On 25 Jan 2015, at 09:45, Jason Resch wrote:
On Sat, Jan 24, 2015 at 12:43 PM, meekerdb <[email protected]>
wrote:
On 1/24/2015 1:23 AM, Jason Resch wrote:
On Fri, Jan 23, 2015 at 10:15 PM, meekerdb <[email protected]>
wrote:
On 1/23/2015 8:16 AM, Jason Resch wrote:
On Friday, January 23, 2015, John Clark <[email protected]>
wrote:
>
> On Thu, Jan 22, 2015 at 4:46 PM, Jason Resch
<[email protected]> wrote
>
>>> >> Do we know that? Do we know that such a digit exists?
>>
>> > It follows from the axioms that there is a certain definite
digit.
>
> They show you how to generate terms in a sequence and if you add
up enough of them you'd get the the 10^(10^(10^100))th decimal
digit of pi; but it assumes that there is no barrier that makes
doing that impossible and states that assumption with 3 little
dots (...). I don't know for certain but those 3 little dots
*might* be saying something that is logical nonsense, I do know
for certain that the first mathematicians who used those 3 little
dots knew nothing about quantum mechanics or the computational
limit of the universe, and that gives me pause.
>
What I explained is that if you think there is a largest number
*that* is what definitely leads to logical nonsense.
Say there is a largest number N, such that N+1 is not a bigger
number, but is still N. That means N+0 = N+1. Now subtract N from
both sides.
Why should subtraction of the biggest number not obey special
rules, e.g. Subtracting N from any normal number yields -N.
Subtracting N from any Big number yields zero.
I still don't know if that would escape the problems. Let's say M
is the number right before the biggest number (or any "Big
numbers"). Then using your rules you find that:
M - (M+1) = 0 --- or is it -N?
but
(M - M) + 1 = 1
It might be possible to come up with axioms that allow you to have
a biggest number that operates in a consistent way, but I think it
would be very difficult, and probably not very useful.
It may be useful in computer science, c.f.
http://www.impan.pl/~kz/files/MKKZ_ArFM.pdf
Nor do I see the point of hobbling a theory (supposedly about the
infinite natural numbers) by declaring some aspects of that theory
to be strictly off limits and beyond the possibility of discussion.
I'm not proposing that anything is off limits. I'm proposing that
there are different axiomatic systems which may equally correspond
with our finite observations,
How can it be said that any physical observation corresponds with
entities that exist within an axiomatic system? Do you see the goal
of physics being to provide mathematicians with a correct axiomatic
system?
therefore it is not justified to pick one of them and claim that it
is empirically proven. Axiomatic systems may contradict one another
and that is why "X provable in A1" only implies "X true relative to
A1" and not "exists X". Plato was led to suppose that perfect
forms existed because in his day it seemed there was only one
possible mathematics which was something like arithmetic plus Euclid.
In our day we know that no axiomatic system can be perfect, and
accordingly that axiomatic systems can at best approach/approximate
rather than prescribe/define truth. In that way they're not much
different from physical theories we develop about the objective
physical universe: axiomatic systems are theories about objective
mathematical objects.
Without doubt for the separable part of mathematics. The situation is
more problematic for analysis, set theories. With computationalism it
is simpler to already put analysis, like physics, in the
epistemological.
Even if a large part of (constructive) real numbers can be conceive
and talk about directly in PA, soon or later PA might add a new axiom
to extends its "perception" of the arithmetical truth.
The (re)discovery by the humans of the universal machine is a
recurrent theme in the arithmetical reality.
Nature did already with the DNA, then again, with the brain, and
before with the quantum vacuum. Those are abstract big-bangs. There
are "Creative" (in Post sense) Explosions. There are transfinite, when
viewed from inside arithmetic by arithmetical creatures.
The arithmetical reality is full of life, and of gods (non computable
sets talking about the truth about machines, that those machine cannot
prove, know, observe, etc).
Perhaps too much, so that a digital brain cannot filter the
computational histories+oracles, and some primitive matter would have
a (non Turing emulable) role (We would be less free, I think. It looks
ad hoc, but only the measure theory, and the testing can decide).
Abstractly, computationalism leads to a Many-Types, No-Token, theory.
Token becomes indexicals pertaining on equivalence classes for first
person non-distinguishability relations.
Bruno
Jason
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