# Re: Entropy and information

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On 29 Feb 2012, at 02:20, Alberto G.Corona wrote (to Stephen):```
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```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.
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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.
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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.
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Why? If they can be finitely described, then I don't see why they would be non mathematical.
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What is usually considered  genuinely mathematical is any structure,
that can be described briefly.
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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.
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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.
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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.
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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).
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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.
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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.
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Precisely not. Kolmogorov complexity is to shallow, and lacks the needed redundancy, depth, etc. to allow reasonable solution to the comp measure problem.
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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.
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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.
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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.
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Bruno

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```.
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```Dear Albert,

interesting things, "But life and natural selection demands a
mathematical universe
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```Could it be that this is just another implication of the MMH idea? If
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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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