Scale Is Different: From Mathematical Descriptions to Theories of Things by Daniel Porter

Daniel,

You state the issue well, especially the footnote on page 8, "Though perhaps a puzzle without edges, a definite number of pieces, or a guarantee that we even can have all the pieces."

That is a subtle way of stating that we may not be permitted to know everything, without saying why.

You mention Kuhn in an earlier footnote, but Kuhn's concepts are the primary basis of your closing paragraph, "-paradigm shift". I cited Kuhn in my essay, 1294, but you apply Kuhn's concepts on a broader scale. The last sentence, "By understanding the limits of our mathematical descriptions, we can begin the search for a novel description of the universe that will carry us into the next stage of scientific progress.", is pointing toward a possible solution.

I would have added one small addition, "... or learn to apply mathematical descriptions in a different way... ." The IEEE paper cited in my topic, as a solution to an erroneous assumption, presents a different way to apply mathematics to physical law, and it is simple, but not too simple.

Frank,

Thank you for your great comment, and your interesting points.

You are right that the question of whether or not we are permitted to know everything about the universe is one that I do not explicitly address in my essay. Though my argument runs up against this issue, I tried to stay away from that particular sort of epistemological claim.

I am inclined to say that a description that exists within a particular system (as it seems logical that all our descriptions must) cannot completely capture the complexity of the system without being the system itself. That is, the only system that completely describes the universe is the universe itself.

That said, we obviously have descriptions that are "less complex" than the systems they describe, so it seems like we should be able to "gain" a little information with our descriptions. Perhaps a complete description, one that is less complex than the universe itself, could then be possible. One way this might be possible is if the "extra information" in the universe is all at a level of accuracy that is higher than is possibly experimentally. Then, if we can have one description that "covers" one particular domain -- describes it as accurately as possible -- we should be able to have a "patchwork" of descriptions that covers all possible domains. Perhaps we need an infinitude of descriptions, perhaps even that would not be enough. I do not yet know how to approach this particular question.

In response to your second comment about Kuhn, I do try to apply him on a broader scale. Though I certainly can't say what my "novel" description mentioned in the end will be, I think it will by necessity involve a new system of recording and transmitting knowledge. One of Kuhn's main arguments is that our "paradigms" are partially encoded and passed on in the tools we use to teach -- textbooks and aging professors. Perhaps if we had a knowledge system that is more complex, more fluid, and more inclusive than our current means we could depart from scientific knowledge that progresses in the way Kuhn describes.

I unfortunately have no good conception of what this might look like.

Your essay and the paper you describe raise an interesting question about units. I think that sort concrete argument about commonalities in and amongst our descriptions could be a very enlightening way of learning about relationships between them -- critical for further evaluation and, potentially, their combination into the sort of "patchwork" I described.

-Daniel

Quick skim of your essay outlines the problem of "scaling" nicely.. What works small doesn't work big. Seems that one needs a model that scales sizes without any change of the model.

In my To Seek Unknown Shores

聽聽 http://fqxi.org/community/forum/topic/1409

is propose the equal "activities" of spherical and tetrahedral structures co-joined as proposed hold from a point to a cosmos.. Mass has a 6.25% probability to exist, while energy divides into 1-D and 2-D versions..

Might you destroy idea so I can sleep?

I'm tired, whatever, it is fun.

Good Luck to you..

Daniel

Have you seen?

http://fqxi.org/community/forum/topic/1554

Dear Daniel,

What is your opinion about Scale dimension and SPF symmetry of physical laws at different levels of matter? Similar problem are investigated in the Theory of Infinite Hierarchical Nesting of Matter (my essay).

Sergey Fedosin Essay

Yuri,

Thank you for passing on this reference. I think the sort of scaling analysis can be very useful. This is the sort of research I suggest there should be more of at the end of my essay.

As an undergraduate I did research investigating the fluid dynamics of droplet breakup (http://pre.aps.org/abstract/PRE/v85/i4/e041701), and there scaling laws of exactly the sort you mention are a very powerful tool for understanding the dynamics of droplets from the millimeter scale to sub micrometer scales. I argue that fields like complex matter physics will be as important as elementary particle physics in our search for a "fundamental" understanding of the universe.

