On Oct 7, 9:21 am, Bruno Marchal <marc...@ulb.ac.be> wrote:
> On 06 Oct 2011, at 23:14, Craig Weinberg wrote:
> > On Oct 6, 12:04 pm, Bruno Marchal <marc...@ulb.ac.be> wrote:
> >> On 04 Oct 2011, at 21:59, benjayk wrote:
> >>> The point is that a definition doesn't say anything beyond it's
> >>> definition.
> >> This is deeply false. Look at the Mandelbrot set, you can intuit that
> >> is much more than its definition. That is the base of Gödel's
> >> discovery: the arithmetical reality is FAR beyond ANY attempt to
> >> define it.
> > Can't you also interpret that Gödel's discovery is that arithmetic  
> > can
> > never be fully realized through definition?
> The usual model (N, +, *), taught in school, and called "standard  
> model of arithmetic" by logician fully "realize" it, and is definition  
> independent.

What is it that is taught if not definitions?

> > This doesn't imply an
> > arithmetic reality to me at all, it implies 'incompleteness'; lacking
> > the possibility of concrete realism.
> The word "concrete" has no absolute meaning. Comp is "many types---no  
> Token".

It doesn't need to have an absolute meaning. A relative meaning makes
the same point. Incompleteness says to me 'lacking in completeness',
not 'complete beyond all reckoning'.

> >>> So, the number 17 is always prime because we defined numbers in the
> >>> way. If
> >>> I define some other number system of natural numbers where I just
> >>> declare
> >>> that number 17 shall not be prime, then it is not prime.
> >> No. You are just deciding to talk about something else.
> > I think Ben is right. We can just say that 17 is also divisible by
> > number Θ (17 = 2 x fellini, which is 8.5),
> 8.5 is not a 0, s(0), s(s0)), .... You are just calling "natural  
> number" what we usually call rational number.

It's not 8.5, it's Θ. It doesn't matter what we usually call it, now
we are calling it a natural number. The fact that we feel
uncomfortable with this illustrates that our basis for arithmetic
truth is sensorimotive, and not itself purely arithmetic. We feel that
natural numbers are 'natural', but there is no arithmetic reason for
that. It's sentimental. I brought up the idea earlier of a number
system without any repetition. A base-∞ number system which would run
0-9 and then alphaumeric, symbolic, pictograms, names of people in the
Tokyo phonebook, etc. This would be closer to an arithmetic system
independent of sensorimotive patterning. The familiarity of the digits
I think functions like a mantra, hypnotically conjuring the dream of
an arithmetic reality where there is none. There is a sensorimotive
reality and an electromagnetic 3-p side to that reality, and there are
1-p arithmetic computations with which the sensorimotive can model 3-p
isomorphic experiences for itself, but there is no truly primitive
arithmetic reality independent of subjective observers.

>You illustrate my point.  
> You talk about something else, and you should have disagree with the  
> axioms that I have already given.

Not sure what you mean.

> > and build our number system
> > around that. Like non-Euclidean arithmetic.
> That already exists, even when agreeing with the axioms, of, say,  
> Peano Arithmetic. We can build model of arithmetic where we have the  
> truth of "provable(0=1)", despite the falsity of it in the standard  
> model, given that PA cannot prove the consistency of PA. This means  
> that we have non standard models of PA, and thus of arithmetic. But it  
> can be shown that in such model the 'natural number' are very weird  
> infinite objects, and they do not concern us directly.  But "17 is  
> prime" is provable in PA and is thus true in ALL interpretations or  
> models of PA. Likewise, the Universal Dovetailer is the same object in  
> ALL models of PA.
> All theorems of PA are true in all interpretations of PA (by Gödel's  
> completeness theorem).

I'm not saying that arithmetic isn't an internally consistent logic
with unexpected depths and qualities, I'm just saying it can't turn
blue or taste like broccoli.

> > Primeness isn't a reality,
> > it's an epiphenomenon of a particular motivation to recognize
> > particular patterns.
> They have to exist to be able of being recognized by some entities, in  
> case they have the motivation. The lack of motivation of non human  
> animal for the planet Saturn did not prevent it of having rings before  
> humans discovered them.

Rings from whose perspective? Without something to anchor perceptual
frame of reference, there would be no difference between the ringlike
visual qualities of them and the crunchiness of the oceans of ice,
dust and rocks that make them up, or the tiny nubs of light on either
side of a speck in a distant sky, or the nothing at all that it would
be in the absence of visual qualia.  Who says Saturn has rings at all?
Only our eyes, through telescopic extension, and our sensorimotive
feedback loops of our brains with their observations and experiences
in applied astronomy. The rings are part of the human story of the
Saturn, not necessarily Saturn's story of itself.


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