"The Principle of Shrinkage"

[Jonathan Post]

Coincidently, I am writing a paper which starts with some Spencer-Brown
notions, in this case "imaginary truth values."  His work is fascinating,
and many have extended and interpreted his work in different ways, with
mixed success. Another motivation of the paper I abstract below is
explaining the deeper structure of many OEIS sequences involving
concatenation of digit strings, and digital reversal, the latter
generalizing to adjoint operators which reverse the order of multiplication
of products (abc...xyz)* = z* y* x* ... c* b* a*. Spencer-Brown ideas are
often philosophically profound, but hard to apply rigorously.

Star-Algebras of Imaginary Boolean Strings Closed
under Add-without-carry, Convolution, Complex
Conjugation, and Reversal of Finite and Infinite
Sequences

Jonathan Vos Post
Version 3.0 of 13 September 2006
19 pages, 5200 words, corrects typos in 1.0 and 2.0,
adds base-3 data and nth root of NOT, some additional
references

ABSTRACT:

A class of models of multivalued propositional logic
is presented which have the apparent form of
structures known as star-algebras. These models
interrelate "imaginary" truth values that oscillate
over time [Kauffman, 2002; Varela, 1979;
Spencer-Brown, 1969], modular arithmetic, complex
conjugation, and reversal of finite and infinite
sequences. Although finitely generated, these models
lead naturally to transfinite ordinals. Such models
also allow for electronic logic devices that behave as
"nth root of NOT" for any positive integer value of n,
generalizing the Deutch Gate which is a quantum
computing implementation of "square root of NOT."

[End Abstract]



On 9/14/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> Really, the incompleteness theorem means that this can't be true.
>
> It isn't very hard to show that the general question of whether a
> function converges is undecideable.
>
> This means that the domains of functions where the principle
> applies are all trivial (e.g., rational functions), or else membership
> in them is itself undecideable - which isn't very useful.
>
> The phrase "naturally occurring function" seems to be trying to
> avoid this, but it doesn't work that  way.  You can't rule out
> the exceptions without also excluding the interesting cases.
>
> In this case "He challenged me to say I really believed that the
> error might suddenly explode outside such bounds after such
> extensive evidence of shrinkage."  Suddenly explode?  No.
> Gradually diverge?  Unlikely, but possible.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: eclark at math.usf.edu
>
> ... In the proof GSB used
> a new axiom ("the principle of shrinkage") which Tim Chow
> formulated as follows:
>
>   If f:R -> R or f:N -> R is a "naturally occurring function" and
>   lim(x->oo) f(x) appears to exist, then it does.
>
> Then Tim asks the question:
>
> Can anyone give an example of a conjecture of the form "lim(x->oo) f(x)
> exists" for which there was extensive numerical evidence and which
>
>
>
>
>
>
>
>
> ________________________________________________________________________
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