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This is a note to myself as well as anyone else who may find such things interesting. I’m not sure how I overlooked this, but this “paper book” looks interesting.
Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold
Authors: Anastasios Mallios, Ioannis Raptis
It reminds me of one of my all time favorite papers from grad school
Geroch, R., Einstein algebras, Communications in Mathematical Physics, 26, 271 (1972).
which they do site.
Warning: As much of an ardent fan I am of the subject, the rhetoric might be a bit over the top.
The first three chapters can probably be skipped completely.
Section 4.1.7 (page 97) might be worth a look:
The categorical imperative: the category of differential triads, its properties and versatility compared to the category of smooth manifolds
this “paper book” looks interesting.
After scanning over tens and tens of pages of pure text which does not make a reassuring impression, the interspersed formulas that I saw didn’t give me the impression that there is much going on here, behind the text. At this point I tend to advise you to drop this and use your time on something else.
If I am wrong and there is well hidden something useful in this document I’d be happy to admit it if somebody gives a decent concise 1-page summary.
[cough]
@Eric: for someone who lambasted some work of Witten by saying (July 21)
The opening paragraph is pure rubbish
I’m a little surprised that you don’t seem to think this “paper-book” is just a tad self-aggrandizing.
Witten’s remark was as aggrandizing or more than this.
I suppose I wasted some ascii by saying the first 3 chapters could be skipped AND warning about the rhetoric.
Has anyone mentioned arrogance around here lately?
Has anyone mentioned arrogance around here lately?
You have (and just did). But I don’t understand why.
Your last comment suggests that maybe you need to calm down a bit before making another comment.
I suppose I wasted some ascii by saying the first 3 chapters could be skipped AND warning about the rhetoric.
Nope. I quite did look at the article myself.
Eric, since you are being blunt, it is maybe indicated that I start being more blunt myself: this is getting well into crackpot territory here. I advise you to not get carried away with such things. Better than wasting time on this would be to spend your time on learning some of the things that you had set out learning a while ago.
I had a fresh look today at one of Tim’s papers I enjoy a lot:
As a habit, I checked the SPIRES “cited by” and this paper by Mallios & Raptis showed up. In the copy of Tim’s paper he sent me via email last year, I see an acknowledgement:
Acknowledgement
I would like to thank Ioannis Raptis for very useful conversations on the subject of this paper and also for his encouragement. Thanks are also due to Rafael Sorkin for pointing out an error in an earlier version and to Jonathan Gratus for some suggestions for improvements to an earlier draft.
I have been interested in papers by Ioannis Raptis and Rafael Sorkin since I was in grad school. Although this paper-book is admittedly heavy on prose, I think much of the prose is singing to choir here. The paper-book summarizes progress made over many years and provides references to the technical papers where all the formulas you could ever want can be found.
As I edit this (thoughtfully), I see Urs has labeled Raptis’ work as crackpot. Nice.
As I know Ioannis a bit let me chip in here. I liked the overall approach in his work but felt that some of his collaborators were better salesmen than judges of the importance of results. With a maths physicist a few years back we worked hard trying to see the substance of this stuff. There is some and some of it is very nice but the claims are out of proportion to the actual stuff proved. Some of that was a fairly routine approach to differential graded algebras in a discrete context, and some were reinventing the wheel as they did not know a whole lot of the dga stuff from rational homotopy, nor, if I remember rightly, very much about the cohomology of partially ordered sets. There were however some very neat ideas that deserved more attention, buried in the papers.
The main criticism I had was that there was a lot too much verbiage at the start of several of the papers. (This, I felt, was perhaps a cultural thing and Ioannis did not do the same in some of his other collaborations.)
Thanks Tim. I think that is a totally fair assessment. Physicists are often not mathematicians (obvious). Something that may seem trivial to a mathematician, e.g. differential graded algebras in a discrete context, can seem completely mind boggling to a physicist. I think they can be forgiven for getting overly excited.
I remember a life-changing experience happened to me when I first read
Introduction to Noncommutative Geometry of Commutative Algebras and Applications in Physics
in Recent Developments in Gravitation and Mathematical Physics, Proceedings of the Second Mexican School on Gravitation and Mathematical Physics, eds. A. Garcia, C. Lämmerzahl, A. Macias, T. Matos, and D. Nunez, Science Network Publishing 1998, ISBN 3-9805735-0-8
You can get so much physics just from algebra?!?! No calculus? No continuum? Just algebraic relations? This paper set me onto months and months and pages and pages of algebraic gymnastics.
If you haven’t done it, treat yourself to some beautiful light reading with
by Robert Geroch. Nothing heavy. Just simple sweetness.
