On 20 Aug 2017, at 19:38, David Nyman wrote:
On 20 Aug 2017 17:23, "Bruno Marchal" <marc...@ulb.ac.be> wrote:
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.
It's taken me years to appreciate this fully. The funny thing is
that when you say this to people they often object that this is the
case only in mathematics or logic, not in the 'real' world. But
actually we can never avoid the fact that we are always reasoning in
terms of the assumptions of some theory or other, whether it's
explicit or not (usually not). So we need to agree on what we agree,
indeed.
The beauty of the Church's thesis, is that it entails by
"theoremata" the existence of the emulation of all computations in
elementary arithmetic.
We can indeed agree to agree on that.
(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).
Indeed.
The question is not "is math real", but do you believe that 2+0= 2,
and a bit of logic.
Difficult to disagree with.
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.
About time too.
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.
Easier said than done though.
Well, strictly speaking, it is done for the propositional level. The
difficulty is more in finding people capable and knowing and being
serious in logic, metaphysics, and physics.
Logicians hates metaphysics (and there are historical reason from that).
Physicists misunderstand logic (not just Penrose).
And metaphysicians run away from logic and physics, in the best case
where they do not mystify them.
Today, we can smash protons at quasi-the speed of light, but to
confine a logician, a metaphysician and a physicist near the
blackboard might require much more work and ingenuity!
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.
Possibly in Airstrip One.
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).
Good.
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).
Truth is certainly undoubtable.
Only the truth you known is undoubtable.
Only the truth know it is the truth, except for consciousness where
truth and you get locally identified.
For my consciousness here and now, I would say that I am 100% sure,
despite the vagueness and indexicalness of "here" and "now".
But for 24 is composite, although I would really like to say that I am
as much sure, I remember that dream where I get totally convinced that
the modus ponens rule alone was inconsistent, and the worst, was that
the argument was a window hanging (!). When I wake up, I understood
that the feeling of being totally convinced by an argument can be
mimicked at will, independently of the argument, (that explains also
the utter strangeness of the salvia experience, and strange dreams).
And that is enough for making me not sure that I am saying something
true when saying that 24 is composite (let me verify: 24= 2 * 12, or
3*8, or 4*6, it looks quite composite, but I can only "philosohically"
be only 99,999999999999999999999999999999999998 % sure.
Consciousness seems immune to this, because, even in a dream, you
can't doubt of the consciousness "here and now", I can't even conceive
a dream making me wrong on this, and no amount of salvia has led to
this. There is a case of a mental pathology, where people asserts that
they are dead, but I have not found any of them asserting that they
are unconscious. It seems it is more like a loss of all emotional
value, than a lack of consciousness.
I have many selected papers on consciousness pathologies, but it is
not an easy literature, and also a place where causation and
correlation are not always validly related.
Best,
Bruno
David
Bruno
David
Bruno
David
Bruno
https://futurism.com/the-most-important-question-underlying-artificial-general-intelligence-research-is-math-real/
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