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I am "Prototype John" as to what the Platonists can wreak.This started n the summer of 1957, when I lectured on "Foundations of Mathematics" as part of a National Science Foundation Institute for High School Mathematics Teachers. Some of my lectures critically outlined what became known in 1958 by that silly label, "The New Math". (I say "silly" because some recommendations concerned useful "old math" eliminated in the 20's by "progressive" non-mathematician educators.)

But I noted that "standard"

set theorydemands writing the factor set of, say, 30, 60, 90, 150 as {2, 3, 5}, ignoringmultiple tokens of an element, thereby treating 60, 90, 150 as the same as 30.Thus, I kenned that "The New Math" does not support "The Old Math"! (Such comments in my lectures resulted in The National Science Foundation treating me as a nonperson, ignoring any grant requests I made from that time on.)

However, my literature search revealed that

only the philosophical prejudice of Platonism precluded multiple tokenage. Later I stated at a MAA meeting talk that "Platonists" seem to believe that "Less is elegant!" and "God is my thesis adviser".That is, a

set theoryallowingmultiple tokenagemust

- admit "atoms", which Platonists considered inelegant in their pretty system;
- more seriously,
set-theoretic atomsVITIATE the two most powerfulnonconstructive methods of proof-- namely,tertium non datur (proof by contrdiction)and theAxiom of Choice. By theChoice Axiomyou can claim (for example) that "the Real Numbers can bewell-ordered", although none knows how to do so. However, if God be your Thesis Advisor, you can sleep soundly with the Axiom under your pillow.(My 1974 Fable, "The Battle of the Frog and the Mouse", on this subject and the foundational controversy between

formalistDavid Hilbert andintuitionistL. E. J. Brouwer, taking my Fable's title from a contemporary comment by Albert Einstein. This Fable was reprinted in a recent book on mathematics,Pi in the Sky, by British astronomer, John Barrows.)I discovered three ways of extending

set theory:

- axiomatically
- by
Cartesian productofconstructive ordinals- as a
graded algebra.Adapting terms "typological" and "ordinal" from

the theory of measurement scales, I labelstandard setsast-sets(recognizingtype, notorder) andextended setsaso-sets(also recognizingorder). I further speak oft-logicando-logicand, in general,t-mathematicsando-mathematics. Thus, "a square-free" number (singletons of each prime factor), such as 30, is at-number, but 60, 90, and 150 areo-numbers.If you know the

Möbius function, m(n), a newnumber-theoretic functioncan be constructed asindicator of t-numbers and o-numbers: m(n) = 0 for "nonsquare-free numbers" (o-numbers), +1 for "squarefree numbers" with an even number of factors, -1 for those with an odd number of factors. By taking theabsolute valueofm, the latter two components coalesce into one component of anindicator. By subtracting from 1,the indicator can be reversed. (For a look at the mathematics of this indicator, click here.)I was motivated to extend from

t-mathtoo-mathbytwo TRADITIONAL subjects of o-math.One is

factor theory, which I've already discussed, since so many numbers we encounter arecompositeor what I've designated aso-numbers. Clearly, teachingt-set theoryin "The New Math"disconnects this subject from "The Old Math"-- that is, from thefactor theory of arithmetic.The other TRADITIONAL subject is in

lattice theory. (I've a mathtivity for children which involveslatticeswithout actually mentioning them: "Pecking Order". If you don't understandlattice theory, a brief sketch will help.The

factors of a t-number form a COMPLEMENTED DISTRIBUTIVE LATTICE. Butfactors of an o-number form a DISTRIBUTIVE LATTICE, with some elements not COMPLEMENTED. The German mathematician, Richard Dedekind (1831-1916), created thiso-extensionto encompassfactors of integers(o-numbers).Elsewhere I describe THE PYTHAGOREAN REPERTORY, a collection of mathematical systems, which are ABSTRACTLY THE SAME: one TRANSFORMS BETWEEN THEM MERELY BY CHANGING THE LABELS, just as an actor in a repertory company changes roles in plays, or a little girl changes the costumes of a Barbie Doll, or a boy changes the uniforms of a G. I. Joe Doll. I named this "Pythagorean" since

factor theory(motivated by the Pythagorean distinction of "prime" and "composite") is the PROTOTYPE of this repertory. Sets and lattices also belong to this repertory. And so does STATEMENT LOGIC. Hence, I developed O-LOGIC, in which the TERTIUM NON DATUR or "double-negative" rule of logic works only for certain members.(Tertium-failures correspond to exceptions recognized by L. E. J. Brouwer -- the "frog" of my Fable, cited above. Brouwer said: "If a criminal, in committing a crime succeeds in destroying the only evidence which can be used to convict him, must I therefore find him innocent? No! I demand a third verdict, 'Not proven'." Scottish jurisprudence includes this, as seen in the 1949 David Lean film, "Madeleine", based upon a famous 19th century murder case. In 1998, during the Senata "trial" of President Clinton, Sen. Arlen Spector (PA) suggested going by Scottish jurisprudence and considering the thid verdict, "Not Proven".)

