On 16 May 2017, at 23:32, John Clark wrote:
On Tue, May 16, 2017 at 2:44 AM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
>> There is no mathematical reason time or space or
anything else can't be continuous, nor can mathematics find
anything special about the numbers 1.6*10^-35 or
5.4*10^-43 , but physics can. And mathematics can produce
paradoxes but physics never can.
> You confuse a mathematical reality, like the standard model
of arithmetic, or a group, with the theory about such object.
You're right, I am confused why you believe a model of something
is entirely different from a theory of the something.
Have you heard of model theory? It is also called semantic. A model is
the math version of a reality. A theory is a finite object (a set of
formula) pointing of a reality. Physicists used model in the sense of
theory. Logician use model in the sense of the semantic of a theory.
Logician used model in the same sense as a painter. the model is the
naked person that we draw, or the panorama, and in that image; the
drawing is the theory. Physicist use model in the sense of the little
kids: the model of a car is a little toy.
Exemple: take RA or PA. The (standard) model is the mathematical
structure (N, 0, +, *). Or take the axiom of group. Those axioms give
the theory, and a model is any group.
By incompleteness, no theories can prove the existence of a model of
itself. That would be equivalent (by Gödel-Henkin completeness
theorem) to proving its own consistency.
Completeness theorem: a theory is consistent if and only if the theory
has a model.
Note that this is false for some higher order logic.
> You cannot produce a contradictory mathematical reality, only
a false theory,
So what is the correct purely mathematical theory that would
avoid the paradoxes I described? After 2 minutes is that Zeno lamp
on or off? How many balls are in that box, a infinite number or
zero? According to you the fact that nobody can physically make such
a lamp or box makes no difference, so there is an answer, so let's
have it!
You need to make it mathematically clear before. If not, it is like
saying a false mathematical proposition, or a senseless one. We can do
that with physics too.
> Mathematics is not a language.
Most mathematicians would disagree.
?
Most mathematician agree with that idea, especially after Gödel proved
his incompleteness theorem, but all non go theorem in math, like
sqrt(2) is not a ratio can be used to develop that intuition. Read
hardy "a mathematician's apology".
Then logicians made that clear, by distinguishing precisely the
languages from the theories, and the models of a theory from that
theory.
> We can build reversible computer, so we can implement all
computation without erasing memories
True.
> and without using energy.
False. You still need energy, you can make the amount of
energy arbitrary small with a reversible computer if you can
also keep the computer arbitrarily well isolated from unwanted
interactions, but the less energy you use the slower the
calculation and if you use zero energy the calculation will
grind to a halt. And besides energy you certainly need matter
to perform calculations.
I need energy to impolement the computation, but the computation
itself will not need energy. Of course, I will need energy for the
read and write, and any use of the computation.
That is true in arithmetic already, so energy is itself an indexical
coming from self-reference. The need of energy in our physical
implementation of computer is thus not relevant, and only an emerging
(logically) aspect of arithmetic when viewed through the "material
points of view" (like Bp & p with p sigma_1).
> Information would be physical if Everett was false,
I don't see what Everett has to do with it.
If everett is false, and if QM is correct, information and mind have
to be physical to reduce the wave packet at a distance, and many magic
thing like that. Of course, that is the main reason to believe in
Everett, or more correctly, to disbelieve in any wave packet reduction.
> and von Neumann correct, like with a reduction of the wave
packet by consciousness. But then we get 3p [blah blah]
Screw peepee screw baby talk.
Well, this is not convincing, especially when you don't put the quote
entirely.
>> In mathematics you start with certain axioms
> Hoping they are true on the reality we intend to reason about.
There are a infinite number of axioms that mathematicians could
have started out with, why did they pick the particular ones that
they did?
It depends on the application, or, in pure mathematics, from their
taste. The mathematical reality is ample. it contains many structures
obeying to different theories.
Because there were best ones they could find that conformed with the
word that they saw around them.
That is the particular case of (theoretical) physics.
In every case physics leads and mathematics follows.
If that is true, computationalism is false. That's the point. You
present physicalism like if it was metaphysically proved. Obviopusly,
that is not the case, and there are no evidences at all for it, only
evidences of the contrary, unless you eliminate consciousness, person,
etc. I think you beg the question.
>> and agree to follow certain rules on how to manipulate those
axioms, and if you follow those rules without error then we say the
result is mathematically true,
> No. Not only we don't do that, but we cannot do that. G does
not prove []p -> p. You confuse formal mathematics, which is
like asking a machine, and doing mathematics oneself, which is an
informal task.
Mathematicians get hunches just like everybody else but they
don't publish hunches, they try to manufacture their hunch by
manipulating the axioms using the agreed on rules of the game,
and if they find way to do that then and only then they publish.
Many mathematicians have a hunch that Goldbach's conjecture is
true but nobody has been able to construct it by manipulating the
fundamental axioms, and that's why it's called a conjecture and not
a theory and that's why they say it hasn't been proved.
OK. sale for any science.
>> but all we're really saying is that in the language of
mathematics the result is grammatically correct.
> Not that is plain false. Read any book in logic.
Formal logic just like mathematics starts out with axioms and ways
to manipulate them
Mathematicians and logicians proves their theorem informally, without
any axioms. Logicians are interested in formalism, but, like any
mathematicians, they make the proofs informally, in english, relying
on their intuition and familiarity of a domain. You confuse the
informal theories about formal theories and machine and the formal
theories themselves. It is akin to the confusion between brain and
mind, or theories and models.
and logicians learned which ones to use and which ones not to use by
observing how the world works. For example, logicians observe that
in our physical world things usually continue, and that's how they
know about induction.
You defend conventionalism in mathematics.
Conventionalism says that axioms in mathematics
should be chosen for the results they produce not for their
coherence with the physical world, and I'm saying the opposite of
that.
> It is simply refuted by the incompleteness theorems,
It says that no language, not even mathematics, can
describe everything about the world in a way that is free of all
paradoxes, I don't see how that supports your view.
It says that no theories can describe everything already about the
arithmetical reality.
Bruno
John K Clark
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