> >> I have still a question for Russell, what is the meaning of
> >> giving a "physical existence" to a mathematical structure ? This is
> >> not at all a clear statement for me.
> >
> >I am using it in the sense of Tegmark - essentially saying that some
> >structure really _does_ exist. A bit like saying Platonic forms really
> >_do_ exist.
>
> OK. But I reserve platonic existence to numbers and their (intricate
> enough) relations. The "whole of math" is not a well defined
> being, it seems to me. See also my post to Chris.
> And there is a difference which remains to explain between the
> atemporal universal Turing machine and the apparently
> contingencial concrete universal computer I am using right now.
A bit like whoever it was who said that "whole numbers are the work of
god - all else [of mathematics] is the construct of man". I have some
sympathy for this view, but suspect that that debate would occupy many
centuries of discussion.
>
> A lot of people seems to take mathematical reality for granted.
> They should realise that the very mathematical discovery of the
> computer (by Post, Turing, Church, Markov, etc.) comes from
> works in the foundation of math, when all attempt to define the
> "whole math" appears to be inconsistent ... or incomplete.
>
Incomplete I agree with, however I don't believe incompleteness to be
a problem per se.
> Regards,
>
> Bruno
>
>
I have thought about your last week's post some more, and believe
there is much more to tease out here. It appears that you are
concerned by incompleteness of theories built on finite axiom sets. It
is true that all of the embedded theories (Tegmarkian ensemble members
embedded in Schmidhuber's) I was referring to are RE. A finite axiom
set defines a partition of wffs into three sets (true, false and
undecidable). If an additional independent and consistent axiom is
added, then the true and false sets are expanded, whilst the
undecidable set is reduced. This in effect defines a partial ordering
on theories T1 and T2:
T1>T2 iff true(T2) \subset true(T1) and false(T2) \subset false(T1)
and undec(T1) \subset undec(T2).
However, T1 and T2 are quite distinct theories. In your view, (N,+,*)
is somehow the limit of such a partially ordered chain - this appears
to fly against the prevailing opinion of defining (N,+,*) to be the
incomplete result of some finite set of axioms, with its three-way
partitioning of theorems.
Now for my purposes, all of the incomplete theopries (T1,T2 etc) can
be mapped into the Schmidhuber ensemble by my method (and hence are
given physical existence). One needs to include a program the
recursivley enumerates all theorems, and sorts them into canonical
order to ensure that isomorphic theories are indeed mapped to the same
set of indistinguishable programs. The completed theories can also be
mapped onto bitstrings (however the bitstrings contain infinite
information, so would have measure zero) - however, can all the
theorems be recursively enumerated to ensure that all isomorphic
complete theories are mapped into same set of indistinguishable
programs? Intuitively, it would appear possible, but infinity is a
tricky beast.
In any case, it would appear unimportant, as the complete theories
must have measure zero, so we wouldn't be living inside a universe
described by a complete mathematical theory.
This is from considerations of Goedel's incompleteness
theorem. Another line of thought that emanated from your criticism was
theorems that exist in the "completion" of the RE list of
theorems. For example, lets assume that Goldbach's conjecture (that
any even number can be written as the sum of two primes) was true, but
that no finite manipulation of our currently agreed axioms of the
whole numbers could prove it. Now GC can be proved false by
counterexample, so GC can be proven true in countably infinite steps
by testing each even number. So, in this sense, GC will lie in the
"completion" of the set of all theorems.
Is this a problem? Certainly, if this were the case, GC would not be
listed in the RE of all theorems by a UTM. However, all the
enumeration is important for is to identify uniquely the set of
equivalent mathematical theories with a set of equivalent UTM programs.
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
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Received on Sun Nov 14 1999 - 16:21:39 PST