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.
> > 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.
>
>
> > 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.
> > 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.
>
>
>
> > 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 . The fact that strange, but relatively simple
mathematical structure (M theory) include islands of macroscopic laws
that are warm for life. 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.
>
>
> > 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 . 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.
> > 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. I mean that any structure with
contradictions has longer description than the one without them.,so
the first is harder to discover and harder to deal with.,.
> 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
>
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>
> http://iridia.ulb.ac.be/~marchal/
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