If there was some way to wave the wand of category theory over all this and express it is some totally simple manner, that would be jaw-droppingly awesome.
As I said, if somebody extracts a 1-page summary of actual content, that would be good.
Eric: it is not good to judge content by appeal to authority. And if the pope himself has cited this article, it still does not make it look better.
If you want to prove me wrong, don’t appeal to some authority or what you take as such. Instead, take up the exercise that I proposed: extract from the article the genuine content. Then we will look at it.
Perhaps there is a use to be served by having some pages on discrete approaches to that sort of Physics, and with an evaluation of what it achieves and what it fails to do. This might be of a different nature to other pages with well posed critiques forming part of it.
If you want to prove me wrong, don’t appeal to some authority or what you take as such.
[cough]
If you don’t see what you’re turning into, please have someone close to you give an objective assessment. Of all people, it is saddening.
Instead of letting this discussion deteriorate, let’s look back at what Tim said:
I liked the overall approach in his work but felt that some of his collaborators were better salesmen than judges of the importance of results. With a maths physicist a few years back we worked hard trying to see the substance of this stuff. There is some and some of it is very nice but the claims are out of proportion to the actual stuff proved. Some of that was a fairly routine approach to differential graded algebras in a discrete context, and some were reinventing the wheel as they did not know a whole lot of the dga stuff from rational homotopy, nor, if I remember rightly, very much about the cohomology of partially ordered sets. There were however some very neat ideas that deserved more attention, buried in the papers.
Putting this together with what Urs said, the whole idea would be to get to what substance and nice neat ideas there are. Eric, if you are all excited about the paper, then perhaps you ought to be the one enthusiastically digging out the nuggets. If Urs doesn’t feel inclined to root through the over 400 pages (filled with rhetoric and fluff and hype) with nothing more to go on than someone’s vague say-so that there’s interesting stuff buried in there, that’s his prerogative isn’t it?
Let’s try to keep this discussion intellectual and focused.
Edit: Tim, I intended no disrespect to you with the “vague say-so”; the whole situation is getting a little uncomfortable right now.
Here is my original comment reproduced in its entirety:
This is a note to myself as well as anyone else who may find such things interesting. I’m not sure how I overlooked this, but this “paper book” looks interesting.
Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold
Authors: Anastasios Mallios, Ioannis RaptisIt reminds me of one of my all time favorite papers from grad school
Geroch, R., Einstein algebras, Communications in Mathematical Physics, 26, 271 (1972).
which they do site.
Warning: As much of an ardent fan I am of the subject, the rhetoric might be a bit over the top.
The first three chapters can probably be skipped completely.
Section 4.1.7 (page 97) might be worth a look:
The categorical imperative: the category of differential triads, its properties and versatility compared to the category of smooth manifolds
Innocent enough, right?
The opening line said:
This is a note to myself as well as anyone else who may find such things interesting.
If anyone else does not “find such things interesting” I have no problem with that and I would never be unhappy if someone didn’t want to wade through 400 pages of verbiage that I even warned was a bit difficult to swallow. Even starting in chapter 4 is a bit tough to swallow. Criticizing the paper (preferably on substance more than style) would have been enough, but the suggestion that I am not capable of determining for myself what to spend my limited time on?
At this point I tend to advise you to drop this and use your time on something else.
What is the point of saying that? I’ve been reading Raptis’ stuff for years.
The subject matter is interesting. By that, I mean interesting to me. What other possible meaning could that sentence have? Can I declare something to be interesting to anyone else?
I thought there may be someone here (lurking or otherwise) who may find the paper (and references therein) interesting as well.
this is getting well into crackpot territory here
There really is no excuse for the crackpot reference.
Putting this together with what Urs said, the whole idea would be to get to what substance and nice neat ideas there are. Eric, if you are all excited about the paper, then perhaps you ought to be the one enthusiastically digging out the nuggets.
I’m happy to try, but I never said I was excited about the paper. There is not much there I haven’t seen before. My original notice was fairly subdued I thought.
Dear All,
I have never printed out the 400 pages nor read them. (How many trees is that?) I merely am saying that Raptis in discussions and in his shorter papers has made some interesting points (interesting to me).
As I said, I would like to see some (non-inflammatory) discussion of non-continuous models for that part of physics. At the Dagstuhl meeting on this area some years ago there was some discussion about Sorkin’s work (he was there) and I was left not knowing what to think. My own background (in shape theory) makes me biased to think that continuity is non-physical and smoothness even more so. (It is possibly not necessary as an assumption.) The smooth models are just models and are useful, but if they are saying things about points of singularity of functions and interpreting that then I have no feeling for what that can interpret as, being ’brought up’ believing in Quantum Theory.
