On 29 Feb 2012, at 15:47, Alberto G.Corona wrote:
On 29 feb, 11:20, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 29 Feb 2012, at 02:20, Alberto G.Corona wrote (to Stephen):
A thing that I often ask myself concerning MMH is the question
about
what is mathematical and what is not?. The set of real numbers is a
mathematical structure, but also the set of real numbers plus the
point (1,1) in the plane is.
Sure. Now, with comp, that mathematical structure is more easily
handled in the "mind" of the universal machine. For the ontology we
can use arithmetic, on which everyone agree. It is absolutely
undecidable that there is more than that (with the comp assumption).
So for the math, comp invite to assume only what is called "the
sharable part of intuitionist and classical mathematics.
I do not thing in computations in terms of "minds of universal
machines" in the abstract sense but in terms of the needs of
computability of living beings.
I am not sure I understand what you mean by that.
What is your goal?
The goal by default here is to build, or isolate (by reasoning from
ideas that we can share) a theory of everything (a toe).
And by toe, most of us means a theory unifying the known forces,
without eliminating the person and consciousness.
The list advocates that 'everything' is simpler than 'something'. But
this leads to a measure problem.
It happens that the comp hypothesis gives crucial constraints on that
measure problem.
The set of randomly chosen numbers { 1,4
3,4,.34, 3} is because it can be described with the same
descriptive
language of math. But the first of these structures have properties
and the others do not. The first can be infinite but can be
described
with a single equation while the last must be described
extensively. . At least some random universes (the finite ones)
can be
described extensively, with the tools of mathematics but they don´t
count in the intuitive sense as mathematical.
Why? If they can be finitely described, then I don't see why they
would be non mathematical.
It is not mathematical in the intuitive sense that the list of the
ponits of ramdomly folded paper is not. That intuitive sense , more
restrictive is what I use here.
Ah?
OK.
What is usually considered genuinely mathematical is any
structure,
that can be described briefly.
Not at all. In classical math any particular real number is
mathematically real, even if it cannot be described briefly.
Chaitin's
Omega cannot be described briefly, even if we can defined it briefly.
a real number in the sense I said above is not mathematical. in the
sense I said above. In fact there is no mathematical theory about
paticular real numbers. the set of all the real numbers , in the
contrary, is.
OK. Even for Peano Arithmetic, in fact. Basically, because a
dovetailer on the reals is an arithmetical object.
It looks like you define math by the "separable part of math" on which
everybody agree. Me too, as far as ontology is concerned. But I can't
prevent the finite numbers to see infinities everywhere!
Also it must have good properties ,
operations, symmetries or isomorphisms with other structures so the
structure can be navigated and related with other structures and the
knowledge can be reused. These structures have a low kolmogorov
complexity, so they can be "navigated" with low computing resources.
But they are a tiny part of bigger mathematical structures. That's
why
we use big mathematical universe, like the model of ZF, or Category
theory.
If maths is all that can be described finitelly, then of course you
are right. but I´m intuitively sure that the ones that are interesting
can be defined briefly, using an evolutuionary sense of what is
interesting.
I agree with you. The little numbers are the real stars :)
But the fact is that quickly, *some* rather little numbers have
behaviors which we can't explain without referring to big numbers or
even infinities. A diophantine polynomial of degree 4, with 54
variables, perhaps less, is already Turing universal. There are
programs which does not halt, but you will need quite elaborate
transfinite mathematics to prove it is the case.
So the demand of computation in each living being forces to admit
that universes too random or too simple, wiith no lineal or
discontinuous macroscopic laws have no complex spatio-temporal
volutes (that may be the aspect of life as looked from outside of
our
four-dimensional universe). The macroscopic laws are the
macroscopic
effects of the underlying mathematical structures with which our
universe is isomorphic (or identical).
We need both, if only to make precise that very reasoning. Even in
comp, despite such kind of math is better seen as epistemological
than
ontological.
There is a hole in the transition from certain mathematical
properties in macroscopic laws to simple mathematical theories of
everything .
Sure. especially that if we start from the observations, all theories
are infinite extrapolation from finite sample of data.
The fact that strange, but relatively simple
mathematical structure (M theory)
If you call that simple, even relatively ...
include islands of macroscopic laws
that are warm for life.
With comp, such picture is false. If we take it seriously, it leeds to
a reductionism so strong that it eliminates consciousness and persons.
It is contrary to the fact, if you agree that you are conscious <here-
and-now>.
With the computationalist hypotheses, based on an invariance principle
for consciousness, (yes doctor), we see that we have to justify the M-
theory, or whatever describing correctly the physical reality, from a
theory of consciousness (itself justifiable by the machine, for its
justifiable part).
I do not know the necessity of this greed for
reduction. The macroscopic laws can reigh in a hubble sphere,
sustained by a gigant at the top of a turtle swimming in an ocean.
It is an open, but soluble problem. If this is correct (which I doubt)
then the hubble sphere sphere sustained by a gigant at the top of a
turtle swimming in an ocean (of what?) has to be derived from logic,
numbers, addition and multiplication only.
That's the point.