Dear Daniel,

I enjoyed your essay. I think that the subject and focus are quite important and timely; never before have we had so much experimental evidence of the importance of scale as we have today. I think the problem is actually bigger than relativity and quantum theory; after all, relativity works on large scales only if you fudge in dark matter and dark energy, which may be the right way to go, but aren't exactly on a solid footing at present. I think that there are at least five evident scales, where the different "forces" dominate: the weak-strong scale, electromagnetic scale, ordinary gravity scale, dark matter scale, and dark energy scale. Likely enough there are more, but they are too small or too large for us to see. I discuss this a bit further in my essay here On the Foundational Assumptions of Modern Physics if you're interested.

Regarding the possible inadequacy of mathematics: as you know, math has its own completeness/consistency issues, as pointed out by Godel, etc. If the universe is infinite, then it will at the very least exhibit the same issues involving undecidability etc. as the natural numbers. I do think math still has a lot more to offer physics, however. Take care,

Ben Dribus

Ben,

Thanks for the comment. The points you raise are excellent ones; the problem is most certainly "bigger" than general relativity and QM. I chose these examples because, as you mention in your essay, "these two theories represent our deepest understanding of fundamental physics." I think the difficulties we encounter with these two theories are indicative of a broader methodological difficulty. It may even be beyond the discussion of length and time scales (complexity and emergence, for example also appear in systems where the number of components goes through a "scale transition" but the size of the system does not).

I think Godel's incompleteness and associated ideas are intimately linked to my ideas, you're right. In some sense, this tells us that mathematical descriptions are not, and cannot be, "complete." It doesn't tell us if the universe is or not. If it is not, then the best we can do are a bunch of descriptions that get close in certain situations. If it is, then we need a description that is more than mathematics. Either way, I completely agree that math has plenty more to offer physics. I simply point out that we as a research community do not get caught up in mathematics as the ONLY description that has a lot to offer physics.

Thanks again, and best of luck!

Very interesting confirmation slogan of Anderson ...

Dear Daniel. You touched on an interesting topic. What kind of science, physics? Mathematically possible to describe physical phenomena? Mathematics - one method. Abstract - another method. Abstract description of the car explains all the cars. Mathematical model abstraction. Formula - abstraction. Numbers and numbers - mathematics. Planck's constant - multiples, making it possible to write an abstract model. The universe can not be described, but you can write a mathematical model is not the entire universe, and its area. Details can be found in my essay and comments. Good luck in the contest.

Information-Energy Quantum Balance by Vasily Kletshkin

14 posts • created by Vasily Kletshkin • Aug. 30, 2012 @ 12:08 GMT

Daniel,

Yes, in any case, the physical ideas ought to come first, and the math ought to be whatever is necessary to get the job done (if possible). Also, I appreciate you looking at my essay. Take care,

Ben

For better clarification my approach

I sending to you Frank Wilczek's 3 keen articles

http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits388.pdf

http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits393.pdf

http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits400.pdf

All the best

Dear Daniel,

I have read your paper - and I can share some of your thoughts, especially your final statement, that is, understanding the limits of our mathematical descriptions, we can begin the search for a novel description of the universe that will carry us into the next stage of scientific progress.

Metaphysics is the most criticized discipline inside modern physics, but it is - as conceived by me - the most promising one, because it deals with something that is fundamental by its very nature. This entity is commonly defined as an all-embracing and unconditioned foundation. But this foundation seems to be scientifically ungraspable because it can neither be described mathematically nor be detected experimentally.

Just this transcendent nature of the metaphysical foundation turned out to be a scientific barrier that could not be overcome until today. But there is a way to do this. Actually a universe with a transcendent foundation must be described in a very specific way. This transcendent description points in a direction that is very similar to yours.

In the FQXI-Contest 2009 I have presented this idea of a transcendent description. The title of my essay: Taming of the ONE.

Good Luck for Your Paper.