I do not think that most of the more discrete methods put forward so far have any hope of going really deep into the heart of physical phenomena, but that is for foundational reasons. I am however more or less ignorant on this so would welcome some balanced discussion.
Here is a link to the Dagstuhl meeting web site:
Tim said:
I do not think that most of the more discrete methods put forward so far have any hope of going really deep into the heart of physical phenomena, but that is for foundational reasons.
I’d love to hear your reasons for this. The reasons could provide some guidance for future research.
With my background in numerical analysis, the general thought among most practitioners is that we are approximating some continuum model. When you take the finite model seriously, however, (like “taking categories seriously”) there are tangible benefits. For example, if your numerical simulation of Maxwell’s equations satisfies
then charge is conserved exactly on the lattice (neglecting rounding errors). A naive numerical method may not satisfy and this leads to instabilities as fictitious charges accumulate in the grid. When I say it like that, it might sound obvious. Why didn’t the engineers do this earlier? Well, very few engineers have ever heard the word “cohomology”. Perhaps not even “exterior derivative”. And I am talking about extremely good and extremely intelligent engineers.
So when building finite models of fundamental physics, it is a good idea to take clues from the continuum, but you can not expect to replicate the continuum model exactly. For example, you can define a finite differential graded algebra, but the product will not be skew commutative at the cochain level (prior to taking cohomology classes). This failure to be skew commutative depends on the grid spacing and vanishes in the continuum limit.
It took a while for me to recognize this as a feature rather than a bug. It draws in all kinds of parallels to quantum mechanics.
In the paper
Introduction to Noncommutative Geometry of Commutative Algebras and Applications in Physics
in Recent Developments in Gravitation and Mathematical Physics, Proceedings of the Second Mexican School on Gravitation and Mathematical Physics, eds. A. Garcia, C. Lämmerzahl, A. Macias, T. Matos, and D. Nunez, Science Network Publishing 1998, ISBN 3-9805735-0-8
I was blown away that finite differences appear from noncommutative relations.
My continued goal has been (and will continue to be) to reproduce as much of differential geometry as possible without ever introducing a continuum space. That is precisely the idea behind abstract differential geometry and is precisely why I find the subject interesting.
The problem with the ’discrete’ approach is that it still seems to be looking at points. I am not a physicist and do not know of what is thought of as happening beyond the Plank scale. (Continuum limits are to me problem.)
It is a bit beyond me just like the question ’ what happened before the big bang?’ although there was a TV programme with something like that as a title recently.
I understand the finite models but it is the comparison of two finite models that bugs me. I do not think straight on that!
The problem with the ’discrete’ approach is that it still seems to be looking at points.
Interesting :)
My gut response to this is that I think points are “ok” as long as we don’t try to “measure” a point.
Just thinking out loud…
For any kind of finite space we can dream up, there will always be a corresponding dual space. I think these two are partners in a dance and you cannot neglect one. The question is about measurement and being able to resolve points. I don’t think any measurement should be allowed to resolve a point. Perhaps, the best resolution is constrained to within the dual cell?
Then there is the other duality, i.e. Fourier duality. To resolve an infinitesimal point requires infinite amount of energy.
So I agree that we should not be able to resolve an infinitesimal point experimentally, but I’m not sure if that means we cannot allow points in our space (especially if we interpret them as being dual to n-dimensional cells).
Maybe one thing prior efforts in finite models has been missing is a model for measurement?
Maybe one thing prior efforts in finite models has been missing is a model for measurement?
I think i agree with that.
In the introduction to my dissertation, I discussed measurement and how it relates to integration as a motivation for considering forms in computational physics.
Here is a snippet:
The ultimate test of a simulation is how well it agrees with physical measurements. With this in mind, it is worthwhile to consider what types of electromagnetics quantities are in fact measurable, for these are the quantities that will be of most practical value when formulating numerical methods.
[snip]
Although this standard prescription [edit: in terms of vector fields] lends itself to mathematical analysis, these vector field quantities are not directly amenable to physical measurements. The reason being that vector fields constitute an assignment of a single vector to each point in space-time. Measuring such pointwise quantities would clearly require an infinitesimal probe. Instead, what is typically measured is the integral of the projection of these vector fields over the region of some finite probe.
[snip]
In this way, the physical process of measuring electromagnetic quantities can be associated to the mathematical process of integration on manifolds.
[snip]
In the discretization process, as in the measurement process, it is the integral of the projections of these vector fields over domains of integration that are of relevance. The fact that both the measurement process and the discretization process involve the integration of the projection of vector fields over domains of integration should not be seen as a coincidence, but rather as a guiding principle.
I wonder if it is sufficient, or maybe even a step in the right direction, if the finite model had a theory of integration along with it?
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