Four Color Theorem and beyond!
Locked
 
|
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: [hidden email]
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.
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....
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
Re: Four Color Theorem and beyond!
|
|
I agree with you about the "numerological" or anthropomorphic feel of this attempt to unify disparate subjects with a common pattern. But I can only speak to the bias I see in example 3. At this point, I'm sure I sound like a broken record. So, I'll merely raise the point again and leave it be unless others chime in. The discretization into 4 types (set, class, set member, class member) is violated in lots of mathematics as it's practiced, namely in impredicative definitions (sets defined by a quantification over the set being defined). This is indirectly related to the openness of practical math raised by Feferman and the demonstrations of the practical utility of formal systems that are both complete and consistent (i.e. "simple" enough to escape the GIT, but complex enough for engineers to use to good effect). Aczel helped to formulate this rigorously and demonstrated a foundational math where a set can be a member of itself, which means the magic number would not be 4, but 3 (or perhaps 2). So, the bias toward 4 is situational, I think. That does NOT mean the idea isn't interesting, though. On 04/27/2013 08:28 AM, Steve Smith wrote: > 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.... > ------------------------------------------------------------------------ > > ----- Original Message ----- > *From:* Jack K. Horner <mailto:[hidden email]> > *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: >[...] > 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. > [...] > [5] Körner S. The Philosophy of Mathematics: An Introductory Essay. > 1968. Dover reprint, 1986. > [9] Fraenkel A and Bar-Hillel Y. Foundations of Set Theory. North > Hollnad. 1958. > > > Jack K. Horner > P.O. Box 266 > Los Alamos, NM 87544 > Voice: 505-455-0381 > Fax: 505-455-0382 > email: [hidden email] <mailto:[hidden email]> > ------------------------------------------------------------------------ > -- glen e. p. ropella http://tempusdictum.com 971-255-2847 ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
«
Return to Friam
|
1 view|%1 views
| Free forum by Nabble | Edit this page |