Kind Regards

Helmut

Dear Daniel,

It was nice reading you essay. You write clearly and get to the point without unnecessary complications. We have many ideas in common. The ideal of unification is one of my deepest ideals as a physicist. But it does not imply reductionism. Like you, I also think there are times we have to fit the theories together instead of thinking that one can be reduce to the other. I also believe we will get to a hierarchical structure of theories where one theory is the base for the others but the others cannot be reduced to this one. For sure, understanding concepts like mind and conscience would require a new level in this hierarchy. In my article, "On the Nature of Reality", I describe it as a hierarchy of final theories. Here I prefer the concept of final theory than theory of everything. I also believe that our current mathematical theories are not enough for unifying physics, actually I'm working on new mathematical frameworks and the connection between math and language. You might find my essay interesting: The Final Theory and the Language of Physics . There I discuss on the nature of physical theories discussing the its many aspects: mathematical formalism, interpretation, language, postulates, ets. Take a look and rate it please.

Best regards! and Good Luck!

Frederico

After studying about 250 essays in this contest, I realize now, how can I assess the level of each submitted work. Accordingly, I rated some essays, including yours.

Cood luck.

Sergey Fedosin

Dear Daniel Porter,

As the scale cannot emerge from single point, assigning the fundamental matter as a sting-segment between two points that is described in Coherently-cyclic cluster-matter paradigm of universe, is supportive of your concept of 'uni-field' of universe; whereas in this paradigm, continuum of matter is expressional that is unified.

With best wishes

Jayakar

Dear Daniel,

I like this essay - understanding the hierarchy of structure is indeed what physics should be about, not just understanding the foundations. Well done.

Best wishes

George Ellis

If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process.

Sergey Fedosin

Dear Daniel,

I just discovered and read your essay and found it to be clearly and well written. I liked how you "zoomed out" at each stage of your paper, it reminded me a bit of the "Powers of 10" and the beginning of Ohanian's intro text book.

I found several similarities between the ideas in your paper and in mine. In particular:

1. That scale has something to do with the differences in the descriptions of nature we have to give. I already see the mathematical fact that equal-shaped objects of different size have different ratios of area per volume as a basic hint of this, since this means one can take smaller objects to be not only smaller but also more 2-dimensional than larger equal-shaped objects.

2. That emergence plays a more fundamental role in nature than currently appreciated. You may not agree with this, but I strongly suspect that dimensionality itself is emergent

3. That the apparent incompatibility between GR and SM may reflect a limited ranges or domain of applicability of these theories, not that they are fundamentally wrong.

4. That an all-encompassing description of nature transcends our usual conception of a "theory": You used the term "description", I used the term "Metatheory" in my paper

5. That the evolution of science seems to proceed largely in accordance with Kuhn's observations (this last point is not mentioned explicitly in my paper, but reflects my perspective on progress in science).

The one point at which our views may differ is the range of applicability of mathematics in describing nature, but it is possible that the difference is not as large as it at first might seem.

You see, in my view it is not that mathematics is inadequate to model nature, but that we have failed to capture the most fundamental aspect of physical things within our mathematical frameworks. I believe the most fundamental aspect of any physical object is that it exists, yet existence is not part of our current conceptual inventory in physics, and therefore not thought to be expressible using the language of mathematics.

But if you think about it, many of our most fundamental problems are deeply intertwined with this concept. For instance, with respect to gravity you might ask, how does an object know that a another mass exists nearby? Newton would say that it "feels" the other object's gravitational force, and Einstein that spacetime in its vicinity becomes curved. In quantum theory, this is even more apparent: How is it possible for any object to exist in a superposition of mutually exclusive states? Why is it that we can only detect the existence of a quantum object at a particular location probabilistically?

So, perhaps your suggestion that mathematics may not be adequate to model all of nature is in reality just a call to consider concepts which today we think of as unrelated to or indescribable by mathematics to become associated with it?

If so, we actually agree on this point as well because that is precisely what I believe.

But regardless, I am glad to find someone like-minded, you may find my essay of interest, too.

All the best,

Armin