Limits of mathematics in cosmology

Kirk, somehow I don't believe you wish to be "that guy". ;-)

How big is the Universe?

Size and geometrical shape of the universe depends on mathematical models we use for its description. By using Euclid infinite flat space universe is infinitely big. We could travel with a light speed for ever and would never come to the border of the universe. By using Riemann spherical finite space universe becomes finite. By traveling with the light speed after many of years we would come back into the same place.

In Euclid geometry is known that "infinite distance 100 kilometers" is still "infinite distance". When we use Euclid geometry for describing universal space and we say "Universe is infinite" this does not mean a concrete measurable distance.

Regardless which geometry we use for describing universal space Universe is too big to be fully comprehended by the rational mind. In order to know universe deeply we have to use consciousness as a scientific research tool. Consciousness reveals us infinite nature of the universe that reaches far beyond rational understanding.

So here goes...

If the general topic is whether Mathematics offers sufficient tools to encompass Cosmology, and the specific emphasis is whether the Physics of the expanding universe can be explained thereby, I have some thoughts. I have not read your essay yet Lev, but reading the question (short and long form) and comments I already have quite a lot to say. So I'll offer a few preliminary comments.

While it may be a valid claim that conventional Math can't deal well with some things we observe, perhaps like an expanding Cosmos, there is unconventional Math and unconventional Physics which uses more conventional Math in novel ways. So the real issue seems not to be whether Math is ultimately good enough, but how far into the frontiers do we need to go, to find the right stuff for the job. Since there are plenty in my knowledge who have grappled with these questions in one way or another, I'll cite some examples.

I'll start with Brouwer's intuitionism which became Constructive Math and Geometry. This does incorporate in some ways the idea of procedural evolution whereby geometric features and topology emerge. This idea has also come out in recent work marrying Twistors and Strings, which was inspired by a meeting of Penrose and Witten and has more recently been championed by Arkani-Hamed and others. And of course, there is a notion of evolving spaces in NCG. Alain Connes said in one paper "Noncommutative measure spaces evolve with time" so this gets into some of the same territory you are exploring.

I'll leave off here, and go to some Physics examples next time.

Regards,

Jonathan

Since it seems I have not gotten my point across, I will try to describe the situation again.

If we rely on the conventional mathematical concept of space or ANY of its possible modifications (including those described by Jonathan), any expansion of space cannot be accompanied by the generation of structurally novel entities inside, simply because, by any definition of space, the basic structure of space has to be preserved (otherwise it would not be an expansion of that space).

Can we agree on this point?

Lev,

We agree. However, one is reminded that we only allow "the structure of space" as a time-distance measure among spatial coordinate points, and we only allow knowledge of the expansion of spatial coordinate points by changes in position among _mass_ points.

So it is not the basic structure of space that we are concerned with preserving; it is the basic structure of _spacetime_. You're right -- space has no structure of itself (we construct points and lines to make sense of it.)

The reason that I think your idea of "structs" is brilliant, is that it makes a clear formal distinction between time-dependent events which are irreversible, and spacetime events, which are reversible in the language of classical physics.

A time dependent system implies self organization of random fields in a scale invariant universe. Would not random fields be identical to unstructured space?

Tom

Soooooooo, if we agree on that, I suggest that we should not look right now for a new concept of space (since it will not look like anything we know), AND worry about it later, when we understand better how the space is instantiated based on some new structural/temporal representational formalism, e.g. ETS.

NOW this is a much more interesting/productive topic, as far as I'm concerned.

Let me repeat it again. It seems to me (and to some physicists and philosophers) that the concept of space is derivative, and more importantly, we can now--in contrast to Leibniz and Whitehead--intelligently guess how it could be 'grown' based on the information provided by the structure of corresponding events in the struct (which is being instantiated). What would be also nice to discuss is to which extent ETS offers a better/worse scenario for space instantiation as compared to some current attempts in quantum gravity, e.g. Fotini Markopoulou and Renate Loll.

Yes, Brendan, it's relevant, since it appears that a particular kind of discreteness might be absolutely primary as far as the space generation is concerned.

By the way, here is a simplified visual illustration (Windows Media Player) of the process of spatial instantiation of the struct representation for the Bubble Man example from Part III of the main ETS paper.

Lev,

I think Brendan's point re CDT is that the arrow -- the orientation -- of the programming object, the triangle, is a "by hand" convention ("causal gluing"). That is, we only get 4 simply connected dimensions from 2 in a dynamically evolving algorithm if time is oriented identically on each discrete object. The result is a 4-dimensional continuum, the physical world that we live in.

If you take your (and my) conclusion that time is identical to information, the "by hand" operation is obviated in favor of a fundamental quantum of information that is unavoidably oriented by what "information" _means_. Because quantum information is time dependent, if quantum information is all that the world is made of (Jacobson-Verlinde) then time actually structures space, so that time flow over a manifold constrains the quantum vacuum "inside." I do agree with you -- we don't know anything about this internal structure or "structurelessness" as it were; we do know, however, that the field constrained by the time metric contains information that we want to extract and can only extract from the surface (manifold). (This is also the basis of holography.)

