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:jhor...@cybermesa.com> > *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: jhor...@cybermesa.com <mailto:jhor...@cybermesa.com> > ------------------------------------------------------------------------ > -- 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
