Limits of mathematics in cosmology

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. ;-)

Lev,

Right on. This very same insight, that points lack the degrees of freedom to participate in an evolving system, led me to conclude that the algebraically closed C* provides the means for non-ordered dimensionless points of definite values to self-organize, allowing time dependence.

So I truly grok what you mean when you say that information string geometry is richer -- the transformation of random fields into ordered structures and novel forms reminds one of Darwinian random mutation and natural selection.

In my NECSI ICCS 2006 paper ("Self organization in real and complex analysis") I put it this way:

"2.0 The Geometry of Counting

2.1 Let us introduce some interrelated tools that differentiate our continuous experience of time, from the discrete metric that imposes a moment of time onto our experience:

2.2 Suppose time is an independent physical quantity, a 0-dimensional point on a random self-avoiding walk in n dimensions. The singularity imposes the limit. (Therefore, just as in discrete counting we say that the set is not empty, we say that the point is not dimensionless. An important distinction, since a series of zero-dimensional points is not self-limiting.) Suppose the point is self-similarly extended on a 1-dimensional metric whose (arbitrarily chosen) endpoints define relative positions of the evolving point and its

complex conjugate on the complex (Riemann) sphere.

2.3 The Euler equation, e^ipi = - 1, therefore describes the least path such a point travels in the complex plane, as the Euler Identity: e^ipi 1 = 0 . (The unit radius is therefore self similar to a point; we will explain, in 5.11, this fact of analysis.)"

The "indpendent physical quantity" that we call time, then, has to be identical to quantum information bits creating structures from unstructured relations.

Tom

I think we should be patient. Brendan has already joined the conversation, with an important piece of clarification, and he is certainly one of the involved physicists.

This is pioneer territory for all of us, Lev, and while I think you're doing a fine job of marking the trail, you might want to pause and let the rest of the party get within shouting distance. :-)

Tom

Tom, that's exactly what I was planning to do.

I thought it might be useful to restate once more the topic of the forum. ;-)

My claim is that we need a fundamentally new formalism for modeling the physical reality of expanding universe.

Why? Because there is nothing in the conventional math. which would allows us to come to grips with such process of expansion that is accompanied by the ceaseless CREATION OF QUALITATIVELY DIVERSE NEW 'STUFF'. No existing concept of space can offer any help in this respect (and, frankly, I am somewhat surprised at how this situation has been overlooked).

I also would like to discuss some proposals for dealing with such process (including my own). Thus, we can address the wish, mentioned in one of my posts, by Frank Wilczek that "more meat to be put on inflation", including "structure" and "mechanism".

Lev,

I am disappointed, as I know you must be, that the topic hasn't generated the kind of interest that (at least I think) it deserves.

Suppose the question were framed:

Does there exist a computable non-numeric representation of evolving novel forms that accurately describes the evolving physical world?

Then suppose that rather than assuming the positive, we try to rule it out.

I'll ask first: what do we mean by "computable?"

Tom

Tom,

Let's be patient and wait (after all it's summer), and let's also be optimistic about it. ;-)

Thanks for your continued interest! I hope you will stay with the topic.

Here is yet another summary of our topic. ;-))

As I stated above, in contrast to the line of generalizations of the concept of 'space'--reals, vector space, metric space, topological space--we have absolutely no concept with which to start the corresponding line of generalizations for the concept of 'evolving' space. And there are perfectly sensible (formal and informal) reasons for this situation: the present mathematics has developed in such a way that it is not 'equipped' to deal with evolving structures, except by means of formulas and sequences. But those cannot deal with the underlying mechanisms responsible for the structural 'evolution'. I submit that it is the ubiquitous spatial/ point representation that, on the one hand lies at the foundation of all math. concepts, and on the other hand is standing in the way of the new formalism which can adequately deal with the evolving structures: since the 'point' cannot evolve, it must be replaced by some dynamic/evolving structure, which requires complete rebuilding of mathematics as we know it. Evolution of the universe is about creation of novel structural entities, and such entities cannot be built out of non-structural entities, i.e. 'points'.