I have been following your, Jacobson's, Verlinde's, Markopoulou's and the Loll group's results with much interest -- I have strong reasons to think that the next breakthrough in theoretical physics (and quantum computing) will come from a merging of these ideas.

Tom

Hello again,

I made a brief addendum to the above comment stream which is now in the overflow zone. But this topic looks very interesting. I enjoyed the comments on CDT and related topics. Would like to mention 't Hooft's CA based QG model, and other stuff. So I'll likely take up later down here at the bottom of the page, when there is time.

Regards,

Jonathan

"I have strong reasons to think that the next breakthrough in theoretical physics (and quantum computing) will come from a merging of these ideas."

OK Tom, but this is exactly what we want to discuss!!

First of all, as far as the starting point is concerned, I have a hunch that we need something like ETS, because it is a MUCH more complete formalism/math. for structural representation. It might be useful for some to see why this is so first, since accepting ETS (or something 'similar') as the starting point should suggest the more concrete formal language and direction in which to proceed with the development of QG. As has always been historically the case, without such new formal language the progress becomes simply impossible: the right language is the only royal road.

"Measurement and generation of spaces are both entirely procedural in constructivist Math."

Jonathan,

I'm somewhat surprised that you disagree with one of the basic silent points in mathematics: historically (and this cannot be changed overnight) the concept of space--and indeed any basic math concept--in order to qualify as such must delineate certain defining (fixed) 'features' that cannot be changed without destroying the original concept. That is why there are no concepts of evolving spaces, groups, etc, but there sequences of spaces, groups, etc.

Lev,

I think even Brendan might agree that structs are a higher level of programming object than identical triangles. For one thing, instead of being simply connected, structs have the capability of forming systems that evolve at different rates (like natural biological evolution) and thus bring complex system, non-hierarchical, nonlinear dynamics to the fore -- just the way that nature appears to actually evolve. CDT gives us classical spacetime (which may itself be a struct).

Nevertheless, CDT is a solidly impressive experimental result, because it unequivocally shows the importance of orientation in a scale invariant spacetime; i.e., if we take seriously the notion that time structures space, the implications that time drives system change and that time is therefore identical to information, follow.

I see the advantages of a class-based event space of discrete objects in which a dynamical model produces quantum coherence and decoherence on the large scale. I also see the advantages of recovering smooth spacetime from self similar discrete objects. At the end of the day, a self organized complex network depends on the duality of these models.

Tom

Of course, it would nice to hear the involved physicists, but probably this is too much to ask ;-))

Gee Whiz Lev,

Perhaps I need to inform you that I have been exploring precisely the alternatives, or roads beyond, the tacet assumptions in Math and Physics regarding fixed nature of which you speak - for quite some time. I regard fixed nature to be in the realm of convenient assumptions. Reality is dynamic, and perhaps dynamical. Perhaps you need to read MY contest essay, or my paper in Quantum Biosystems Vol. 1, no. 1 - to understand where I'm coming from on this. Briefly; I feel that our choice of first principles must support what is real or observed.

Mikhail Kovalyov gave a wonderful talk at FFP10 where he reminded us that in Physics the real story is often told by nonlinear equations, but in general they are insoluble. So we make a few convenient assumptions to make a soluble equation where we can actually plug in numbers and see if we have a good fit. But at the point we forget it is a linearized nonlinear equation, a solution that works only in a limited range of variation, we are in 'la-la' land. But some people never learn that it is a simplified expression with limits.

I was first inspired to begin investigating alternative to conventional Math seriously about 23 years ago, when I had conversations with Benoit Mandelbrot, regarding the idea of a Cosmology inspired by the Mandelbrot Set. Because of what I found, I have been exploring the idea of 'evolving spaces' for years now - and it appears there is a mathematical basis.

More Below,

Jonathan

BTW,

I know of plenty of exceptions or areas in which the boundaries of definition are being stretched. One could create an evolving group fairly easily in Category theory, if I understand the concepts. In that framework, a group is merely a category with a single object, and all morphisms invertible. So what happens if you choose one morphism to be directional or time-like? There you have it. I've simply defined an evolving group using existing Math. Is that what's called a groupoid? I don't know. Category theory is not something I know; but it exists.

All the Best,

Jonathan

Hello again,

I left another 2 comments above, and I'll return later with more thoughts. There are so many interesting threads to this conversation already. So I'll be back, but I need to get some work done still today.

Regards,

Jonathan

Jonathan,

Category theory or not, the basic language of mathematics--since it is based on the POINT/spacial representation of reality (remember the set theory itself)--does not allow for GENUINELY EVOLVING objects. The bottom line is that point representation cannot evolve: in order to evolve you need an evolving object representation.

Thus, it appears that the only way out is to replace the point by a new kind of structural object, and moreover, in order to deal with 'evolution' in a natural manner, that structural representation has to embody temporal information. That has been exactly our motivation for the development of the ETS formalism during the last two decades.

Before that, when I began to study the conventional string representation, i.e. 'accda', and the associated string distances--the shortest way to transform one string into another relying on the chosen substring deletion, insertion, and substitution operations--even at that early stage the difference with the (numeric) vector spaces became clear: the geometry of string spaces is 'richer', simply because the set of distance defining operations is richer. That was the initial hint. ;-)