On Oct 8, 10:26 am, Bruno Marchal <marc...@ulb.ac.be> wrote:
> On 07 Oct 2011, at 22:33, Craig Weinberg 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.
All of the problem solving abilities (which are not taught but rather
guided - the actual learning is developed subjectively through
sensorimotive exploration), concepts, consequences, and concepts are
dependent on the definition of the arithmetic systems. My point was
that arithmetic is not definition independent. It's a language, like
any other, but it is a universally generic language so that it can be
applied to anything which is generic (i.e. not subjectivity, which is
non-generic and proprietary).
> >>> 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.
That's why I worded it that incompleteness 'says to me', because I'm
giving you what I think is an unintentional clue to interpreting the
concept. The observation that arithmetic propositions cannot be
completely captured by an axiomatic theory can be interpreted either
your way; that arithmetic truth is greater than arithmetic proof -or-
it can also be interpreted my way at the same time; that the failure
of arithmetic to prove itself demonstrates that the complete truth can
never be expressed arithmetically. There is always more than one way
to interpret profound truths.
> >>>>> 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.
Those four concepts - simplicity, illustration (metaphor), intuition,
and finiteness are all sensorimotive references. Those are natural and
mandatory to awareness. Numbers are a learned language of counting.
They are completely optional. If we didn't have 10 digits on our
hands, our natural numbers could have been anything - binary,
hexadecimal, or, as I suggested, non-repeating names that go on
indefinitely. Innumeracy is quite possible, but Inexperiential is not
> > 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
Turing universality supervenes upon sensorimotive phenomenology. The
whole idea of something which has a continuous purpose propagated
through discrete iterations, with sequential read/writes (energy,
events), memory (tape), coding and encoding...this assumes a whole
universe of underlying bootstrap ontology underneath it. Machineness
doesn't just arise out of the plenum, it needs a bunch of foundational
sensemaking phenomena to support it first.
> > 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)
> 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).
Sure I agree, but so what? Those ideas are no more primitive than
'music in a minor key sounds introspective and sad', or 'orange is a
mix of red and yellow'. The concept of 0 is no more universal than the
concept of invisibility or silence.
> >>> 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.
I don't understand the point of assuming anything? I assume as close
to nothing as I can.
> >>> 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?
Yes, of course. A turtle doesn't know what any of those things are.
Well maybe the moon, who knows? This is not to say there are no other
entities besides us which experience the universe this way, but
without a perceptual frame of reference to orient the observation,
there can be no form or content to an object. The form and content
arises purely from the relation between the two entities and their
inertial-relativistic frames (how their perceived worlds intersect).
> How ironical. I am supposed to be the idealist, but apparently I am
> more physical realist than you.
Yes, my position embraces the absolutely idealist and absolutely
realist at the same time. It's only the relation between the two that
make any 'sense'.
> 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.
lost me there.
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