On 29 feb, 18:35, Bruno Marchal <marc...@ulb.ac.be> wrote:
> On 29 Feb 2012, at 15:47, Alberto G.Corona wrote:
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> > 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.
>
My goal is the same. I start from the same COMP premises, but I do not
not see why the whole model of the universe has to be restricted to
being computable. I start from the idea of whathever model of an
universe that can localy evolve computers. A mathematical continuous
structure with infinite small substitution measure , and thus non
computable can evolve computers. well not just computers, but problem
adaptive systems, clearly separated from the environment, that respond
to external environment situations in order to preserve the internal
structures, to reproduce and so on.
> 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.
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> >>> 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.
>
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> >>> 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!
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> >>> 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.
>
that is not a problem as long as diophatine polynomials don´t usurpate
the role of boolean logic in our universe, and the transfinite
mathematics don´t vindicate a role in the second law of Newton. ;)
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> >>> 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.
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> >>> 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...
>
> 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.
As I said above I do not see why a model of the universe as a whole
has to be restricted to the requirement of simulation. I see (local)
and macroscopic computability as an "antropic" requirement of Life,
but not more.
>
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> >>> 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 ..
I can define a set and arbitrary ioperations with contradictions. I
can say True ´AND True is False half of the time. and True the other
half.depending on a third stocastic boolean variable that flip
according with some criteria. I can define multiplication of numbers
in weird ways so that i break the symetric an distributive
properties in certain cases . and so on. All of them can be defined
algorithmically or mathematically. In the broader sense, these
structures will be mathematical but with larger kolmogorov complexity
than the good ones.(and useless).
.
>
> leer más »
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