And our very notion of what is intuitively considered mathematical:
"something general simple and powerful enough" has the
hallmark of
scarcity of computation resources. (And absence of contradictions
fits
in the notion of simplicity, because exception to rules have to be
memorized and dealt with extensively, one by one)
Perhaps not only is that way but even may be that the absence of
contradictions ( the main rule of simplicity) or -in computationa
terms- the rule of low kolmogorov complexity _creates_ itself the
mathematics.
Precisely not. Kolmogorov complexity is to shallow, and lacks the
needed redundancy, depth, etc. to allow reasonable solution to the
comp measure problem.
I can not gasp from your terse definitions what the comp measure
problem is .
Do you understand the notion of first person indeterminacy? Have you
read:
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html
It is a deductive reduction of the mind-body problem into a body
problem in arithmetic. It gives the shape of the conceptual solution,
and toe.
Comp has the advantage of having already its science, computer
science, which makes it possible to translate a problem of philosophy-
theology in precise technical terms.
The shape of the conceptual solution can be shown closer to Plato's
theology, than Aristotle's theology, used by 5/5 atheists, 4/5 of the
Abrahamic religion, 1/5 by the mystics, and large part of some eastern
religion.
What i know is that, kolmogorov complexity is critical
for life. if living beings compute inputs to create appropriate
outputs for survival. And they do it.
Yes, it can have many application, but it is very rough, and computer
science provides many more notion of complexity, and of reducibility.
Brent cited a paper by Calude showing specifically this, notably.
Kolmogorov complexity might be the key of the measure problem, but few
people have succeeded of using it to progress. It might play some role
in the selection of some particular dovetailer, but it can't work, by
being non computable, and depending on constant. I don't know. I'm
afraid that the possible role for Kolmogorov complexity will have to
be derived, not assumed. or you might find an alternative formulation
of comp.
That is, for example, may be that the boolean logic for
example, is what it is not because it is consistent simpleand it´s
beatiful, but because it is the shortest logic in terms of the
lenght of the description of its operations, and this is the reason
because we perceive it as simple and beatiful and consistent.
It is not the shortest logic. It has the simplest semantics, at the
propositional level. Combinators logic is far simpler conceptually,
but have even more complex semantically.
I meant the sortest binary logic.
Classical logic is not the shorter binary logic. In term of the length
of its possible formal descriptions.
I mean that any structure with
contradictions has longer description than the one without them.,
?
None logic get contradictions, with the notable exception of the
paraconsistant logics.
Intuitionist logic is a consistent (free of contradiction) weakening
of classical logic. Quantum logic too.
Note also that the term logic is vague. Strictly speaking I don't need
logic at the ontological level. I put it here for reason of simplicity.
so
the first is harder to discover and harder to deal with.,.
You preach a choir. Classical logic is the one with the simplest
semantics. It is the common jewel of both Plato and Aristotle, and
with comp, it forces to distinguish proof from truth. Which I think is
the essence of science and religion (not of human academies and
churches, alas).
It looks we agree on some things: the importance of classical logic.
The need to restrict ourselves to the separable part of math.
Now, if you are interested in the mind-body problem, you might
understand, with some work, that the comp hypothesis reduces the
problem into a body problem, which takes the form of a relative
measure problem on computations. It shows that the physical laws have
an arithmetical origin, or a theoretical computer science origin.
I don't pretend that comp is true, just that it leads to that kind of
reversal for the global toe. It can be considered as strongly
reductionist ontologically, but it can be shown to prevent any
normative or complete theory for the persons and any other universal
numbers.
There might also be a flaw in the argument. Don't worry at all if you
find one. UDA1-7 is "easy".
The step 8 is too much concise in sane04, and you can find here a more
detailed explanation:
http://old.nabble.com/MGA-1-td20566948.html
The list was already discussing the "measure problem" related
naturally with the everything-type of theories. Comp adds a lot,
notably by making important the distinction between first person and
third person view, which is the key for the mind-body problem.
The second part (of sane04) is technical and thus more difficult
without the study of Mendelson, Boolos, or Davis 65 (the original
paper of Gödel, Church, Kleene, Post, Rosser). But you don't need that
to understand the comp mind-body problem, and the comp "reversal".
Information and complexity are important concepts, but there are many
other one, and in fine, it depends on what we are interested to solve
or clarify.
Bruno
But the main problem of the MHH is that nobody can define what it is,
and it is a priori too big to have a notion of first person
indeterminacy. Comp put *much* order into this, and needs no more
math
than arithmetic, or elementary mathematical computer science at the
ontological level. tegmark seems unaware of the whole foundation-of-
math progress made by the logicians.
Bruno
.
Dear Albert,
One brief comment. In your Google paper you wrote, among other
interesting things, "But life and natural selection demands a
mathematical universe
<https://docs.google.com/Doc?docid=0AW-x2MmiuA32ZGQ1cm03cXFfMTk4YzR4cn
...
somehow".
Could it be that this is just another implication of the MMH
idea? If
the physical implementation of computation acts as a selective
pressure
on the multiverse, then it makes sense that we would find ourselves
in a
universe that is representable in terms of Boolean algebras with
their
nice and well behaved laws of bivalence (a or not-A), etc.
Very interesting ideas.
Onward!
Stephen
http://iridia.ulb.ac.be/~marchal/
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