Monday, January 6, 2014

On Logistic, 1 of 8; Logistic Defined

          On Logistic
          By Nathaniel Hellerstein

1.     Logistic Defined
2.     Reduction
3.     Einstein Addition
4.     Mediant
5.     Carrollian Laws
6.     Logistic Fixedpoints
7.     Means by Logistic
8.     Beyond Logistic

          1. Logistic Defined

          Define a logistic function  as a function on the interval [0,], defined from constants, variables, addition and reciprocal; all continuous on [0,].
          Logistic addition is defined on the closed interval [0,], including the right endpoint, infinity. In logistic addition, 1/0 + 1/0 = 1/0; this is perfectly fine for 1/0 considered as positive infinity; but problematic if 1/0 equals its opposite, for then 1/0+1/0=0/0. For now we shall consider only zero, infinity, and positive reals; and in that context, infinity equals limit behavior for large positive numbers.
          Note these tables of values:
+           0                     1/x
0           0                          
          This has the same form as these Boolean logic tables:
And     T      F                  not x
T          T      F                    F     
F          F      F                    T
          So logistic on zero and infinity is isomorphic to Boolean logic, if you associate + with And, reciprocal with Not, 0 with True, and with False. Logistic is where logic meets arithmetic.
          There is another isomorphism, which associates + with Or, 0 with False and with True; however I prefer the first one, in which the word ‘and’ means both conjunction and addition; but also because, to me, zero feels truer than infinity. Infinity is where the impossible happens; whereas the void possesses clarity. In this Zen-like interpretation of logistic, to decrease in quantity is to increase in truth.
          Logistic deals with many more numbers than zero and infinity; in particular the number one, which solves the equation
                    x        =        1/x
          Or in other words:
                    x        =        not x.
          So in logistic, unity corresponds to the Liar paradox.

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