On 21 Aug 2017, at 00:16, Brent Meeker wrote:
On 8/20/2017 9:23 AM, Bruno Marchal wrote:
On 20 Aug 2017, at 17:24, David Nyman wrote:
On 20 Aug 2017 2:46 p.m., "Bruno Marchal" <[email protected]> wrote:
On 19 Aug 2017, at 01:21, David Nyman wrote:
On 18 August 2017 at 18:13, Bruno Marchal <[email protected]>
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.
Are you claiming that there is a one-to-one map between true
statements in mathematics and what I experience??
Not at all. I am saying that the laws of physics can be found by
introspection. Introspection lead to the opposite of what you say: we
can only experience (1p) a tiny fraction of the physical reality and
of the psychological reality and all what is "in the head" in the 1p
view is far bigger than the head itself (in this metaphorical image).
Only the "outer God" might see, or be defined, by the "whole
(arithmetical) truth".
The problem with everythingism is that one doesn't experience
everything.
Indeed. But that is a very general problem, and you could say "the
problem with physicalism is that we don't experience primary matter,
nor the whole physical reality. After Gödel, we know that even for the
arithmetical reality, even an immortal being can only scratch it
infinitesimally. Reality is big.
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.
I doubt the infinity of standard arithmetic.
Good. That is why I do not assume it. Only 0, 1, 2, ... all the
objects assumed to exist are finite. Mechanism is a Finitism. Like
Plotinus' God, Infinity does not belongs to the beings. It is
responsible (only) of the existence of the (finite) being. This does
not prevent us to use infinity at the meta-level, like all scientists
do all the time.
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.
All the more reason to classify them as fiction - not part of the
really real.
OK. But then Mechanism has to be assumed false, or you too would not
exist (and I hope you believe that you exist, in a way or another).
Bruno
Brent
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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