Of course, the ease with which it became possible to add adjective "evolving" has partly to do with the light hand of Darwin and the modern biology. But if Darwin, as a non-technical fellow, can be forgiven for this, in mathematics and physics such attitudes are standing in the way of real progress, which, by the way, would also have a transforming influence on biology itself.

The hypothesis, which would be interesting to discuss, is that 'space' (and its 'content') is being incrementally instantiated on the basis of the more basic informational mechanism, which builds the space based on the relational/structural information captured in the structural "event" (see informal introduction in my essay). In particular, such event has sufficient information as to where to attach the space 'patch'.

Lev,

Ever since being introduced to your work, I have wondered if I could be persuaded that a more fundamental structure than the integers underlies the idea of order. I was taken by your coinage, "struct", and its description, almost immediately. Structs seem to fit naturally with the kind of "black box" relation that characterizes a system of components evolving at different rates (like the Ashby/Bar-Yam multiscale variety)*; the internal evolution of the black box is unavailable, but the time dependency of the network forces a visible relation among hubs of coordinated activity**, such that even though the system state shows little change in the aggregate of elapsed time, at any two particular adjacent time measures, the locations of central hub activity may differ radically. (Cf. Gould-Eldredge punctuated equilibria^ for a connection to evolutionary biology, and self-organized criticality^^ in the extended model of evolution.)

If black boxes are structs and structs are nodes, network vertices are time paths, which leads to the important conclusion that you and I share: time is identical to information. In such a network, information is physical (quantum information)accounting for dynamic activity. In other words, the time we measure is independent of the internal "black box" time; the struct is self contained and independent of the network and only enters via a time-dependent relation of measured quantum information.

I understand the lack of evolutionary potential in your "non-structural point." However, what always hangs me up when I engage with your work, is that points of the complex plane are not "non-structural." Points are analyzed as lines in complex analysis, and the complex plane compactified with one point at infinity (Riemann sphere) _does_ give us space and content; i.e., the algebra is closed and the space has dimension 2. From the chaotic field of non-ordered complex numbers, we do get ordered relations in real time and space, from analysis on the Hilbert space. I still don't grasp what in your concept would obviate such a mathematical approach.

Tom

*Bar-Yam, Y. [2003] "Multiscale Variety in Complex Systems," NECSI Technical Report 2003-11-01 (Nov.)

**Braha, D. & Bar-Yam, Y.[2006]. "From Centrality to Temporary Fame: Dynamic Centrality in Complex Networks." Complexity vol 12, no 2, pp 59-63

^Eldredge, N., & Gould, S. J. [1972]. "Punctuated equilibria: an alternative to phyletic gradualism." In: Models In Paleobiology (Ed. by T. J. M. Schopf). Freeman, Cooper and Co.

^^Bak, P. [1996]. How Nature Works: The Science of Self-Organized Criticality. Copernicus.

"I understand the lack of evolutionary potential in your "non-structural point." However, what always hangs me up when I engage with your work, is that points of the complex plane are not 'non-structural.' "

Tom,

Points of the complex plane, or of any vector space, are just these, points. They can be algebraically decomposed via other vectors, that's true, but this decomposition is not unique and hence cannot really be called structural in the temporal sense:

11 = 2 plus 9 = 7 plus 4 = 5 plus 6 etc.

We have no (temporal) information on how number '11' (or any vector in a vector space) was *actually* formed.

Also remember set theory (set = collection of points) as the foundation of mathematics.

For temporal/formative information to be present in the object representation, you need a fundamentally new representational formalism, hence our ETS.

Lev,

We're on the same page. I dealt with that issue of decomposability in my NECSI ICCS 2006 paper here specifically in section 5.0.

I do not rely on the set theory of classical mathematics. A time dependent complex network method frees us of that constraint.

Tom