On Apr 29, 2007, at 12:01 PM, Steve Newcomb wrote:
> Ingvar Johansson <ingvar.johansson@xxxxxxxxxxxxxxxxxxxxxx> writes:
>>> Steve Newcomb schrieb:
>>>
>>> If this is your example of a "fact", then I remain unsatisfied.
>>> Mathematics is grounded in itself. So what?
>
>> Some questions to Steve N:
>> (1) Is it a fact that mathematics is grounded in itself? Or:
>
> I see that I misspoke. As I recall, Russell says that the
> correspondence between mathematics and reality can neither be
> proven nor disproven, because we lack access to the catalog of all
> things, and therefore we cannot know whether that catalog contains
> a listing for itself. (01)
This is just Russellian hash, I'm afraid. You seem to be mixing up
Russell's views about how mathematics applies to reality (a very
interesting and vexing philosophical problem) with Russell's
*paradox*. The latter  and in particular the idea of a "catalog of
all things"  has NOTHING WHATEVER to do with the former. The idea
of an exhaustive catalog has to do with (one version of) the famous
paradox that bears Russell's name, which he discovered in the work of
the great German logician Gottlob Frege. The problem of paradox in
the foundations of set theory was solved (or, at least, avoided) by
the move from socalled "naive" set theory to axiomatic set theory in
the early 1900s. (02)
> I should have said that mathematics cannot itself provide its own
> grounding, and that such grounding cannot be proven to exist. (03)
It's far from clear what you have in mind by "grounding" here (a
proof?). But there is quite a lot of literature on how basic
arithmetic and set theory are "conceptually" grounded in our
perception and manipulation of objects  for example, that bringing
together two (relatively stable and welldefined) objects with three
others always yields five objects. The universality of these
experiences and the universality of the resulting basic mathematics
(in every culture that has any mathematics at all) provides good
reason for thinking that the ability to recognize mathematical truths
is a universal human trait, not an ephemeral cultural contingency. (04)
> (In that light, the fact that mathematics is such a large and
> imposing edifice can be seen as a naive kind of evidence that it
> provides its own grounding, Gödelian misgivings notwithstanding. (05)
Gödel's work justifies no "misgivings" whatever about mathematics or
its "grounding", whatever exactly that might mean. His famous
theorems are rigorous, exceptionally wellunderstood mathematical
propositions about the limitations of formal systems. That we
understand those limitations, and can formulate them ways that are
themselves amenable to mathematical proof, if anything, should
reinforce our faith in the power of mathematics. (06)
I highly recommend Torkel Franzen's marvelous little book _Godel's
Theorem: An Incomplete Guide to Its Use and Abuse_ (http://
tinyurl.com/2r8wqd) to anyone who wants both a good layman's
understanding of Gödel's incompleteness theorem(s) and a fascinating
tour of the astonishing variety of philosophical/theological/mystical
theses into which it has been transmogrified and which it is claimed
to entail  among which, I'm afraid, we can include "Gödel's
Theorems justify misgivings about the grounds of mathematics." ;) (07)
Chris Menzel (08)
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