In the deafening silence of Doug's withdrawal to his private vacation
cottage, I submit this for your "FRIAMic Consideration", as it were.
This colleague of mine has a penchant for his own level of weight in his
postings... he might put the most obscure and obtuse of us to shame.
His postings of this nature are, however, always thorough, footnoted,
and referenced. He also publishes a weekly "kitchen science" column in
the Espanola Rio Grande Sun. And no, he is not "little". And he lives
halfway between myself and Doug (geographically).
My own commentary *follows* the posting.
------------------------------------------------------------------------
----- Original Message -----
*From:* Jack K. Horner <mailto:[email protected]>
*To:* X
*Sent:* Friday, April 26, 2013 8:04 AM
*Subject:* Re: "The Notorious Four-Color Problem"
Jeremy Martin's KU mini-course (see thread below) on the Four-Color
Theorem (FCT, "Every planar map is four colorable", [1]) promises to be
a spectacle.
It's hard to overestimate the importance of the FCT, and on any
dispassionate reckoning, it would have to ranked among the 100 most
important theorems of mathematics.
A "color", in the sense of the FCT, is any nominal distinguishable
property; "red, green, blue, and yellow" work as well as any.
Given this meaning of "color", the FCT, at the heart of which is the
notion of "four-foldness", is much more than a cartographic
curiosity. To sketch a few:
1. The Prague School of linguistics maintains that meaning in all
natural languages can be represented in a system that makes no more than
*four* kinds of distinctions (applied indefinitely/recursively) between
"adjacent" meanings ([2], [3]). It turns out that these
meaning-relations can be represented in a planar map. We can thus think
of the FCT as a representation of the structure of the meaning
of anything that can be expressed in a natural language.
2. The dances of the indigenous peoples of the upper Rio Grande
(e.g., the Corn Dance, the Deer Dance) turn out, one and all, to be
generatable from a set of exactly four fundamental dance moves. The
belief systems of these cultures places fundamental emphasis on the
"four-foldness" of the world. In light of (1) and the FCT, these
dances, whatever their nominal semantics, may be "essays" on the
meaning of 'meaning' ([8]).
3. Adherents of the logicist program in mathematics ([5], esp.
Chaps. II-III) hold that all of mathematics *could* be expressed in set
theory (together with a "logic" and a raft of "mere" definitions). In
its most rigorous form, set theory presumes a four-fold set of
distinctions ("is a class", "is a set" (a restriction of a class), "is a
member of a class", and "is a member of a set" ([9]). This view of
mathematics is thus equivalent to a set-theoretic version of the FCT.
4. The structures of the derivations (proofs) all theorems in
mathematics can be represented in a planar map. The FCT guarantees, in
effect, that no more than four kinds of distinctions need to be
made between adjacent "steps" in the totality of all derivations in
mathematics.
5. The Book of Kells ([4]), a medieval Irish religious manuscript,
is densely illuminated with images of Celtic knots. Most if not all of
the knots in the Book of Kells are, or are composable from, the simplest
Celtic knot, the trefoil knot, which the authors of the Book of Kells
likely regarded as a symbol of the the trinity -- the irreducible
three-in-one. The structure of the trefoil knot is representable in a
planar map, and therefore, by the FCT, the structure of the trefoil
knot is four-colorable. One could (though in practice no one would)
take a (set-theoretic) description of the trefoil knot as something to
be "unpacked" by more derivative mathematics, and in the course of that
investigation, be driven to the FCT.
6. According to modern genetic theory, a set of four nucleic acids
(A, C, T, G) is *sufficient* to encode the genetics of all terrestrial
life ([10]). But as astonishing is that *exactly* four distinct
building blocks (regardless of their specific chemistry) are
also *necessary* to optimize the integrity of the transmission of
information ([7]) in noisy environments over long times (e.g., across
mutiple generations; [6]).
