On 20 Sep 2016, at 06:18, Brent Meeker wrote:
On 9/19/2016 7:26 PM, Russell Standish wrote:
On Sun, Sep 18, 2016 at 11:54:04PM +0200, smitra wrote:
https://www.youtube.com/watch?v=nK6XawDE8_U
Just finished watching Norm's video, and one thing really struck
me. The process of factoring numbers of the form 10^n+23 is
generating
vast amounts of complexity, as n increases. I hadn't really thought
about things that way before, but I have to say this really
constitutes a direct counter example to my oft stated dictum that
evolutionary processes are the only way to generate complexity.
Food for thought.
But is the algorithmic complexity high? A program has to run a long
time to find some large prime factor, but the program is fairly
simple. In Bruno's Platonic view these numbers and relations just ARE
OK.
and their computation is irrelevant.
?
Their computations are among the relations which just ARE, and they
are relevant.
Indeed the computations are, basically, the sigma_1 true relation.
(They are also probable by any Turing universal theory). They are the
domain on which the (8) logics sof self-reference eventually are
applied.
But I see no reason why one cannot axiomatize an unltrafinitist
arithmetic - that's essential what computers do. Then those "dark
numbers" will not exist.
Not from the small numbers/machine FIRST person pov: as the small
program is also implemented by the infinities of the dark giant
numbers, and that might play a rôle in the measure calculus.
The bodily and practical means of the numbers are rooted in the
neighborhood of zero. The fate of the number's soul and experienced
are driven in the neighborhood of infinity.
That's a simple consequence of the inability of the first person to
detect relative delays in the computations which support them (in
arithmetic).
But doctrinal ultrafinitism does not make sense even from a pure 3p
pov, in the sense that it deprives itself of the means to explain what
it means. Robinson arithmetic is consistent with the proposition
"there is a biggest natural number", but we get the consistency by
adding that biggest natural number above all others in some contrived
model.
A genuine ultrafinist will never argue for ultrafinitism.
He/she can't do that without betraying that he/she disbelieves in it
at the meta-level. He/she will only say "what?" when you name a number
too much big for him/her.
Bruno
Brent
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