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From: <hal.domain.name.hidden>

Date: Fri, 9 Jul 1999 22:22:13 -0700

Bruno Marchal wrote:

*> My answer was that I don't see how Tegmark can make this challenge
*

*> effective because the collection of mathematical structures is
*

*> not definable in mathematical terms.
*

[I was on vacation this past week so I am catching up rather slowly

on ~600 emails...]

Does Tegmark attempt to define rigorously what is a "mathematical

structure"? I recall his chart showing different mathematical objects

and their relationship, groups, rings, fields, integers, etc.

Perhaps we candefine a mathematical structure as a formal system?

A formal system consists of axioms and rules of inference. Its

theorems are those things which can be proven by starting with the

axioms and applying the rules of inference.

At a somewhat lower level, it consists of a language, which is the set

of letters or characters in which theorems are written, and possibly

some rules about which strings of characters are well formed. The

axioms are then strings, and the rules of inference tell how to turn

one string into another.

We can enumerate all possible theorems by starting with all axioms and

applying all possible rules of inference in turn. This procedure is

somewhat like the dovetailing universal computation we have discussed

which runs all programs at once.

There are obvious similarities between formal systems and Turing Machine

computations. In each case we start with an initial state and have rules

for evolving the state forward. The set of possible successor states

is implicitly determined by the initial state and rules, but usually

the only way to actually learn what they are is to work the procedure

forward and see what states (theorems) develop.

Can this analogy be made tighter? Can we say that for each TM there

is a formal system whose theorems are the successor states of the TM

(tape + head)? Contrariwise, can we say that for each formal system

there is a TM which implements it? I'm sure these questions have been

studied but I am not familiar with the literature.

If the answers turned out to be affirmative, it would unify the

Schmidhuber and Tegmark approaches and move us closer to the goal of a

model which is so simple that it must be true.

Hal

Received on Fri Jul 09 1999 - 22:26:20 PDT

Date: Fri, 9 Jul 1999 22:22:13 -0700

Bruno Marchal wrote:

[I was on vacation this past week so I am catching up rather slowly

on ~600 emails...]

Does Tegmark attempt to define rigorously what is a "mathematical

structure"? I recall his chart showing different mathematical objects

and their relationship, groups, rings, fields, integers, etc.

Perhaps we candefine a mathematical structure as a formal system?

A formal system consists of axioms and rules of inference. Its

theorems are those things which can be proven by starting with the

axioms and applying the rules of inference.

At a somewhat lower level, it consists of a language, which is the set

of letters or characters in which theorems are written, and possibly

some rules about which strings of characters are well formed. The

axioms are then strings, and the rules of inference tell how to turn

one string into another.

We can enumerate all possible theorems by starting with all axioms and

applying all possible rules of inference in turn. This procedure is

somewhat like the dovetailing universal computation we have discussed

which runs all programs at once.

There are obvious similarities between formal systems and Turing Machine

computations. In each case we start with an initial state and have rules

for evolving the state forward. The set of possible successor states

is implicitly determined by the initial state and rules, but usually

the only way to actually learn what they are is to work the procedure

forward and see what states (theorems) develop.

Can this analogy be made tighter? Can we say that for each TM there

is a formal system whose theorems are the successor states of the TM

(tape + head)? Contrariwise, can we say that for each formal system

there is a TM which implements it? I'm sure these questions have been

studied but I am not familiar with the literature.

If the answers turned out to be affirmative, it would unify the

Schmidhuber and Tegmark approaches and move us closer to the goal of a

model which is so simple that it must be true.

Hal

Received on Fri Jul 09 1999 - 22:26:20 PDT

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