Re: Tegmark's TOE & Cantor's Absolute Infinity

From: Osher Doctorow <osher.domain.name.hidden>
Date: Sat, 21 Sep 2002 23:35:47 -0700

From: Osher Doctorow osher.domain.name.hidden, Sat. Sept. 21, 2002 11:38PM

Hal,

Well said. I really have to have more patience for questioners, but
mathematics and logic are such wonderful fields in my opinion that we need
to treasure them rather than throw them out like some of the Gung-Ho
computer people do who only recognize the finite and discrete and mechanical
(although they're rather embarrassed by quantum entanglement - but not
enough not to try to deal with it in their old plodding finite-discrete
way).

Mathematics and Physics are Allies, more or less equal. I prefer not to
call the concepts of one inferior directly or to indirectly indicate
something of the sort, unless they really are contradictory or something
very, very, very close to that more or less. As for a computer, maybe
someday it will be *all it can be*, but right now I have to quote a retired
Assistant Professor of Computers Emeritus at UCLA (believe it or not,
bureaucracy can create such a position - probably the same bureaucratic
mentality that created witchhunts and putting accused thieves' heads into
wooden blocks so that they could be flogged by passers-by in olden times),
who said: *Computers are basically stupid machines.* We knew what he
meant. They're very vast stupid machines, and sometimes we need speed,
like me getting away from the internet or I'll never get to sleep.

Osher Le Doctorow (*Old*)


----- Original Message -----
From: "Hal Finney" <hal.domain.name.hidden>
To: <everything-list.domain.name.hidden>; <Vikee1.domain.name.hidden.com>
Sent: Saturday, September 21, 2002 7:18 PM
Subject: Re: Tegmark's TOE & Cantor's Absolute Infinity


> Dave Raub asks:
> > For those of you who are familiar with Max Tegmark's TOE, could someone
tell
> > me whether Georg Cantor's " Absolute Infinity, Absolute Maximum or
Absolute
> > Infinite Collections" represent "mathematical structures" and, therefore
have
> > "physical existence".
>
> I don't know the answer to this, but let me try to answer an easier
> question which might shed some light. That question is, "is a Tegmarkian
> mathematical structure *defined* by an axiomatic formal system?" I got
> the ideas for this explanation from a recent discussion with Wei Dai.
>
> Russell Standish on this list has said that he does interpret Tegmark in
> this way. A mathematical structure has an associated axiomatic system
> which essentially defines it. For example, the Euclidean plane is defined
> by Euclid's axioms. The integers are defined by the Peano axioms, and
> so on. If we use this interpretation, that suggests that the Tegmark
> TOE is about the same as that of Schmidhuber, who uses an ensemble of
> all possible computer programs. For each Tegmark mathematical structure
> there is an axiom system, and for each axiom system there is a computer
> program which finds its theorems. And there is a similar mapping in the
> opposite direction, from Schmidhuber to Tegmark. So this perspective
> gives us a unification of these two models.
>
> However we know that, by Godel's theorem, any axiomatization of a
> mathematical structure of at least moderate complexity is in some sense
> incomplete. There are true theorems of that mathematical structure
> which cannot be proven by those axioms. This is true of the integers,
> although not of plane geometry as that is too simple.
>
> This suggests that the axiom system is not a true definition of the
> mathematical structure. There is more to the mathematical object than
> is captured by the axiom system. So if we stick to an interpretation
> of Tegmark's TOE as being based on mathematical objects, we have to say
> that formal axiom systems are not the same. Mathematical objects are
> more than their axioms.
>
> That doesn't mean that mathematical structures don't exist; axioms
> are just a tool to try to explore (part of) the mathematical object.
> The objects exist in their full complexity even though any given axiom
> system is incomplete.
>
> So I disagree with Russell on this point; I'd say that Tegmark's
> mathematical structures are more than axiom systems and therefore
> Tegmark's TOE is different from Schmidhuber's.
>
> I also think that this discussion suggests that the infinite sets and
> classes you are talking about do deserve to be considered mathematical
> structures in the Tegmark TOE. But I don't know whether he would agree.
>
> Hal Finney
>
Received on Sat Sep 21 2002 - 23:48:35 PDT