Re: Entropy and information

On 29 feb, 18:35, Bruno Marchal <marc...@...> wrote:

On 29 Feb 2012, at 15:47, Alberto G.Corona wrote:

On 29 feb, 11:20, Bruno Marchal <marc...@...> 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.

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. ;)

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.

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).