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.
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.
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.
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.
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.
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.
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.
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? Ifthe 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 theirnice and well behaved laws of bivalence (a or not-A), etc. Very interesting ideas. Onward! Stephen--You received this message because you are subscribed to the Google Groups "Everything List" group.To post to this group, send email to [email protected].To unsubscribe from this group, send email to [email protected] . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en .
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