[FOM] two questions about Godel
Nik Weaver
nweaver at math.wustl.edu
Fri Feb 17 03:26:19 EST 2006
Harvey Friedman:
> Explaining why you disagree so condidentally with Kurt Godel
> is a good and fair question that I now reiterate. Where and
> why did Godel go wrong?
It's a fair question if you have in mind some particular
argument of Godel's in favor of platonism. You don't specify
one. The argument of his that I'm familiar with is the one
about having an intuitive perception of the objects of set
theory that is analogous to sense perception. I can't tell
you *why* Godel went wrong on this --- that seems like a
psychological question --- but I can tell you what I think
is wrong with the argument.
I don't believe we do have any perception of the actual objects
of set theory, as I think these supposed objects are fictitious.
What Godel is thinking of is our intuitive perception of the
structure that is supposed to be embodied in these fictional
objects, not any direct perception of the objects themselves.
When I intuitively perceive that 2 + 2 = 4 it is not the case
that my mind is in some way reaching out into the netherworld
and grasping hold of some abstract entities, "2" and "4". Rather
I have, say, a mental picture of two dots approaching two other
dots and becoming four dots. The actual picture involved can
vary and it might not need to be involve visual images at all, but
I think Godel's analogy with sense perception is telling because
no matter how one intuitively perceives that 2 + 2 = 4, undoubtedly
some brain structures used in sense perpection will be involved.
I imagine the same is true in all instances of mathematical
intuition. Some brain structures used for sensory processing
will always be involved, and that is why we have the feeling of
direct perception of mathematical truth that is so like sensory
perception. This is no evidence for the actual existence of
non-physical abstract mathematical objects.
> Do you think that he ever subscribed to one of the controversial
> stopping places for predicativity?
The question doesn't make sense; you're confused about my work
on predicativism. You know that Feferman and Schutte proposed
Gamma_0 as a stopping point for predicativism. You are also
aware that I have challenged this proposal and argued that one
can predicatively access ordinals beyond Gamma_0. You have
evidently read the title of my paper "Predicativity beyond
Gamma_0", but I can see you haven't read the paper itself
because you've somehow got the idea that I have put forward
some other "stopping place" for predicativism.
In fact I have proposed formal systems which (I argue) are
predicatively valid and go well beyond Gamma_0; the strongest
one gets up to phi_{Omega^omega}(0). However, predicativism
doesn't stop there. It would be easy to strengthen that
system a little bit in a predicatively legitimate way and get
a little further. The open challenge is to find a systematic
way of (predicatively legitimately) strengthening it which goes
substantially farther and captures a larger proof-theoretically
significant ordinal.
Moreover, I extensively argue in the Gamma_0 paper that it is
highly implausible that one could ever precisely identify any
ordinal as the "stopping point" of predicativism. So your
declarations that predicativism is too vague to be identified
with a precise ordinal is not as distressing to me as you might
think.
However, your assumption that I have merely come up with a
different "story" whose correspondence with predicativism is
neither better nor worse than the one adopted by Feferman and
Schutte is not correct. I don't have any different "story".
I argue that the principles accepted as predicative by Feferman
and Schutte in fact justify systems which go beyond Gamma_0.
I also argue very directly and at length that there is no
coherent philosophical stance which would lead one to accept
all ordinals less than Gamma_0 but not Gamma_0 itself.
Anyway your question is strange because Godel had become a
platonist long before Feferman and Schutte made their ordinal
analysis. Some time during the 1950s Hao Wang had the idea
that one could predicatively access L_alpha for every recursive
alpha but not beyond. I don't know what Godel thought of that
but he was a platonist by then too. I'm not a Godel scholar
though, and I'm not particularly interested in WWGD (What Would
Godel Do) questions, and I don't think I'll answer any more of
them.
Nik
More information about the FOM
mailing list