spaces, fields, numbers and whatnot. (I was
> very glad to hear a similar statement from Voevodsky in his video.) One may
> prove about these things mathematical theorems like "formal proposition P
> implies formal proposition Con(X) over the base formal theory B". I take such
> theorems to be mathematical theorems proper on equal footing with PP. But at
> this point mathematics ends and the following speculative arguments is put
> forward: FPP *represents* or *expresses* PP in such a way that studying
> properties of FPP (which is an object!) one may come to know something
> essential about PP. This speculative assumption is often strengthen by saying
> that PP as it stands is somewhat unclear and that FPP clarifies the content of
> PP.
> My guess is that such an assumption is not justified. Just think how different
> are PP and FPP. FPP is a pulpable combinatorial (syntactic) object; one can
> play with it like with elementary geometrical constructions. This is where the
> sense of precision and clarity comes from. But what this object has to do with
> PP which is a *theorem* but not a mathematical object, about which one may
> prove this or that theorem?
> FPP likely implies Con(PA) over some weak base formal theory B. So if PA is
> in-consistent FPP is false. This does not imply that PP is false and that
> Perelman's argument is flawed.
> In order to claim that FPP is equivalent to PP people use various
> non-mathematical arguments. I cannot imagine a *mathematical* argument that
> could fill this gap.
Thank you... I think that explanation helps clarify for me how your
point of view differs from that of some others on this list.
I should stop here, but I can't resist making another remark.
One might imagine a (superficially) similar argument made by a
geometer in regard to the trisection of the angle. Namely, that the
trisection of the angle should be considered an open problem, because
the purported "proof" that the trisection is impossible is not
geometric: the impossibility proof hinges on replacing geometrical
constructions with certain formal objects representing geometrical
constructions (polynomials over a field), and according to the
geometer's point of view, this replacement (of a geometrical
construction by its formal counterpart in terms of polynomials over a
field) has done such violence to the original question of whether the
angle can be trisected that the problem, from the geometrical point of
view, should be considered open.
It could be that you sympathize with the geometer in this story. Or
it could be that I am overlooking a crucial difference between your
point of view and the geometer's.
Kevin
More information about the FOM
mailing list