On 20 Aug 2017, at 17:24, David Nyman wrote:
On 20 Aug 2017 2:46 p.m., "Bruno Marchal" <marc...@ulb.ac.be> wrote:
On 19 Aug 2017, at 01:21, David Nyman wrote:
On 18 August 2017 at 18:13, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 18 Aug 2017, at 15:39, David Nyman wrote:
He points at a mug and says that 'representations' (meaning
numbers) aren't to be confused with things themselves.
He confuses a number and a possible representation of a number.
LIke many people confuse the (usual, standard) arithmetical reality
with a theory of the arithmetical reality. Yet after Gödel we know
that no theories at all can represent or encompass the whole of the
arithmetical reality.
It is not much different that confusing a telescope and a star, or
a microscope and a bacteria, or a finger and a moon, or a number
and a numeral ("chiffre" in french).
But in math, it is quite frequent. In logic, such distinction are
very important. In Gödel's proof, we need to distinguish a
mathematical being, like the number s(0), the representation of the
number s(0), which is the sequence of the symbol "s", "(", "0",
")" (and that is not a number, but a word), and the representation
of the representation of a number, which, when we represent things
in arithmetic will be something like
2^3 * 3^4 * 5^5 *7^6, which will be some
s(s(s(s(s(s(s(s(s(s(s(s(s(s(s( ....(0)...). (very long!).
But what is the 'thing itself' at which he points?
A mug. I guess.
Just so.
The question will be "what is a mug in itself". A materialist would
say that it is a structured collection of atoms, but a mechanist has
to say something like "a common pattern pointed at by some normal
(in Gauss sense) machine sharing some long (deep) histories.
Something like that.
Yeah, something like that. I enjoyed Frenkel's talk actually. I like
his enthusiasm for mathematics. It's funny though he doesn't seem to
appreciate his implicit assumptions, or indeed that he is in fact
expressing a particular metaphysical position. Is math real? I mean,
really real? Trouble is, people assume that the answer is obvious,
whether they think it's yes or no.
We need only to agree on what we agree. The beauty of the Church's
thesis, is that it entails by "theoremata" the existence of the
emulation of all computations in elementary arithmetic.
(Just that fact, and computationalism, should make us doubt that we
can take a primary physical reality for granted: it is the dream
argument with a vengeance).
The question is not "is math real", but do you believe that 2+0= 2,
and a bit of logic.
I do not claim that the whole of philosophy or theology can become
science, but I do claim that if we assume mechanism, then by Church's
thesis, philosophy and theology becomes a science, even in the usual
empiricist sense.
There is something funny here. The theology of the machine is ultra-
non-empiricist, as the mystical machine claims that the whole truth
(including physics) is "in your head and nowhere else". ("you" = any
universal machine). But that is what makes the machine theology
testable, by comparing the physics in the head of any (sound)
universal machine with what we actually observed.
Math is real? Which math? I doubt that sincere people doubt
arithmetic, and I have never heard of parents who would have taken
their kids out of a school for the reason that hey have been taught
that 2+2=4; neither in the Western nor Eastern worlds.
Now, for limit and real numbers it is much less obvious. here
intutionist and classical philosophy diverge. With Mechanism, it is
better to considered analysis (and eventually physics) as universal
machine mind tools. Gödel's incompleteness justifies partially why the
machine needs to invent infinities to better figure out themselves.
Before Gödel, most mathematician, like Hilbert, were hoping that with
the finite and the symbolic we could justify the consistency of the
use of the infinities, but after Gödel we know that even with the
infinities we cannot circumscribe and justify the consistency of the
finite and the symbolic. The root of the undecidability is the Turing-
universality. With the conceptual discovery of the universal machine,
we got the tools to understand that we have no idea what they are. Nor
what they are capable of doing. A universal machine can defeat all
effective theory about itself, and it knows already that its soul
(first person) is not a machine.
So, to be clear: is *arithmetic* real? I think so. Fundamentally (up
to the Turing equivalence).
Is analysis real, yes, but only as a a phenomenological simplification
of the digital, which has still its laws.
But here we have no Church thesis, and no real notion of "standard
model". Should we teach infinitesimals whose consistency follows some
work in non standard model of arithmetic? Should we use intuitionist
analysis? with or without the intuitionist Church's thesis (not really
related to the classical).
On RA, there is unanimity (among humans today).
On PA there is unanimity minus one (Nelson)
On Analysis, or set theories, there is no unanimity, but a clear
classical "mainstream", and a lot of different, but easily related
options reflecting taste and personal opinions.
I doubt less that 24 is composite than any assumption in physics,
metaphysics, theology or whatever applied sciences.
That 24 is composite is among the 3p notions which are the closest to
the non communicable 1p certainty of consciousness here and now (the
only non doubtable thing).
Bruno
David
Bruno
David
Bruno
https://futurism.com/the-most-important-question-underlying-artificial-general-intelligence-research-is-math-real/
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