by one "Clarence Williams," now contains a link to the new Dharwadker site by "Robert Stewart."  Neither of these names links to a homepage.

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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#reply-tree 
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 clear-cut 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 clear-cut 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.

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Update of August 8, 2005:

Further remarks by "Robert Stewart" on Dharwadker's "proof" may be found at the Wikipedia discussion of the four-color 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 two-element subgroup K of S₃.  Note that r₁ and m₂ belong to distinct left cosets but do not belong to distinct right cosets.



To check this table, replace

r₁ by (abc),
r₂ by (acb),
m₁ by (ab),
m₂ by (ac),
m₃ by (bc).

Then we have, for the right cosets above,

m₁r₁=(ab)(abc)=(ac)=m₂,
m₁r₂=(ab)(acb)=(bc)=m₃,
m₁m₁=(ab)(ab)=1,
m₁m₂=(ab)(ac)=(abc)=r₁,
m₁m₃=(ab)(bc)=(acb)=r₂,

and for the left cosets above,

r₁m₁=(abc)(ab)=(bc)=m₃,
r₂m₁=(acb)(ab)=(ac)=m₂,
m₁m₁=(ab)(ab)=1,
m₂m₁=(ac)(ab)=(acb)=r₂,
m₃m₁=(bc)(ab)=(abc)=r₁.

"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?"

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