UNIFICATION has been a PRIME MOTIVE in PHYSICS since -- BY HIS LAW OF "UNIVERSAL GRAVITATION" -- Newton unified "the physics of earth" and "the physics of Heaven".But BIFURCATION has been a PRIME MOTIVE in MATHEMATICS since Plato's Academy (as noted elsewhere) declared ARITHMETIC and GEOMETRY incompatible.
After 19 centuries, this "Platonic" BIFURCATION (which prolonged slavery!) ended, to be replaced by another BIFURCATION which inhibits future developments< which I call "The TO-STYMIE".
And the great dependence of physics upon mathematics means that DIVERSE MATH interferes with the drive toward UNIFICATION in PHYSICS.
Math Establishment Ideology MATHEMATICS IS BIFURCATED INTO TWO SECTORS -- which I call "YIN" and "Yang". (This classification overlaps with t-logomath and o-logomath, cited elsewhere).
The difference between Yin and Yang involves OPERATIONS WHICH ARE AUGMENTABLE (NONABSORPTIVE) and OPERATIONS WHICH ARE NONAUGMENTABLE (ABSORPTIVE). (An OPERATION is AUGMENTABLE if the AUGEND INCREASES IN VALUE.)
A Yin system has ar least one primary operation which is Augmentable, but nonAbsorptive, whereas a Yang system's primary operations are nonAugmentable, but Absorptive.
Yin and Yang Systems Please consider the system [E, #, =, ¹]E is a set of elements; where # is a binary, associative, comutative operation; where =, ¹ are equivalence, nonequivalence relations.
- If (for e in E) e # e ¹ e, then operation # is Augmentable, nonAbsorptive, and the system is Yin.
- If e # e = e, then # is nonAugmentable, Absorptive, and Yang if this is so for all primary operations.
Yin systems have a Yang subsystem, for natural reasons. And Yang-like measures are found in many physical systems, for good and useful purpose.
But the Yin-Yang Bifurcation in Mathematics exists only because of 2500-year-old philosophical prejudices!
"Yin-Yang" in Whitehead Alfred North Whitehead noted this Bifurcation in his Universal Algebra, 1900, citing arithmetic and "Boolean algebra" as representing the two sectors. And Whitehead speculated about a "universal algebra" which would be -- in my language, given below -- POLYTYPAL-POLYORDINAL IN BOTH SECTORS.)
Yin-Yang Spectra THE YIN-YANG BIFURCATION CAN BE DESCRIBED VARIOUSLY, making MATH GO SCHIZOID! IN MANY WAYS!
Bifurcation, Yin-Yang, and "The New Math" Before finding Whitehead's book, I had became aware, in 1957, that this aspect of what was to become (in 1958) "The New Math" would neglect "The Old Math". Thus, you cannot use the language of SET THEORY to describe FACTOR THEORY, a basic part of ARITHMETIC, since Pythagoras made the distinction between PRIME and COMPOSITE NUMBERS. Thus, the prime factor set of30 (= 2 x 3 x 5) is {2, 3, 5}. But, 60 = 2 x 2 x 3 x 5 has the same factor set, since {2, 2, 3, 5} = {2, 3, 5} (since multiple tokens in sets are ignored). And similarly with 90 = 2 x 3 x 3 x 5 and 150 = 2 x 3 x 5 x 5, etc.
I discovered, from "literature", that only PLATONIC IDEOLOGY forbade the extension of set theory to encompass DEGREE (ORDER) as well as KIND (TYPE). So I published to-mathextensions, locally for classroom use; then, in publications of the American Mathematical Society in the late 60's and early 70's. Then I read, in The Art of Computer Programming, v. 2, Seminumerical Algorithms, by Donald Knuth, about "multisets", which had developed in the 60's. (Knuth cited them for repeated steps in an algorithm, a matter of DEGREE.) But a MULTISET is HYBRID -- A MAPPING FROM SETS TO INTEGERS, whereas I remained within SET THEORY.
Schizoid in The Flats of Babylon Elsewhere, I show MATH that is BABELIZED, again due to philosophical prejudice, but -- in the case of Olinde Rodrigues (1791-1854) -- also involving political and ethnic bigotry.
SCHIZOID MATH makes for FLATLAND SCIENCE.