Re: Devil's advocate against Max Tegmark's hypothesis

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