Ashay Dharwadker,
The Four-Color Theorem,
and Usenet Postings


by Steven H. Cullinane
 July 25, 2005 -
July 30, 2008

July 25, 2005

In 2000, Ashay Dharwadker claimed to have proved the four-color theorem by an argument that involved the Steiner system S(5,8,24).  Since my own work involved a connection, via the MOG of R. T. Curtis, to this Steiner system, I wrote a rough critique of Dharwadker's claim.  That critique has led* to my being called

a pathological liar,
a sociopath,
a crank,
a nut,
someone who should be "locked up in gitmo,"
a moron,
a lunatic,
a fraudster,
paranoid,
an idiot,
stupid,
a pompous fool,
and evil.

The above may be of some use to students of crankery.

* For the trail that leads from my critique of Dharwadker to the above list of epithets, see

Non-computer proof of 4 color Theorem,
2000 Oct. 13-Nov. 30,
sci.math, 23 posts

Open Directory Abuse,
2002 Oct. 2-Oct. 14,
sci.math, 8 posts

Open Directory Abuse,
2002 Oct. 2-Oct. 15,
comp.misc, 2 posts

Steven Cullinane is a Liar,
2002 Nov. 1-Nov.16,
geometry.research, 2 posts

Four-colour proof claim,
2003 Aug. 10-Sept.1,
sci.math, 9 posts

Proof of 4 colour theorem No computer!!!,
2003 Aug. 10-Aug. 20,
alt.sci.math.combinatorics, 8 posts

Steven Cullinane is a Crank,
2005 July 5-July 21
sci.math, 70 posts


Update of August 3, 2005:

The name "Bob Stewart," apparently a pseudonym, appears in postings to the above topic "Steven Cullinane is a Crank."  The "Stewart" postings there contain the epithets "lunatic," "fraudster," "paranoid," "idiot," "stupid," "pompous fool," and "crank."

A possible connection of "Bob Stewart" to Dharwadker --
Today I noticed a new site,
 
Ashay Dharwadker's Proof of
 The Four Color Theorem - A Review


at http://fourcolourtheorem.tripod.com/.

The site's author is "Robert Stewart."

A related site that has existed for some time,

On Dharwadker's Magnificent Proof,

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#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.


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.

The image “http://www.log24.com/log/pix05B/050808-cosets.png” 
cannot be displayed, because it contains errors.

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?"


Updates of September 7-11, 2005:

Remarks related to the above discussion of cosets--

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.uni-muenchen.de/~schauen/papers/haemc.pdf --

The image “http://www.log24.com/theory/Dharwadker/CosetReps.jpg” cannot be displayed, because it contains errors.

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 0-12-498550-5."



Update of May 31, 2007:

L'Affaire Dharwadker continues...

Blitz by anonymous
New Delhi user

From Wikipedia on 31 May, 2007:

Shown below is a list of 25 alterations to Wikipedia math articles made today by user 122.163.102.246.

All of the alterations involve removal of links placed by user Cullinane (myself).

The 122.163... IP address is from an internet service provider in New Delhi, India.

The New Delhi anonymous user was apparently inspired by an earlier blitz by Wikipedia administrator Charles Matthews....

Related material:

Ashay Dharwadker and Usenet Postings
and Talk: Four color theorem/Archive 2.
See also some recent comments from 122.163...
at Talk: Four color theorem.

May 31, 2007, alterations by
user 122.163.102.246:

  1. 17:17 Orthogonality (rm spam)
  2. 17:16 Symmetry group (rm spam)
  3. 17:14 Boolean algebra (rm spam)
  4. 17:12 Permutation (rm spam)
  5. 17:10 Boolean logic (rm spam)
  6. 17:08 Gestalt psychology (rm spam)
  7. 17:05 Tesseract (rm spam)
  8. 17:02 Square (geometry) (rm spam)
  9. 17:00 Fano plane (rm spam)
  10. 16:55 Binary Golay code (rm spam)
  11. 16:53 Finite group (rm spam)
  12. 16:52 Quaternion group (rm spam)
  13. 16:50 Logical connective (rm spam)
  14. 16:48 Mathieu group (rm spam)
  15. 16:45 Tutte–Coxeter graph (rm spam)
  16. 16:42 Steiner system (rm spam)
  17. 16:40 Kaleidoscope (rm spam)
  18. 16:38 Efforts to Create A Glass Bead Game (rm spam)
  19. 16:36 Block design (rm spam)
  20. 16:35 Walsh function (rm spam)
  21. 16:24 Latin square (rm spam)
  22. 16:21 Finite geometry (rm spam)
  23. 16:17 PSL(2,7) (rm spam)
  24. 16:14 Translation plane (rm spam)
  25. 16:13 Block design test (rm spam)

The deletions should please Charles Matthews and fans of Ashay Dharwadker's work as a four-color theorem enthusiast and as editor of the Open Directory sections on combinatorics and on graph theory.

There seems little point in protesting the deletions while Wikipedia still allows any anonymous user to change their articles.

-- Cullinane 23:28, 31 May 2007 (UTC)


Update of July 30, 2008:

Theories of Everything

Dharwadker now has a Theory of Everything.

Like Garrett Lisi's, it is based on an unusual and highly symmetric mathematical structure. Lisi's approach is related to the exceptional simple Lie group E8.* Dharwadker uses a structure long associated with the sporadic simple Mathieu group M24.

GRAND UNIFICATION

OF THE STANDARD MODEL WITH QUANTUM GRAVITY

by Ashay Dharwadker
Abstract
"We show that the mathematical proof of the four colour theorem [1] directly implies the existence of the standard model, together with quantum gravity, in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem, at the most fundamental level. We preserve all the established working theories of physics: Quantum Mechanics, Special and General Relativity, Quantum Electrodynamics (QED), the Electroweak model and Quantum Chromodynamics (QCD). We build upon these theories, unifying all of them with Einstein's law of gravity. Quantum gravity is a direct and unavoidable consequence of the theory. The main construction of the Steiner system in the proof of the four colour theorem already defines the gravitational fields of all the particles of the standard model. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with t'Hooft's table [8]. We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles."

Good luck, Garrett Lisi.

* See, for instance, "The Scientific Promise of Perfect Symmetry" in The New York Times of March 20, 2007.


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