Jack
---
[1] Appel K and Haken W. Every Planar Map is Four Colorable. American
Mathematical Society. 1989. As Martin notes, the original proof
was completed in 1976. Minor corrections to the proof were added over
the the following decade.
[2] Jakobson R and Halle M. Fundamentals of Language. Mouton. 1971.
[3] van Schooneveld CH. Semantic Transmutations: Prolegomena to a
Calculus of Meaning: The Cardinal Semantic Structure of Prepositions,
Cases and Paratactic Conjunctions in Contemporary Standard Russian.
Physsardt, Bloomington IN. 1978.
[4] Book of Kells. MS A. I. (58). Trinity College Library, Dublin.
Circa 800.
[5] Körner S. The Philosophy of Mathematics: An Introductory Essay.
1968. Dover reprint, 1986.
[6] Petoukhov SV. The rules of degeneracy and segregations in genetic
codes. The chronocyclic conception and parallels with Mendel's laws.
Advances in Bioinformatics and its Applications, Series in Mathematical
Biology and Medicine 8 (2005), 512-532.
**
[7] Cover TM and Thomas JA. Elements of Information Theory. Wiley. 1991.
[8] Putnam H. The meaning of 'meaning'. In H Putnam. Mind, Language,
and Reality. Cambridge. 1975. pp. 215-271.
[9] Fraenkel A and Bar-Hillel Y. Foundations of Set Theory. North
Hollnad. 1958.
[10] Hartwell L, Hood L, Goldberg M, Reynolds A, and Silver L.
Genetics: From Genes to Genomes. McGraw-Hill. 2010.
----- Original Message -----
*From:* *Z*
*To:* *Y*
*Sent:* Thursday, April 25, 2013 7:52 PM
*Subject:* "The Notorious Four-Color Problem"
Apropos of our discussion at Saralyn's loft on the evening of April
17, in which I brought up the Four-Color Problem and Jack gave a
cogent description of it for the layman, I discovered that one of
the classes being taught at the Mini College this June covers the
same topic. Here is the description from the Mini College schedule
(https://kuecprd.ku.edu/~clas/minicoll/schedule/index.shtml
<https://kuecprd.ku.edu/%7Eclas/minicoll/schedule/index.shtml>),
which does a pretty good job of explaining the problem and hints at
the method of solving it:
The Notorious Four-Color Problem
. Jeremy Martin
<https://kuecprd.ku.edu/%7Eclas/minicoll/speakers/index.shtml#82>,
Mathematics
How many colors are required to color a map so that no two adjacent
regions (say, Kansas and Missouri) are given the same color? It
turns out that every map can be colored with at most four colors, a
fact suspected to be true since 1852, but not confirmed until 1976
(with the aid of intensive computation, an unprecedented approach to
research at the time). In the century-long attempt to solve the
map-coloring problem, mathematicians have developed theories of
unexpected power and beauty: for example, problems about optimal
routing and scheduling (and even Sudoku puzzles!) can be expressed
as graph coloring problems. This course will explore both the
history and the mathematics of the four-color theorem, including its
practical applications, the many failed attempts to solve the
problem, the debate over the validity of computer-assisted proofs,
and the theoretical research for which mathematician Maria
Chudnovsky was recently awarded a MacArthur "genius" grant.
Jack K. Horner
P.O. Box 266
Los Alamos, NM 87544
Voice: 505-455-0381
Fax: 505-455-0382
email: [email protected] <mailto:[email protected]>
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SAS commentary
I have not taken the time to follow all of Jack's references and this
expose' verges on numerological argumentation, at least half of the
bullet points below are convincing to me on their own merits.
The position is that "4" is a certain kind of magic number in a
topological sense, relevant to some fundamental things like Cartography,
Language, Aboriginal Cosmology, Mathematics, Genetics, and most
oblique... the Celtic Knot.
Reminds me of the anthropic posit-ion that we live in 3 (perceptible)
spatial dimensions because it is the lowest number of dimensions where
all graphs can be embedded without edge-crossings. Can't remember the
source of this....
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