- Contemporary messages sorted: [ by date ] [ by thread ] [ by subject ] [ by author ] [ by messages with attachments ]

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

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

Received on Thu Mar 16 2000 - 15:53:57 PST

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

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

Received on Thu Mar 16 2000 - 15:53:57 PST

*
This archive was generated by hypermail 2.3.0
: Fri Feb 16 2018 - 13:20:06 PST
*