Thus,

o-logomathrecognizes distinctions of ORDER (DEGREE) as well as of TYPE (KIND). Other instances occur in my extension of "Venn diagrams" (t-diagrams) andindicator tablestoo-diagramsando-tables. (I got the idea foro-diagramsfrom serving as a World War II Army Air Force weather forecaster, drawing concentric "rings" of isobars for HIGHS or LOWS. In ano-diagram, the numbers 2, 4, 8 are built up as concentric "rings", respectively included, similar to tographic lines of elevation on a land map.)BUT PLATONISM, like opponents of Civil Rights for minorities, "stands in the doorway" of admittance to

o-logomath. To me, a clear threat against our "ordinal" future!For example, the

mathematical theory of evolutionhas led toblood and tissue typing, decoding DNA, etc. But this theory presently usest-probability measures(based ont-sets). Imagine! The onlyobservable evolutionI find in the literature concerns gradually darkened wings of a moth due to the gradually smoked surfaces of its forest home. But this iso-evolution, a matter of DEGREE, not TYPE. It can be describedrhetorically, but not by standard mathematics.(Incidentally, in 1974, I published an Abstract about my

o-extensionof theHardy-Weinberg Law, called "The Fundamental Theorem of Evolution", but Platonists refuse to recognize this! When you introduce a factor of two degrees, Hardy's proof goes through the same. My lattice of o-probabilities is easily drawn and easily comprehended. This diagram shows the DISTRIBUTIVE LATTICE ON FACTORS OF 12, and in parentheses beside each node are probabilities consistently applied. The probability model fits myo-extension of the Hardy-Weinberg Law, mentioned above, although my proof doesn't appeal to it.)Can you suggest other extensions of t-logomath to o-logomath? Or are you intimidated by the Platonists?

The Platonists of Plato's time bifurcated

geometryandarithmetic, further saying thatarithmetic cannot describe motion, which delayedThe Mechanical Industrial Revolutionfor 1600 years (hence, also the Thermodynamic, Electric, and Electronic Revolutions), thereby extending human slavery. Forced by scientific-technological progress to admit correction, Platonists promote FLATLAND SCIMATH by declaring thatset theory,probabiliity theory, andlogic-- among other subjects -- CANNOT RECOGNIZE DEGREE! I find this A CLEAR THREAT AGAINST OUR FUTURE!If you followed the last hyperlink above to study the

o-lattice on 12 and probabilities, did you notice two INELEGANCY. One is the label "atom" for the primes integers 2, 3 -- a crtical concept for Arithmetic, which Platonists would like to sweep under the rug. The other INELEGANCY is the presence of TWO TYPES OF ORDERING IN THE DISTRIBUTIVE LATTICE --partial orderingandsimple or total ordering-- whereas THE COMPLEMENTED DISTRIBUTIVE LATTICE HAS ONLY ONE TYPE OF ORDERING:partial ordering. The Platonists should be objecting to thisinelegancy of mixed orderingsas much as they object toatoms and multiple tokenage.I know of two reasons for this silence:

- The Dedekind lattice was developed decades before Platonists became aware of the "atomic" problem. They seem to have followed a policy of "don't ask, don't tell".
- The other reasons involves a funny come-uppance for Platonists which also invokes the above policy.
In the 1950's, the most vocal of Platonists were the members of a clique of French mathematicians (and one Polish-American mathematician, Samuel Eilenberg) who wrote "textbooks in modern math" under the collective name, "Nicholas Bourbaki". (The surname rhymes with "settee", as in "BOURBAKI, KISS MY SEtTEE!".)

The come-uppance was revealed in th 70's. xxxxxx was a mathematic professor at The Naval Academy at Annapolis, MD, and author of a book, xxxxx, which taught me much. In 1974, xx organized a Seminar at the Academy on progress in

lattice theory. Among the invited was Garrett Birkhoff, the chief promoter oflattice theoryand the standardizer of the name of this field. In his lecture, printed along with others given during the Seminar, Birkhoff used the expression "Bourbaki -- no friend of lattice theory" to note that many of the Bourbaki books had been revised because this field had become so well known in Mainstream Math. Why? Because, inanalysisand related fields,two limit-chains can result in the same limit without a link in one chain dominating a link in the other. But this is PARTIAL ORDERING.So, although their books take little notice of

lattice theory, as such, the Bourbaki had to sneak inlattice-tricksto bolster more traditional fields of mathematics. How embarrassing!