Anthropic reasoning on requirement of our "mathematical structure"

From: Christopher Maloney <>
Date: Thu, 03 Jun 1999 22:23:27 -0400

As I mentioned in my last email, I'm new to this list. So if
this topic has already been discussed, I apologize.

The argument I want to make is related to the "Doomsday
argument" that I just read on one of your websites (I can't
remember whose). In that argument, one draws conclusions
about one's environment based on the proposition that the
SAS that you are is likely to be an "average" one. That is,
each of us can assume that we are a sort of random sample
among all possible SAS's of similar complexity. That's
the most general form of the statement that I would feel
comfortable making.

It seems plausible that our complexity, or our information-
processing capability, is not an accident. If it were much
less, we would probably not be self-aware. I don't know
why it's not much more than it is, although I (of course)
have a theory about that too.

Anyway, Tegmark, in his paper, made some rather extravagant
claims, IMO, that:
    It appears likely that the most basic mathematical
    structures that we humans have uncovered to date are
    the somae as those that other SASs would find.

He attempted to justify this idea using anthropic reasoning,
which I found awfully dubious. I just don't have the
confidence that he expresses that we humans aren't "missing
something" when it comes to the types of mathematical
structures possible. I would guess that the field of
mathematics just keeps getting more varied, more bizarre,
and more rich, the more it is studied. How could we possibly
know that there aren't more SAS-supporting structures of types
that we cannot yet dream?

On the other hand, I believe that one can reason that the
environment we find ourselves in must be as it is, or at least
must be typical, using the concept that each of us is a
typical SAS.

It's very simple: if the measure of the infinite number of
SAS's allowed by one mathematical structure A is greater than
that in structure B, then we *must* find ourselves A or in
yet another structure with an even greater measure. That is,
we must find ourselves in the structure which admits the
largest number of SAS's of our complexity.

This idea first occured to me when pondering the vast number
of branches allowed by MWI QM. I haven't investigated this
fully, but I think it's of the order of 2^c, where c is the
cardinality of the real numbers. I could be wrong. Anyway,
it's big.

The logical continuation of this idea is that the order of
SAS's allowed in our mathematical structure must be utterly
unbounded. I don't know how to describe this, perhaps
Aleph-infinity. Isn't it true that for any infinite cardinal
number, there is another that is of a higher order? I'm a
little rusty on this - I have to re-read my primer on
infinities. But I just wanted to get this idea out, so my
terminology might be screwed up.

But how could the cardinality of the allowable SAS's in our
mathematical structure be "utterly unbounded". The only way
that I can conceive of this is that the laws of physics
are an infinitely deep well of complexity. That, for example,
when we find a theory of quantum gravity, that it allows for
yet a higher order of SAS's than that allowed by the standard
model. But then we would find that that is still not the
ultimate theory, and when we dig further, we find yet more
possibilities. I can't see any aesthetic reason for assuming
that the "structure" we are in would ever be completely well
demarcated to us, in our frog perspective.


Chris Maloney
"Knowledge is good"
-- Emil Faber
Received on Thu Jun 03 1999 - 19:42:50 PDT

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