GB, Part VIII: Typographical Counting Math

Rob Speer

from G and Bach: a Glorious Braid
(an Imaginary Lipogrammatic Translation
Involving Twisty Loops)

Contents

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Crab Canon and Unusual Loopbacks

A trinity of unusual loopbacks occur in Crab Canon. Swift Warrior and Slow Animal both discuss works of art -- and said works of art turn out to show a kind of isomorphism to that Discussion! (Think of my shock as I, Doug, caught on to this!) Also, Crab says things about a biological form, and that, too, is isomorphic to that Discussion. Now, you could know of crab canons as told by Crab Canon, but oddly fail to know that that Discussion is also a crab canon. This is knowing it by what it says, but not knowing it by its form. You must know Crab Canon in both ways to truly know its unusual loopback.

Kurt G's construction, too, must talk about a thing's form as it also talks about what it says. What things? A formalism that I shall build in this Part -- Typographical Counting Math (TCM). Its twist is that, through an odd mapping of Kurt G's, a form of any string is also a part of this formalism. Now, I'll start to build this odd formalism that has such a capacity to wrap around.

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What You Can Say in TCM

I'll start by citing a typical fact or two in counting math, and I'll show that such facts consist of just a bunch of basic notions, which you mix to build up into big notions. By giving a symbol to all such notions, a formalism is built. I should say now that "counting math" will signify only facts about naturals -- things you can count to, starting with zilch. "Minus" plays no part in this formalism. And it is important -- vital -- for you to distinguish in your mind both TCM, a formalism, and counting math, a familiar old branch of math that is not so formal; this I shall call "CM".

Six typical facts of CM:

  1. 2 is atomic.
  2. 4 has a 2-root.
  3. 2000 is 2 by a natural with a tri-root.
  4. No sum of non-zilch naturals with tri-roots has a tri-root.
  5. Atomic naturals go on to infinity.
  6. 2 factors 6.

Now it may look as if I must build a symbol for all notions such as "atomic", "3-root", or "non-zilch" -- but such non-basic notions go too far. Atomicity, you know, has to do with a natural's factors, which in turn has to do with multiplication. A tri-root has to do with multiplication, too. So now I'll say facts as basic notions:

  1. No natural a is such that a natural b is such that a trumps unity, b trumps unity, and 2 is a by b.
  2. A natural b is such that b by b is 4.
  3. A natural b is such that 2 by b by b by b is 2000.
  4. For all naturals b and c, if b is non-zilch and c is non-zilch, no natural a is such that a by a by a is b by b by b plus c by c by c.
  5. For all naturals a, a natural b that trumps a is such that no natural c is such that a natural d is such that c trumps 1, d trumps 1, and b is c by d.
  6. A natural d is such that 2 by d is 6.

This analysis has got us a long ways towards basic parts of a formalism for counting math. A small quantity of parts show up again and again:

for all naturals b...
a natural b is such that...
trumps
is
plus
by
zilch, unity, two...

TCM has basic symbols for most such parts, but not "trumps", which is not that basic. You can say it with this:

A non-zilch natural c is such that a is b plus c.
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Num-strings

If all naturals had a distinct symbol, it's bad; an infinity of symbols is poor form. So, TCM has a uniform way of giving compound symbols to all naturals -- much as I did with p and q many parts ago. My notation for naturals is:

zilch: 0
unity: S0
two: SS0
and so on.

S's symbol has an isomorphism, though a bit clunky, in Anglo-Saxon: "that natural that follows". So TCM's string SS0 is isomorphic to "that natural that follows that natural that follows zilch". I call a string of this form a num-string.

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Unknowns and Formulas

Obviously, I must show a way to say unknown naturals as strings. How about glyphs of Anglo-Saxon? I start with four glyphs: a, b, c, and d. But only four will not do. Our supply of unknowns must not run out; an infinity of unknowns is crucial, just as my Propositional Calculus had an infinity of atoms. And similarly to my Propositional Calculus, you can just tack on a tick, or any quantity of ticks, to distinguish an infinity of unknowns. So unknowns can look as such:

d
c'
b''
a'''

In a way, it is a luxury to claim four glyphs of Anglo-Saxon, as just a and a tick would do. Long from now, I will actually drop b, c, and d from TCM, and I'll call it "harsh TCM" -- harsh in that parsing big formulas in it is a bit difficult. But for now, I'm luxurious with TCM.

Now what about addition and multiplication? That's trivial: I'm giving both ordinary symbols, + and *. But unambiguous grouping is a must, so I add grouping symbols: ( and ). (As of now, I'm slowly slipping into a formalism for "good strings" of TCM.) Now, you can say "b plus c" and "b by c" as such:

(b+c)
(b*c)

You cannot put in grouping symbols in a lax way; you must put grouping symbols around all groups. If you don't, your formula is not a good string.

And though this part will go on for a good bit, I'll stop now. --Rob

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