Re: On Tegmark: Different most-suitable forms of mathematical logic, information, probability, uncertainty across Level III?

From: Hal Finney <>
Date: Wed, 23 Apr 2003 16:11:32 -0700

Benjamin Udell writes:
> However, I haven't received a response (on-list or off-list) regarding
> the main questions which I posed (corrected version of original post
> reproduced below).

I will respond in part; however, your message is too long to be the
basis for a practical discussion. There are too many different
ideas where we would not agree on the assumptions. I will just focus
on the first part.

> I notice that he does not mention universes wherein unconventional
> (for our everyday purposes) forms of logic, information, or probability
> would hold as most suitable. He does put predicate logic at the base of
> his mathematical structure. But would variations in suitable logical
> structure - different numbers of truth values, for instance - be most
> likely to be found only across Level IV?

Yes, since level 4 includes all possible mathematical structures, that
would include ones with non-standard logic.

> More generally, _wouldn't variations in the suitable form of mathematical
> logic, information, & probability & uncertainty, more plausibly reflect
> something pertaining to quantum coherence, decoherence, etc., which are
> at the basis for the idea of Level III, than anything pertaining to the
> ideas for the other Levels?_ (Not that I know a great deal about quantum
> coherence!) But Level III seems like the place for such variations,
> if anywhere. No?

Level 3 refers to our own universe's quantum physics as expressed in the
many-worlds interpretation. All of the level 3 worlds should share the
same concepts of logic, information and probability. Now, it's possible
that other level 4 worlds, which might have a different form of logic
or probability, might themselves have level 3 parallelism within them.

> Tegmark does discuss how different lightcone structures (reflecting
> variation across Level II) would affect one's capacity for inference, but
> inferential processes in an intelligent system are a different thing than
> the subjects of mathematical theories of logic, information, probability,
> & uncertainty. I mean that differences of lightcone structure would affect
> intelligent (including inferential & other philosophically interesting)
> processes as they would affect cybernetic, stochastic, & sensitive
> dynamic processes. But these differences would not necessarily reflect
> different forms of *mathematical* logic, information, or probability or
> (more generally) uncertainty.

I'm not sure what you're getting at here, but let me make one point.
It's not clear that you can say that our universe is based on a particular
form of mathematical logic. Does our universe in any sense rely on or
embody the principle that we can't have P and not-P? I don't see it.
Our universe is based on atoms and quarks and fields. When we say
that P and not-P can't both be true, the logic is in our heads, not in
the universe.

The sense in which our universe is a mathematical structure is something
like the following. Imagine that we develop a complete theory of
physics; we know all of the particles, all of the forces, all of the
laws. Everything fits together, there are no exceptions or unknowns.
And further let us suppose that we discover the exact initial conditions
of the universe (Tegmark hypothesizes that the universe is initially
in a state of perfect simplicity). Then these laws of physics define
a mathematical structure which is essentially identical (isomorphic)
to our physical universe.

I don't see that these laws would be particularly likely to embody a rule
that says p and ~p is impossible. Newton's laws don't have any such rule.
They just say Force = Mass times Acceleration. Einstein's equations
don't have such a rule; they say that particles follow geodesic paths
and that the shape of space is determined by the stress energy tensor.
Quantum mechanics doesn't have such a rule; it says that the time
evolution of the system is based on the exponential function applied to i
times the Hamiltonian. The point is, these are the kinds of mathematical
statements which have been the basis for physical laws in the past, and it
may be that in its final form, physics will be expressed in similar ways.
The equations won't necessarily build on mathematical logic at all.

For another view of the Level 4 multiverse, I recommend Juergen
Schmidhuber's description of it as the output of a super-computer that
runs all possible computer programs. He has a new page up at, with the introductory
paper at This is
compatible with Tegmark's view if we assume that to every mathematical
structure there corresponds a computer program, and vice versa.

Sorry I wasn't able to respond more substantively to the later portions
of your message. You might want to re-write some parts and describe
your ideas in new messages.

Hal Finney
Received on Wed Apr 23 2003 - 19:13:12 PDT

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