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 ∞ ∞
∞ ∞ ∞ 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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