Amateur Mathematicians in Power 1

by teti : freezotic@yahoo.com : April 2002

"Teti Matrix Family"

The Family of Teti Matrixes is like having a box with group photos of unknown persons, each picture showing the characteristics of a certain combination of operators:

{+, *, ^, ^^, ^^^, ...}

School maths involves adding, multiplying, powers, and their respective inverses, which are the only binary operating functions taught/thought to exist, but the daredevil expansion of this 'operating system' was introduced to me by Conway and Guy in "The book of numbers" (1996).
Making a minor change in the book's ^arrow head^ notation we can call every possible operator O in its order of appearance:

O1(a,b)=a+b

O2(a,b)=a*b=a+a+a+...

O3(a,b)=a^b=a^b=a*a*a*...

O4(a,b)=a^^b=a^ (a^ (a^...

O5(a,b)=a^^^b=a^^ (a^^ (a^^... (always take b copies of a)

Proceeding onwards in the style of the (1928!) Ackermann numbers:

1^1, 2^^2, 3^^^3, ... = O3(1,1),O4(2,2),O5(3,3), ...

we penetrate into the realm where useless numbers are dwelling, inbetween arithmetically meaningful 'used' numbers and Cantor's cardinals starting at infinity.
Curiously enough: if mathematicians talk about the largest number they always opt for the criterium of useful application within a mathematical theory (where largest numbers are mostly upper bounds for a solution of a combinatorial problem, and subject to deflation!)

The 'useless' search for even larger numbers cannot be a goal in itself, but 'largest number inflation' may occur spontaneously and more potent notations may become of future use:

Conway and Guy define their chained arrow notation as:

a -> b -> c', where c' is the number of arrows ^ ,

with two additional rules or axioms (you may skip this, it's all just mind-bogglingly big):

a -> b -> ... -> x -> y -> 1 = a -> b -> ... -> x -> y,

a...x -> y -> (z+1) = a...x -> (a...x -> (a...x -> [ ... *] -> z) -> z

[* on the left y copies of '(a...x ->', on the right y-1 copies of ') -> z']

It comes in handy to locate the champion Graham's number:
between 3->3->64->2 and 3->3->65->2, but less than 3->3->3->3.
Graham's number is an upper bound for a problem in Ramsey Theory, but the actual answer may well be 6 (sic!), says David Wells in his wonderful "The Penguin Dictionary of Curious and Interesting Numbers" (1997).

How to call your operator

I refuse to leap immediately into the abyss of no avail, and propose a more modest and useful notation: Oc(a,b), where the c tells us what kind of operation O is used.
Then my (differing) c (=c'+2) allows the Oc to naturally express the earlier O1: a+b and O2: a*b

My "QFO" notation also handles operator repetition, as in a series:

QF(c,d)=Oc(f1,Oc(f2, ... ,Oc(f(d-1),fd) ))

or specified for all d variables fn, as for example in:

Q(1,3)(5,7,9)=5+7+9=21

If d extends without limit to either side, QcF is an infinite series of copies of operation Oc on variables d fixed by a function F.
Now the QFO-notation is still a mathematical baby, but I feel it has the potential to make the Teti Family flourish.

After the discovery of i=(-1)^(1/2) and the Complex a+b*i, our number system is said to be proven to be complete.
The larger operators Oc>3, with their inverses ('quarkroots' and 'analogarithms'), surely will expand this horizon, starting a never-ending creational flow of new numbers 'B'. Because teti says: "Beautiful numbers for Beautiful people!".

In theory the number c in Oc(a,b) itself can be made subject to meta-mathematical calculations: teti's operarithmetic musings. This renders the discovery of new operators not unthinkable, as the c in Oc can be any element from any set of numbers: NZQRCBA.
I propose to find these operators, put them in my matrix box and make these rascals work, like:

O1½(halfway + & *), O0(simpler than +), O-1(further back), Oπ, Oe & Oi!

Douglas Hofstadter (you knew thát would come) in "Gödel, Escher, Bach" (1979) writes about Gödel's Typographical Number Theory (1931!), in which Kurt Gödel attached a number to each 'mathematical atom' to prove his famous incompleteness theorem. One step further to the computer age!
However, I believe that he overlooked the possibilities of teti's proposed operators (designed to prove just about anything?)

Operator functions can work on 3 or more variables, requiring another (binomial) kind of numbering, and likewise I can imagine operators working on any NZQRCBA number of variables, or go beyond what is strictly called a function and address not one but many (any!) number of solutions.

If operarithmetic is feasible it may be applied to itself by recursion, or in an operarithmetic equation, with the hidden Ox as an unknown in an Oc jungle.
The uselessly big meta-numbers starring in Conway and Guy's book can be called in QFO by counting rows of successive O's with or without recursion. Read this example of a 'small' number:

OOO1(1,1)(2,2)(3,3)=OO2(2,2)(3,3)=O4(3,3)=3^^3=3^(3^3)=3²⁷= 7625597484987

So 'what the teti' is a super operator arithmetic with an unnatural number (not from N+) of recursive O's? Soaperarithmic math soup?

Zero negative geometry

An expansion of operating functions is no good unless we can compare their properties.
The Teti Matrix does just that: making connections between operators discernible as they function together. But a revision of the basic axioms is needed first, to properly interpret the whole of a Teti Matrix. I stumbled into this trying to visualise the negative geometry of the first of the family pictures...

My first Teti Matrix showed me things that were essentially not there: anti-points or negative space objects, with connections spreading into infinite dimensions. This defines a Set antithetically by the part of its universe that its elements do not occupy: an alien form of 'all else'. The crux is that this 'anti-elementarity' is spread in an orderly way: there is infinite structure in the Teti Matrix!

Another observation from my neonatal Teti Matrix was that its void areas must contain a different breed of zeros from some deep zero complex space. This involves a re-evaluation of the axioms of algebra dealing with zeros, such as the cancellation law: a+(-b+b)=a', and the powers of zero and of negative numbers: 0^a and -1*-1
Teti's axioms have to deal clearly and meticulously with all zeros and inverses. And if we give each Axiom a number to be reckoned with, teti says it will happen: "All of creation Anew and All numbers found!" NZQRCBA

These marvelous insights stem from the + * ^ abelian distributive matrix dubbed "1.3T1M2.3".
Starting other dimensions is as simple as that!

I'll show you how at the

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TAOURT
Egyptian Goddess