http://friam.383.s1.nabble.com/Four-Color-Theorem-and-beyond-tp7583027.html
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 -----
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), 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,
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:
[hidden email]
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....
Meets Fridays 9a-11:30 at cafe at St. John's College