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From: Hal Finney <hal.domain.name.hidden>

Date: Sat, 21 Sep 2002 19:18:45 -0700

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 - 19:19:57 PDT

Date: Sat, 21 Sep 2002 19:18:45 -0700

Dave Raub asks:

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 - 19:19:57 PDT

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