Limits of mathematics in cosmology

Happy or not with semi-Riemannian geometry, this is not the point. Not that I don't see any problem with general relativity, but I thought that knowing precisely what particular problem we are debating would be useful. At least for me. I confess that for me is not that obvious as is expected.

Regards,

Cristi

Cristi,

Again, my reason for opening the discussion is that if we wish to accept the reality of the Big Bang, we should realize that the conventional mathematical and physical concepts of space are due for a radical rethinking and that--in contrast to the previous scientific developments--there is absolutely nothing on the mathematical shelf that can be remolded for this purpose. We need to start practically from the beginning (and that is VERY VERY hard).

However, before proceeding with the more interesting and difficult part related to which way to go, we need to see to which extent there is an understanding of the unprecedented situation we find ourselves in.

I just discovered this site; please excuse my naive question. (You can find me using MathSciNet (or Web of Science) or look at the Sept., 2010 issue of the Pacific Journal of Math for a new article.) :-)

It seems to me that Ricci flows, mean curvature flows, etc. include the ideas of "evolving topology" (after pinching off, the topology could change), "expanding metrics" and expanding Riemannian manifolds. Ricci flow is about changing the metric with "time." Geometric measure theory allows stranger objects (e.g. varifolds, currents.)

What am I missing here? (Sorry if this is a dumb question.)

Hi Kirk,

No the question is not really dumb, but I take the word "evolution" more seriously.

First, the short answer, again, is: I don't think there is any basic math. text in which such concepts as, for example, EVOLVING topology is introduced. Probably because we take the term "evolution" seriously.

The longer answer is this. Yes, indeed, in general, one can introduce, a parametric family of metrics (dependent even on several continuous parameters). But--in our context, i.e. evolution of the universe--can one safely assume that such parametric family capture such evolution?

My answer is NO: it appears that the underlying processes driving the evolution are not continuous (QM), and hence, I suggest, the resulting space cannot evolve in a ANY continuous manner. And so the big question I would actually like to discuss is: How does the space evolves?

Hi Lev. I don't claim to be an expert in this area but the criticisms you mention don't seem convincing.

1. If you use GMT flows (e.g. level set method), you don't get "continuity" in some sense. I would have to dig around for references or examples if I wanted to be more specific but "Brakke flows" can exhibit this type of behavior. If you work with BV (or SBV) functions, varifolds or various types of (rectifiable) currents, "discontinuous" behavior can occur.

2. I'm not sure that one can model quantum mechanics very well. If your goal is a "theory of everything (in physics)," then I would start with an easy project like proving the Riemann Hypothesis. :D (What are the existing "good models" of QM? What besides probability measures, functional analysis, PDEs, etc. are involved with these models of QM? This is a vague question. One of my colleagues does QM & QC and has been a "fellow" (?) at KITP; I'll eventually try to formulate my question in a better manner.)

3. Varifolds are Radon measures on the manifold crossed (Cartesian product) with the appropriate Grassmannian. Don't people like Robert Bartnik, Tom Ilmanen, etc. (maybe Gerhard Huisken, Klaus Ecker) use GMT to study general relativity? (Maybe I should throw in Fred Almgren's "multi-valued functions" for fun.)

I don't think "basic math texts" would ever cover this stuff at a serious level. I'm not sure what you mean by "EVOLVING topology" as a concept; it sounds completely trivial. Now "evolution" is not trivial but the idea that topology changes as a parameter (e.g. time) changes is trivial.

What is evolution? Imagine the praise if the evolution of the universe turned out to be the solution of some flow problem and you identified this flow problem. Could QM be included in a "measure-theoretic" flow problem? I have no idea if this even makes sense but it reminds me of "string theory - banes - M theory" and multiverse (meta-universe, metaverse) theory.

I understand that physicists want to understand "reality" and not just some mathematical theory. If you would be so kind as to define "reality" this would help. ;D

Kirk,

First, I'm not trying to "convince" anyone, simply because at this stage in our understanding of the evolution of the universe this would be a foolish undertaking.

Second, as I mentioned above, my reason for opening the discussion is this: it stand to reason that the reality of the Big Bang, in addition to many other reasons I'm planning to give, including those outlined centuries ago by Leibniz, strongly suggest that "the conventional mathematical and physical concepts of space are due for a radical rethinking and that--in contrast to the previous scientific developments--there is absolutely nothing on the mathematical shelf that can be remolded for this purpose"(with my apologies for the repetition). We need to start practically from the beginning (and hence, quite understandably, the tremendous reluctance).

Your line of thinking appears to me as follows: whatever the nature throws at us we can handle it within the conventional formal (continuous) paradigm.

Now back to the "reality" of the Big Bang:

How, within the conventional/continuous paradigm, do you 'grow' the space when all kinds of new "particles" and other entities are emerging and interacting? I don't believe, continuous models, can offer compelling models for the ongoing emergence and interaction of qualitatively different kinds of entities. (I intend to elaborate on this later.)

By the way, even well known physicists, e.g. Frank Wilczek (slide 5), wish to see "more meat to be put on inflation" (i.e. "structure" and "mechanism").

Kirk, I forgot to answer your question about what evolution is. To simplify, I believe it is about the ongoing generation of structurally novel entities.

Lev: It is difficult to argue for or against your statement "the conventional mathematical and physical concepts of space are due for a radical rethinking." Look at the richness Dirac added to mathematics (with the help of L. Schwartz); distribution theory might not exist otherwise. On the other hand, I can't tell if you understand the mathematics that exists now; your claim might be wrong.

