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