Correction:
"there are n-dimensional manifolds, for any POSITIVE integer n".
(For instance, in classical analytical mechanics one models the phase space of a system of n particles by a 6n-dimensional manifold.)

Dear George,

Thanks. I will come back to you on the main points in your excellent post. Busy.

In the meantime:

"Also, sorry, but there are n-dimensional manifolds, for any integer n (even infinite-dimensional) in mathematics"

I am happy to be corrected on this matter. I just had a feeling I had read (misread) something in Penrose (Road to Reality), wherein only certain integers for n were allowed. However, it wasn't that there was a magnitude cap, rather only certain values of n are allowed. I can't find the cite (if it exist!) from a quick skim through his book.

Dear George,

Being omnipotent suggest boundlessness.

Penrose twice, as you did, mentions any positive integer for n is allowed... and zero as a single point of 0-manifold),yet his examples seem more limited. However, Penrose appears to relate 3D Euclidean topology to a 6-manifold based on three dimensions of degrees freedom and three dimensions of the rotational orientation of the body. Also, noting the 6-manifold mensioned necessarily manifests more than 4 dimensions (space-time).

Please excuse brevity.

"...it is thinkable to believe in both a self-sustaining Universe and God, who “created it”."

It is in that very direction I suspect theistic scientists will find themselves latter this century. Perhaps,now, if they are can tolerate the speculative nature of current theories.

A self-sustaining universe can only allude to self-suffiency, it cannot absolutely disprove the existence of God. Such a finding can only suggest that God may not be necessary for the universe to exist. Likewise, if the universe had a Creation Agent (what we call God), we cannot know if the Entity is historical (known to religion(s)) or a-historical God(something else), except with former significant trust is placed in the scriptures: Only there are various conflicting scriptures.

Alternatively, a-historical God might countenance faith/religion, without direct involvement. Of course, the last proposition runs against the First Commmandment in the OT.

Dear Oliver,

>>I had read (misread) something in Penrose (Road to Reality)<<
I own the book (over 1000 pages!) though I cannot claim to have read it cover to cover. It is true that there are certain - usually topological - restriction on a manifold to admit a particular (geometric) structure. For instance, a four dimensional non-compact manifold admits a Lorentzian metric (entering Einstein’s GR) iff its Euler characteristic is 0. Sometimes there are also restrictions on its dimensionality (e.g. only even-dimensional manifolds can admit a complex structure).

These are PURE MATHEMATICAL restrictions irrelevant to what you want to use the manifold for to model - space-time in GR (where the physics requires 4 dimensions), or the phase space in classical mechanics (where the dimensions are 6n) or in the case you refer to (where n= 6), etc. Penrose indeed is not very clear in where he is talking about pure mathematics and where he uses it to model physical reality.

>>It is in that very direction I suspect theistic scientists will find themselves latter this century. Perhaps,now, if they are can tolerate the speculative nature of current theories.<<
I am not sure what you mean by theistic scientists, but I agree that a contemporary Christian scientist can accept only such interpretations of his/her faith that do not contradict established scientific theories, and should potentially accept - you call it “tolerate” - speculative theories PROVIDED THESE SPECULATIONS REMAIN WITHIN THE CONFINES OF SCIENCE. Examples: Davies’ self-sustaining Universe and the related concept of biocosm, emergence, self-organisation, Everetts’s many worlds, multiverse, branes and many big bangs, etc.

>>there are various conflicting scriptures<<
I agree. This is where my analogy stops, since as far as physical reality is concerned there is essentially only one mathematics used to model it. The problem of many religions is a different problem, not easy for a believer, see my reference above to “consciousness being more explicitly involved“.

God “sustaining the world without direct involvement” could refer to deism (that inded contradicts Christian theologies), depending on what you mean by “direct”.

Dear George,

Thank you for your comprehensive relpy. I enjoy our exchanges.

I find it intersting that we often have different ultimate interpretations yet agree on many fundamental frameworks. We think
alike, perhaps more so than others whom play on the same team as yourselves.

More later, when I can find a break.

Cheers.