From: Chris Simmons <cps102.domain.name.hidden>
Date: Thu, 16 Mar 2000 23:44:52 +0000 (GMT)

Hello indeed, I'm new here, having just recently found Max Tegmark's paper
on Theories Of Everything. I was kind of enthralled, because that's
roughly speaking, my 'religion'. I wasn't previously entirely sure how to
phrase it, but having read the aforementioned paper clarified things.

Now Max supposes that 'mathematical existence' is equivalent to 'physical
existence', or at least 'mathematical existence' implies 'physical
existence'. I basically agree, but have a few small(?) problems with it,
but also solutions. My main objection is this - how do we know that *our*
mathematical system is the only such system? What would it mean if there
were other systems? To be a little more concrete, we know that there are
axioms which are independant of the basic set theory axioms (such as the
axiom of choice (right?)) which allows *us* to build diffent mathematical
systems. Isn't this rather strange and worrying for Max's supposition?

As I said, I propose a possible way out of this pickle. I realised that
the important thing about all our mathematics is that their underlying
structure is *invariant* of the precise description chosen. Max gives the
example of boolean algebra with differing number of symbols. So suppose
we presume:

(S) Structure which exist are precisely those structures which are
invariant of their description.

Okay, this is a little vague, but we have to be vague somewhere to bridge
the gap between 'I' and 'everything else'. At least this puts the vague
bits in the same place. We also presume that our universe exists. An
immediate question which would need addressing is:

If the structure X exists, is X a description of X?

I'm not sure this is a good idea, it would need some serious thinking
about... but anyway. Now things get interesting, since we can pretty much
deduce Max's supposition from the above (the weaker version at least) as

Say we have some mathematical structure X which we have *actually*
described (the structure is described in papers/books/electronically or
whatever). The point about such structures is that we can demonstrate
that, as far as it is possible for us to tell, such structures are
invariant of description (if we are talking about isomorphism classes,
perhaps). Also, and importantly, because

(i) Our universe exists
(ii) Our universe contains a substructure which describes X
(iii) X is (as far as we can tell???) invariant of its description

we may deduce, assuming (S) of course that X exists (in it's 'own right').

The good thing about this approach is that it *doesn't* assume there is
only one system of mathematics (if mathematics is the art of describing
structures which are invariant of their description??). Nor does it
presume that because *we* can demonstrate something, it is impossible for
something to be demonstrated (ie proven).

I've got lost more thoughts on this, but lets leave it there for now. I
ought to be tidying my room for my house inspection tomorrow. *doh*.

Comments welcome.

PS okay I'm just going to have to say a little more: suppose there exists
some intelligent entity whose brain is non-finite, in the sense that it
has no trouble carring out, say, infinitely (countably) many arithmetic
operations in parallel in a fixed unit of time, regardless of the
complexity thereof. Then Qu1: Can this being use proof systems which we
could not? (I suspect yes) Qu2: Would Godel's theorem apply to this
system? (I suspect no; since the being could, given any statement about
the natural numbers, compute every instance of the statement to determine
its truth of falsity).

PPS I'm off for real now.

Chris Simmons.
Don't bother, the web page is crap ;)
Received on Thu Mar 16 2000 - 15:53:57 PST

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