by one "Clarence Williams," now contains a link to
the new Dharwadker site by "Robert Stewart." Neither of these
names links to a homepage.
Update of August 5, 2005:
The new web page by "Robert Stewart" discussed above is dated July 31, 2005.
Also on July 31, "Bob," rather than "Robert," "Stewart" posted an
item at the newsgroup sci.math that said he was a "working
mathematician" interested in the
implications of Gödel's incompleteness theorem. Here is my
response to
that posting.
From Math Forum's sci.math postings 
http://mathforum.org/kb/message.jspa?messageID=3874695&tstart=0#replytree
Re: What does Gödel's Incompleteness mean for the Working Mathematician?
Posted:
Aug 3, 2005 1:33 PM
By: Steven H. Cullinane
"Bob Stewart" asked on July 31, 2005, "What does Gödel's
incompleteness mean for the working mathematician?"
"Stewart" supplied as background a passage he plagiarized from Carl Boyer's "A History of Mathematics."
Here is "Stewart"'s version:
"Gödel
showed that within a logical system, propositions can be formulated
that are undecidable or undemonstrable within the axioms of the system.
That is, within the system, there exist certain clearcut statements
that can neither be proved nor disproved. Hence one cannot, using the
usual methods, be certain that the axioms will not lead to
contradictions."
Here is Boyer's version:
"Gödel
showed that within a rigidly logical system such as Russell and
Whitehead had developed for arithmetic, propositions can be formulated
that are undecidable or undemonstrable within the axioms of the system.
That is, within the system there exist certain clearcut statements
that can neither be proved nor disproved. Hence, one cannot, using the
usual methods, be certain that the axioms of arithmetic will not lead
to contradictions." (Wiley paperback, 2nd edition, 1991, page 611)
"Hence"?
This nontrivial "hence" of Boyer's conceals some history of enduring
interest. It suggests the startling conclusion that
"... there is no rigorous justification for classical mathematics."
"Stewart" may, of course, dismiss this conclusion as the ravings of a crank. (See "Stewart"'s recent sci.math postings.)
For more on Boyer's "hence" and "rigorous justification," see
http://hps.elte.hu/~redei/cikkek/ems.pdf.
Update of August 8, 2005:
Further remarks by "Robert Stewart" on Dharwadker's "proof" may be found at the
Wikipedia discussion of the fourcolor theorem.
Here is a sample of that discussion. "Stewart" is attacking mathematician
Jitse Niesen, who has questioned Dharwadker's "Lemma 8" proof:
"It is a trivial exercise to show that distinct left coset
representatives always belong to distinct right cosets. Are you
familiar with elementary Group Theory?"
As the context shows, "Stewart" means to say that "representatives of
distinct left cosets always belong to distinct right cosets."
But, as Niesen points out, this is false. For an example, see the
discussion of elementary properties of cosets at
a web page on quotient groups
titled "Lab6.htm" at the Journal of Online Mathematics. The
figure below, taken from that web page, shows the cosets of a
twoelement subgroup K of S
_{3}. Note that r
_{1} and m
_{2}
belong to distinct left cosets but do not belong to distinct right cosets.
To check this table, replace
r
_{1} by (abc),
r
_{2} by (acb),
m
_{1} by (ab),
m
_{2} by (ac),
m
_{3} by (bc).
Then we have, for the right cosets above,
m
_{1}r
_{1}=(ab)(abc)=(ac)=m
_{2},
m
_{1}r
_{2}=(ab)(acb)=(bc)=m
_{3},
m
_{1}m
_{1}=(ab)(ab)=1,
m
_{1}m
_{2}=(ab)(ac)=(abc)=r
_{1},
m
_{1}m
_{3}=(ab)(bc)=(acb)=r
_{2},
and for the left cosets above,
r
_{1}m
_{1}=(abc)(ab)=(bc)=m
_{3},
r
_{2}m
_{1}=(acb)(ab)=(ac)=m
_{2},
m
_{1}m
_{1}=(ab)(ab)=1,
m
_{2}m
_{1}=(ac)(ab)=(acb)=r
_{2},
m
_{3}m
_{1}=(bc)(ab)=(abc)=r
_{1}.
"Stewart"'s further remarks to Niesen seem relevant:
"You have just shown above that you are not familiar with the elementary
properties of cosets! How are you going to check Dharwadker's proof?"
Updates of September 711, 2005:
Coset Representatives:
Two Opposing Views
by Steven H. Cullinane,
Sept. 7, 2005
From "
Common Systems of Coset Representatives,"
by Ashay Dharwadker, Sept. 2005, at
http://www.geocities.com/dharwadker/coset.html 
"Using the axiom of choice,
we prove that given any group
G and subgroup H, there always
exists a common system of coset representatives
of the left and right cosets
of H in G."

From "
Hopf Algebra Extensions and Monoidal Categories" (pdf),
by Peter Schauenburg, at
http://www.mathematik.unimuenchen.de/~schauen/papers/haemc.pdf 
Update of Sept. 11, 2005:
Dharwadker has changed his claim. He now says,
"Using the axiom of choice, we prove that given any group G and a finite subgroup H, there always exists a common system of coset representatives of the left and right cosets of H in G."

This new claim avoids the difficulty described by
Schauenburg above, since H is now assumed to be finite. It also meets the following conditions stated by Fred Galvin (sci.math, Feb. 20, 2003):
"... if H is a finite subgroup of a (finite or infinite) group G, then
there is a common transversal for the system of left cosets and the
system of right cosets. This is still true for an infinite subgroup of
finite index, but it breaks down for infinite subgroups of infinite
index.
See L. Mirsky, Transversal Theory, Academic Press, 1971, ISBN
0124985505."
