Wednesday, April 30, 2014

Inoculating my Daughter

          Inoculating my Daughter

I have just had the honor of giving my daughter Hannah an inoculation against a common teenage ailment. I refer to Objectivism; for she asked me about Ayn Rand, and I gave her my informed opinion.
Specifically I told Hannah about the infamous railroad scene in “Atlas Shrugged”. I detailed the set-up; the dangerous tunnel, the idiotic bureaucrat’s suicidal demand, the chain of buck-passing, the fatal go-ahead, the complaisent workers, the train’s doom... all of which I described with admiration for Rand’s meticulousness (for once not tiresome) and her passion. A death machine of incompetence and spinelessness, clickety-clicking right there before your eyes! A masterpiece of eldritch horror! That scene was this close to Literature!
But the moment the train entered the tunnel, her writing all went horribly wrong! For Rand then focused on the passengers; a risky move if you do it wrong, and she did. For what if (I asked Hannah) that train had been full of good people? Or perhaps even a few of them super-virtuous by Rand’s elitist Objectivist standards? For them to die alongside the idiot bureaucrat would been supreme injustice; a raw red wound, an affront against humanity! What if Rand had raged and wailed against the sick cruelty of the idiot world? That (I told Hannah) would have been writing!
But no! By the way Rand told it, everyone on the train was an idiot moocher, as bad as the bureaucrat, and so everyone got what they deserved! (Hannah rolled her eyes upon hearing this.) Instant justice, automatic karma, all’s well with the world! Rand got that close to Literature, but at the last moment she turned it into Propaganda!
The trainwreck scene was itself a trainwreck!  And that (I told Hannah) is Ayn Rand.

Tuesday, April 29, 2014

Political Exchange

       Political Exchange

I propose a trade. The Democrats claim Abraham Lincoln as one of their own; and in return the Republicans can have Andrew Jackson. A win-win!

Monday, April 28, 2014

Isopters and the Relativity of Communism

Isopters and the Relativity of Communism

My daughter is inventing, for online-gaming purposes, a fictional race of beings. She calls them Isopters, which is the scientific name for termites; and these fictional beings have a hive mind. It's a classic SF trope; and like all such it is a poetically alienated view of humankind. Us as hive-mind critters.
Part of the fun of fantasy is working out the outlandish implications of taking the premise literally. Thus she and I have worked out that Isopters have the usual social-insect castes; queen, drone, workers, soldiers; that the workers do all the actual thinking, the queen and drones are just the hive's genitalia; that therefore it is a comic gaffe for a visiting human to address the queen.
They communicate by radio; they have a Web of shared thoughts, memories and sensations. They are monocular, but share visual data, so when two are present in a place they have depth perception. They have individual identities, but it’s superficial; underneath they are one. When you talk to one, you talk to all.
Though unified, still Isopters possess sufficient individuality for competition and rivalry. They can disconnect from their Web, for more originality, but less safety. This balance between self-and-all defines Isopter society.
One lovely detail that Hannah came up with is that Isopters are not given names at birth, but numbers. But also they can earn names, either praising or insulting, depending on what they do. Hannah deduces from this that Isopters would be astonished that all humans are given names at birth - and not just one, but two or even three, one of which we don't even use! I chimed in, playing Isopter, “You get names just for breathing? That's Communism!

Friday, April 25, 2014

Lattice Rationals, 10 of 10

Lattice Rationals modulo zeroids and infinities

               Say that x and y are “equal modulo zeroids”, a.k.a.  “x  =0  y”, if and only if:
                              x + 0/n     =     y +  0/n     for some positive integer n
               Therefore we have positive cancellation modulo zeroids:
                              (ac)/(bc)        =0       a/b       if   c>0
               Equality modulo zeroids is an equivalence relation; reflexive, symmetric, and transitive. Therefore it defines equivalence classes, and operations on those classes for well-defined on those classes. For instance:
               If     x   =0   X    and    y   =0  Y     then;
                              x + y    =0    X + Y
                              x - y     =0    X - Y
                              x * y     =0    X * Y
               But reciprocal is not well-defined on the zeroids;
                              0/1     =0     0/2       ;      but    1/0   and   2/0   are not equal modulo zeroids.
               Similarly, reduction is not well-defined for the zeroids.
               The equivalence classes, with operations thus defined, equals the wheel numbers plus the double ringlet of infinities. This is an arithmetic of the rationals, plus alternator, plus all the infinities.

               Reciprocally, we could define equality modulo infinities:
                “x  =1/0   y” if and only if :
                              x 1/+ n/0     =     y 1/+  n/0     for some positive integer n
               This too has positive cancellation:
                              (ac)/(bc)       =1/0         a/b       if   c>0
               Reduction and multiplication are well defined modulo infinities; but reciprocal and addition are not well defined on the infinities. The equivalence classes modulo infinities equals the wheel  numbers plus the double ringlet of zeroids.

Thursday, April 24, 2014

Lattice Rationals, 9 of 10

The Wheel Numbers

               TheWheel Numbers arise from the lattice rationals if you add the axiom:
                              0/n     =    0/1        for     n>0
               This is equivalent to “Positive Cancellation”:
                              (ac)/(bc)        =       a/b       if   c>0
               Note that the alternator is still unequal to zero, and negatives do not cancel.

               The wheel numbers can be divided in mainstream, alternates, and null quotients.
               The mainstream numbers are of the form a/b , in lowest terms, with b>0.
               The alternate numbers are of the form a/b , in lowest terms, with b<0.
               The null quotients are 1/0, infinity; -1/0, negative infinity; and 0/0, indefinity.

               The wheel numbers correspond to a circle surrounding a point. Each wheel number corresponds to the slope from the center to the point. The point at the center corresponds to 0/0; the points directly above and below the center correspond to   +1/0   and  - 1/0;  the points directly right and left of the center  correspond to 0 and @; the right half of the circle corresponds  to the mainstream numbers, the left half of the circle corresponds to the alternate numbers.
                                               -1+@              1
                                             @           0/0            0                        
                                                1+@              -1
                                                             - 1/0

     Note that            - 1/0         =             +1/0  + @
               and                  +1/0         =             -1/0  + @
Reciprocal flips at the infinities between both sign and alternativity.
               The wheel numbers have these laws:
               Addition, reduction and multiplication are commutative and associative.
                              x + 0/1                  =             x  1/+  1/0                       =             x*1/1     =             x
                              x + 1/0  =  1/0   if x is finite
                              x  1/+  0/1   =   0/1 if x is nonzeroid.
                              0  1/+ @           =             1/0 +(-1/0)    =    0 * 1/0  =   0/0
                              X + @                    =             x  1/+ -1/0         =    -x/-1
                              X + @ + @           =             x  1/+ -1/0  1/+ -1/0 =     x
               Triple Distribution:
                              X*(A+B+C)           =             X*A   +   X*B   +   X*C                       if X is finite
                              X*(A 1/+ B 1/+ C)       =             X*A   1/+   X*B   1/+   X*C                          if X is nonzeroid

               Since @+@ = 0,  it’s consistent with exponential arithmetic to posit, for A not equal to 0:
                                             A^@      =    -1
                              and        A^(x+@)   =   -(A^x)
               So in general we can posit:
                                             logA(-1)     =    @
                              and        logA(-x)     =    logA(x) + @
               This is a theory of logarithmic negation without reference to the complex numbers.