On 07 Oct 2011, at 22:33, Craig Weinberg wrote:
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?
The ability to use the definition to solve problems.
And the consequence of those definition.
But in high school we don't give any definition at all. We gives
examples and develop the familiarity with the concepts from that.
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'.
Incompleteness is a technical term in logic. It means that the
arithmetical propositions true in the structure (N, +, *) cannot been
effectively captured by any axiomatizable theory. It means truth is
far bigger than any notion of proof.
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.
Who feels uncomfortable? Arithmetic and arithmetical theories is what
mathematicians agree on the more.
You make complex what is simple. I have still no clue by what
sensorimotive means for you beyond the arithmetical propositions Bp &
Dt & p.
We feel that
natural numbers are 'natural', but there is no arithmetic reason for
that.
They are the simplest illustration of our intuition of finiteness.
It's sentimental.
I can use combinators if you don't like the number. I assume digital
mechanism. The laws of physics (quanta and qualia) becomes independent
of the choice of the initial axiomatic system, as far as it is Turing
universal.
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.
I asked if you agree with:
0 ≠ s(x)
s(x) = s(y) -> x = y
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
I don't use anything else when I mention the numbers (The induction
axioms will be part of the observers, and will be any sound consistant
extension of above, capable of proving its own universality).
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.
Assuming non-comp.
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.
The moon too?
And the andromeda galaxy?
And the galaxies discovered through Hubble?
And the big bang?
How ironical. I am supposed to be the idealist, but apparently I am
more physical realist than you. But I can't explain that in long
computations some object can develop individualities before some LUMs
recognize it. Indeed it is already true for the numbers and the
universal machines.
Bruno
http://iridia.ulb.ac.be/~marchal/
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