THE WHATNESS AND HOWNESS OF "GOOD" MATHEMATICS

"The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete. We go beyond the mathematical formula at our own risk; we may find a model or picture which helps us to understand it, but we have no right to expect this, and our failure to find a model or picture need not indicate that either our reasoning or our knowledge is at fault. The making of models or pictures to explain mathematical formulas and the phenomena they describe is not a step towards, but a step away from, reality; it is like making a graven image of a spirit." Sir James Jeans, physicist and astronomer.
Good mathematics" is "goodies" ("whatness") protected by "good packaging" ("howness"). In the best mathematical formulation, the whatness is patterning, the howness is univalency: that is, 1-1 map of sign to reference, or 1-1 map of sign to meaning. So this univalency must be explained.

Bertrand Russell said that, "[P]ure mathematics is the subject in which we do not know what we are talking about, or whether what we are saying is true," Mysticism and Logic (1919).

Russell presumably meant that mathematics is solely syntax or syntactics, with no semantics, that is, no referential content.

By saying that good mathematical language is univalent, I seem to contradict Russell's statement, since a term can only be 1-1 referent by having a reference. But, valuable as Russell's definitions and dictums were for that time, much has been brought to the fore in the foundations of mathematics and I claim that these advances have given us a subtler understanding of semantics.

The univalency of mathematics can be explained by way of two powerful concepts, the "woof" (well-formed-formula or wff) and closure, since, together they provide perhaps the simplest "packaging" ("howness") of "good mathematics". Thus, we obtain a univalent system so precise that a computer program can produce the whole procedure!

These are used in a mathtivity for teens which I developed and labeled t-Logic. So, differing with Russell, I say that t-logic is talking about t-wff's which enable us to formulate validities (tautologies), which may or may not be true.

Thus, consider,"If it is raining, then the streets are wet; it is raining; therefore, the streets are wet". This is a validity (tautology) known as modus ponens (MP), which is the only "rule of proof" needed for statement logic. But it may or may not be true for the weather outside you window.

As I noted elsewhere, THE PURPOSE OF LOGIC is NOT to discover original truth about the world, but to PRESERVE this truth and to disclose IMPLICIT TRUTHS IN TRUTH ALREADY OBTAINED. In every-day jargon: (1)LOGIC HAS NONLEAKY VESSELS, VESSELS WHICH WILL NOT LEAK TRUTH; (2) LOGIC TELLS US "IN OTHER WORDS ...", that is, IN SAYING ONE THING WE'RE ALSO SAYING ANOTHER.

Hence, Russell is simply being cutesy in the final comment.

Logico-mathematic language is closed language, whereas ordinary language is open-ended. The open-endedness of legal language allows lawyers to charge umpteen dollars an hour looking for "loopholes" either to "close up" or "sneak into". The quasi-closure of much legal language provides that you must spend years and umpteen umpteen dollars learning this sneakiness.

In my GENERATIVE derivation of "The Arithmetic Curriculum", I show how the N-wff of a natural ("counting") number is produced by a recursive definition. (RECURSIVENSS PROVIDES ININITY IN THE FINITE!) This RECURSIVE DEFINITION corresponds to RULE 1 of the t-wff definition, specifying GNOMONS (UNITS) of the FORMAL LANGUAGE.

Next, we need SUBRULES in natural number arithmetic to DESIGNATE OPERATORS for DERIVING N-wffs from N-wffs.

One more "goodie"is needed: "GROUPNESS". When you can say that logomath L forms A GROUP, you have CLOSURE OF THE ENTIRE SYSTEM -- not indivudally for a system-part, and THE ENITRE SYSTEM CAN BE DESCRIBED IN A SINGLE SENTENCE!

GROUPNESS requires just two conditions, one of which we've discussed above: (1)CLOSURE UNDER OPERATION (or TRANSFORMATION); (2)INVERSE FOR EACH OPERATION (or TRANSFORMATION).

"Ay, there's the rub", said Hamlet. THE BREAKDOWN OF CLOSURE FOR INVERSION, and ITS MENDING GENERATES ALL THE NUMBER SYSTEMS OF ARITHMETIC!

Thus, SUBTRACTION -- defined (not by recursion! but by "turn-around") from addition, that is, as INVERSE of ADDIION -- does not always yield a natural number (an N-wff). Thus, 3 - 5 is not a natural or counting number -- so we do not have CLOSURE for subtraction.

Similarly, division -- defined, not by recursion, but from multiplication as INVERSE -- does not always yield a natural number (an N-wff). Thus 3/4 is not a natural number, nor is it an integer -- hence,CLOSURE fails for division.

And EXPONENTIATION (being NONCOMMUTATIVE!) has TWO INVERSES (something apparently taught only here) --namely, LOGARITHM and ROOT EXTRACTION. Neither with CLOSURE, until NEW NUNBER SYSTEMS ARE INTRODUCED!

So we need new number systems (respectively, integers and rationals) to find CLOSURE for subtraction and division. And real numbers and complex numbers, respectively, to find CLOSURE for the OPERATIONS of logarithm and root extraction.

However, once new Number Systems are obtained, this ensures UNIVALENCY-cum-GROUPNESS for a corpus of arithmetic, wherein we are talking about N-wffs and how to obtain validities (not truths!) from arithmetic.

Please contrast this with the case for English grammar. Given its rules, and the same set of data ("correct" and "incorrect" English sentences and sentence-fragments), three different computers will agree with each other more than 95% of the time (surprising many who think English grammar does not make that much "sense") but not 100%. However, three or more computers can be programmed with the rules of statement logic or of arithmetic and the same data and agree with each other 100% of the time!

This is the "whatness" and "howness" of "good" mathematics. But how do we obtain this. Let's look at UNIVALENCY in t-logic, which is easily described, yet "represents the big picture".