Perhaps you could help by explaining your knowledge of mathematics, especially basic (e.g. real analysis) and more advanced (e.g. geometric measure theory, differential geometry) mathematics. For myself, my knowledge of general relativity and QM is limited; I talk to colleagues in physics who do astronomy or QM, study some differential geometry, etc.

To veer wildly off topic, I have become interested in harmonic coordinates; my search for information led me to this site. I believe they were introduced by Einstein and some really nice math papers have appeared starting in 1981 (DeTurck & Kazdan); now Michael Taylor and others use them. Are people here familiar with harmonic coordinates?

Dear Kirk,

A lot of work in harmonic coordinates was done by Vladimir Fock, presented in his book The Theory of Space, Time and Gravitation.

Best regards,

Cristi

Lev,

I'm glad that you reopened this disccusion, because as you might remember from our Email exchanges, I think a lot was left unsaid in the essay contest forum.

For one thing, although it is clear in your essay and other works, that the role you assign to time is identical to that of physical information, you often seem to downplay that aspect in favor of talking about the limitations of mathematics.

Physical information has a mathematical model, however. Could you explain why you don't think that model can be extended enough to overcome your objection to using mathematics in the ontology of physics? You know I am a fan of your "structs" concept -- and that construction is time dependent, is it not? It seems to me that it already has a constructive property suitable to mathematical modeling and computation.

I want to go further, but I'll leave it at that pending your response.

Thanks.

Tom

Kirk,

I also forgot to answer your question about what I might call "evolving topology". (By the way, I'm not suggesting that simply evolving topology is enough, but rather that when we get the right formalism it will NATURALLY exhibit something of this nature, and I intend to discuss later such a candidate formalism.)

Let me take one of (ten or so) equivalent ways of generating topology: via the a base (a subfamily of open sets whose unions can generate all open sets). We probably need both: the underlying set of elements should be growing and the base itself should be changing in non-trivial ways.

The reason I'm having a hard time expressing this in the conventional math. language has to do with fact that in the formalism I have in mind (ETS)we are not dealing with 'point' sets but rather each object has a non-trivial 'discrete' structure that also evolves.

Tom,

I'm also glad you are joining the forum!

1. "the role you assign to time is identical to that of physical information, you often seem to downplay that aspect in favor of talking about the limitations of mathematics."

I didn't quite get what you mean, but I will give it a try. According to ETS formalism, there is no conventional concept of time: the struct embodies the new, richer flow of time.

Although I don't quite get what you mean by "physical information", what I have been stressing is that the fundamentally new in mathematics (and science) concept of structural representation--which as I just mentioned contains irreversible temporal information--comes to the fore.

2. "Physical information has a mathematical model, however. Could you explain why you don't think that model can be extended enough to overcome your objection to using mathematics in the ontology of physics?"

Again, Tom, the main break with the conventional math. is the concept of structural representation. As I mentioned in my essay, mathematics has evolved from the very beginning based on the 'point'/spatial representation. So I am convinced that when you switch to structural representation everything, including something equivalent to 'topology' and 'algebra' also change unrecognizably.

Hi Lev,

I am not fully conversant with ETS formalism, though I am willing to learn. Nevertheless, as I read your essay, I can discern no difference between a "temporal stream of interconnected primitives" which form structs, and a bit stream of primitive states.

Relevant to (1). What I mean is, suppose that time gives structure to space:

Then the orientability which defines the stream (indeed, even allows us to give meaning to the word "stream") is a space-time structure _because_ it is oriented. If we find that all of these primitive spacetime structures -- assuming scale invariance and infinite self similarity -- are oriented the same, we have to allow a 2-dimensional object just on the principle of orientability alone.

So primitives that are featureless unless oriented (streaming) implies both time dependency and spatial domain. Because this domain is 2 dimensional and therefore bounds an infinity of points, and because we can further assume the nonlinear evolution that time dependency implies, whatever structures that emerge in d > 2 are already oriented in the same direction. That is, we see that the 2-dimension topology (S^1) on the manifold of a two ball (S^2) is a three dimension object from any point of S^3, the four dimension sphere on which we live. This leaves the 2-ball itself undefined! We know there's correspondence between a point on the 3-sphere manifold and an interior point of the 2-ball, though, because we map an internal plane self similar to the S^2 manifold, of zero curvature. That's not the "real" structureless interior of the 2-ball though, is it? -- we only know that whatever forms may result on the plane are structured by our singular assumption of orientability.

I won't go into Riemannian geometry (every Riemann surface is orientable), but I think the implications are more or less obvious, and interesting.

Anyway, physical information theory follows the same rules as thermodynamics (Shannon). So I think you can make the connection between the Jacobson-Verlinde treatment of gravity as identical to information, and the conclusion that you and I have both reached by different routes: time is identical to information. If gravity is time dependent, then, there exists at least one quantum gravity model in which time is identical to information, because the classical gravity spacetime trajectory is reversible and the quantum information time trajectory is not. That the time metric can orient to all points of the space gives structure to an otherwise formless object. The field of information in the n-dimension stream, n > 1, is ordered by the 1-dimension time metric.

Therefore, time is more primitive than space -- I would have to be convinced that time is more primitive than the integers, before I would give up on a mathematical ontological model.

Regarding (2): I'm not convinced that we can't treat structs in a topological model using scale invariance. More later?

Tom

Lev: I'd love to be "that guy" (who invents a "new formal language that we have NEVER had before" which proves highly useful for physics.) Wish me luck. :D

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