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Re: [ontolog-forum] {Disarmed} Re: OWL and lack of identifiers

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Sun, 29 Apr 2007 17:25:27 -0500
Message-id: <07E4AFCF-16D9-43D0-AB73-AD8044B00B41@xxxxxxxx>
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 so-called "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 well-defined) 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 well-understood 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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