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From: Russell Standish <R.Standish.domain.name.hidden>

Date: Fri, 17 Mar 2000 11:07:11 +1100 (EST)

Actually most people here would think that these "alternative"

mathematical systems are just as real as the ones we use. More

important is which systems contain self-aware substructures, and of

those, which have greatest measure (i.e. are the most likely to be

sampled from the set of all such systems). It turns out that these are

systems with fewest axioms satisfying the Anthropic Principle. Any

such system is necessarily incomplete, and has Goedel sentences.

In my Occam paper (check my website below), I advance an argument that

this system is a Hilbert space over the complex numbers (ie our

current quantum mechanics theory has it more or less right). At least

to date, noone has shown this argument to be complete crap.

Cheers

*>
*

*> 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
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*> axiom of choice (right?)) which allows *us* to build diffent mathematical
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*> 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
*

*> follows.
*

*>
*

*> 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 ;)
*

*> http://www.york.ac.uk/~cps102
*

*>
*

*>
*

----------------------------------------------------------------------------

Dr. Russell Standish Director

High Performance Computing Support Unit,

University of NSW Phone 9385 6967

Sydney 2052 Fax 9385 6965

Australia R.Standish.domain.name.hidden

Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks

----------------------------------------------------------------------------

Received on Thu Mar 16 2000 - 16:04:59 PST

Date: Fri, 17 Mar 2000 11:07:11 +1100 (EST)

Actually most people here would think that these "alternative"

mathematical systems are just as real as the ones we use. More

important is which systems contain self-aware substructures, and of

those, which have greatest measure (i.e. are the most likely to be

sampled from the set of all such systems). It turns out that these are

systems with fewest axioms satisfying the Anthropic Principle. Any

such system is necessarily incomplete, and has Goedel sentences.

In my Occam paper (check my website below), I advance an argument that

this system is a Hilbert space over the complex numbers (ie our

current quantum mechanics theory has it more or less right). At least

to date, noone has shown this argument to be complete crap.

Cheers

----------------------------------------------------------------------------

Dr. Russell Standish Director

High Performance Computing Support Unit,

University of NSW Phone 9385 6967

Sydney 2052 Fax 9385 6965

Australia R.Standish.domain.name.hidden

Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks

----------------------------------------------------------------------------

Received on Thu Mar 16 2000 - 16:04:59 PST

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