Hi Patrick,

Sorry for having been short, especially on those notions for which some background of logic is needed. Unfortunately I have not really the time to explain with all the nuances needed.


Nevertheless the fact that reals are simpler to axiomatize than natural numbers should be a natural idea in the everything list, given that the "everything" basic idea is that "taking all objects" is more simple than taking some subpart of it. Now, concerning the [natural numbers versus real numbers] this has been somehow formally capture by a beautiful theorem by Tarski, which I guess should be on the net, let me see, "googlle: tarski reals", ok it gives Wolfram dictionnary: so look here for the theorem I was alluding to:

http://mathworld.wolfram.com/TarskisTheorem.html

It is not so astonishing. reals have been invented for making math more easier.

Concerning the white rabbits, I don't see how Tegmark could even address the problem given that it is a measure problem with respect to the many computational histories. I don't even remember if Tegmark is aware of any measure relating the 1-person and 3-person points of view.

Of course I like very much Tegmark's idea that physicalness is a special case of mathematicalness, but on the later he is a little naive, like physicist often are when they talk about math. Even Einstein, and that's normal. More normal and frequent, but more annoying also, is that he seems unaware of the mind-body problem. John Archibald Wheeler "law without law" is quite good too. My favorite paper by Tegmark is the one he wrote with Wheeler on Everett more or less recently in the scientific american. I'm sure Tegmark's approach, which a priori does not presuppose the comp hyp, would benefit from category theory: this one put structure on the possible sets of mathematical structures. Lawvere rediscovered the Grothendieck toposes by trying (without success) to get the category of all categories. Toposes (or Topoi) are categories formalizing first person universes of mathematical structures. There is a North-holland book on "Topoi" by Goldblatt which is an excellent introduction to toposes for ... logicians (mhhh ...).

Hope that helps,

Bruno




Le 23-mai-05, à 12:51, Patrick Leahy a écrit :


Now I'm really confused!

I took Russell to mean that real numbers are excluded from his system because they require an infinite number of axioms. In which case his system is really quite different from Tegmark's.

But if Bruno is correct and reals only need a finite number of axioms,
then surely Russell is wrong to imply that real-number universes are covered by his system.

Sure, they can be modelled to any finite degree of precision, but that is not the same thing as actually being included (which requires infinite precision). For instance, Duhem pointed out that you can devise a Newtonian dynamical system where a particle will go to infinity if its starting point is an irrational number, but execute closed orbits if its starting point is rational.

On Mon, 23 May 2005, Bruno Marchal wrote (among other things):


Le 23-mai-05, à 06:09, Russell Standish a écrit :

 Hence my interpretation of Tegmark's assertion is of finite
axiomatic systems, not all mathematic things.


I don't think Tegmark would agree. I agree with you that "the whole math" is much too big (inconsistent).

Since Tegmark defines "mathematical structures" as existing if self-consistent (following Hilbert), how can his concept be inconsistent? But there may be an inconsistency in (i) asserting the identity of isomorphic systems and (ii) claiming that a measure exists, especially if you try both at once.


It is mainly from a logician point of view that Tegmark can hardly be convincing. As I said often, physical reality cannot be a mathematical reality *among other*. The relation is more subtle both with or without the comp hyp. I have discussed it at length a long time ago in this list. Category theory and logic provides tools for defining big structure, but not the whole.

As I understand it, this is because "the whole" is unquantifiably big, i.e. outside even the heirarchy of cardinals. Correct?

The David Lewis problem mentionned recently is not even expressible in Tegmark framework.

It might be illuminating if you could explain why not. On the face of it, it fits in perfectly well, viz: for any given lawful universe, there are infinitely many others in which as well as the observable phenomena there exist non-observable "epiphenomenal rubbish". The only difference from the White Rabbit problem is the specification that the rubbish be strictly non-observable. As a physicist, my reaction is that it is then irrelevant so who cares? But this can be fixed by making the rubbish perceptible but mostly harmless, i.e. "White Rabbits".

Paddy Leahy

http://iridia.ulb.ac.be/~marchal/


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