"From... elementary theories we build up descriptions of more and more complex systems. But in all these efforts we take for granted that we may use any language we wish and as many [languages] as necessary. That is, we choose whatever mathematical formalism is most useful and then interpret the symbols and measurement operations in very highly developed natural language. To a large degree, the simplicity of natural laws arises through the complexities of the languages we use for their expression."
I did an undergrad in physics decades ago, and it pleases me to no end to see that someone else has made this observation. Especially when dealing with quantum weirdness, the advice of professors to focus "follow the math" always seem to gloss over the extensive interpretation (Copenhagen and otherwise) that gave context and meaning to the math & experimental results.
In my experience, the reason to “follow the math” is that it is the best and perhaps only way to truly understand a physical theory, especially one as strange as quantum mechanics. One can understand the math first, and then develop an understanding of the context and meaning. But this abstract understanding is anchored in math. This is important because when the meaning gets too obscure, one can return to the math to resolve any confusion. The other way around does not work.
Scientific understanding (defined by the ability to make accurate predictions) can be (must be?) anchored in both the math and physical intuitions/observations, and it's very difficult to bridge the two without resorting to natural language (even if it's just you talking to yourself in your head while you look at the equations).
The whole problem with the Navier-Stokes equations is that the math seems to work extremely well, but we have no way to be sure it actually captures every aspect of reality (given suitably accurate initial inputs). You can use the equations to generate pretty convincing simulations, but they certainly do not always predict the fine-grained behavior of real-world turbulent systems.
Feynman's lectures repeatedly stress that physical laws (and the math that formalizes them) are, at best, idealized approximations of reality. Here's one, but you can google "feynman approximation laws" for more: https://www.feynmanlectures.caltech.edu/I_01.html
> Scientific understanding (defined by the ability to make accurate predictions) can be (must be?) anchored in both the math and physical intuitions/observations
I wouldn't conflate intuitions and observations like that. Observations in many physical realms are best described by math, which is used to build up a natural-language approximation for communicating of unintuitive findings.
"This is important because when the meaning gets too obscure, one can return to the math to resolve any confusion. The other way around does not work."
Note: my math is not very advanced, at least not good enough to understand quantum mechanics
But why does it not work the other way? When the math tells me something very wrong, can't the conctext and meaning show where the math modell is wrong?
As far as I understood, every physical modell is only a limited model of reality, so they all have flaws. Meaning the math can be wrong when applied to reality, which one could spot, with the understanding of reality?
> When the math tells me something very wrong, can't the context and meaning show where the math model is wrong?
The point is that when learning a new part of physics, it is far more likely that your intuition was wrong as the math was right, even if it's surprising.
Sure, the mathematical model is not a perfect model, but it can still be _very_ good, so if you disagree with it, you're very likely to be wrong.
GP is indeed wrong. We do use fundamental principles and philosophical positions to figure out what's wrong with our math when the results are wrong or nonsensical. These are in fact part of even an undergraduate physicist's training to be able to tell if some model is violating energy conservation or the second law of thermodynamics or the speed of light etc.
However, if you count the number of times a physicist uses math to resolve conceptual confusion and count when they use physical principles to fix the math, you will find the former many times larger than the latter.
Doubt anyone can understand the reality outside of the mathematics. Tho I will say that I've had math classes taught by physicists and they talk about how the math works/what it means a lot better than the math people, that know the theorems and the logical connections well butn't always develop the conceptual framework implied.
The mathematical formalism of quantum mechanics is (believed to be) central in the network of facts about quantum mechanics, if we draw an edge between two facts when one helps you understand another. Therefore, it makes more sense to learn the formalism first and use it to help you understand the data, predictions and interpretations, rather than the reverse.
The point of follow the math is that on this specific case, the only valid interpretation is the trivial ones that map your numbers into your observations.
The possible explanations for the weirdness are all speculation about how to solve a very important problem. But they neither valid interpretations (because they don't explain anything anybody can see) nor about Quantum Mechanics (they are about an open problem of physics, not about the theory the teacher is explaining).
It is important to speculate on how one can solve problems. That's where solutions come from. But this is not the same thing as interpreting results.
That paper is from 1993. Hard to believe that it took such a long time to find a concrete pathology in a Newtonian system. Now the question is whether these blowups only occur on initial condition sets of zero measure.
John Baez also has a nice, accessible series of articles called "Stuggles with the Continuum" [0]. As a side product, it gives a nice perspective on the development of modern physics as a series of attempts to fix these infinities (only to create more subtle one).
Honestly I feel as though that's one of the most incredible aspects of theoretical physics I've seen. The fact that often times these underlying physics are danced around and even though they're unknown, we can tell something is not quite right.
I believe this guy even talks about it a bit in this video made by the Royal Institution (which I really enjoy watching), discussing the possibility of different fundamental building blocks of nature.
https://www.youtube.com/watch?v=zNVQfWC_evg
Since these equations are something I've never studied and certainly don't hear about on a day to day basis, the thing they bring to mind is the scene in Cryptonomicon when Waterhouse doesn't get that a simple question on an army test is actually a simple question and starts in on some complex math. It's an entertaining read.
[The Navy] gave [Lawrence Waterhouse] an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back?
Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier-Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn’t prove his intelligence, what would?
Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal.
Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.
Reminds me of my (old) college professor who once gave a test where the first question was about a sailboat with an on-board fan blowing into the sail.
It was meant to be an easy question to see if students understood Newton's third law, but one student filled in the entire test with momentum calculations showing that the boat would actually move forward at X velocity because the sail would essentially redirect some % of the air backwards like a reverse thruster (conservation of momentum). He left the rest of the test blank because he blew the whole time limit on the first question.
The professor was perplexed when grading this student's exam and built a "sailboat" out of a pinewood derby car with a dowel rod mast and aluminum foil sail. He taped a handheld fan to the car, pointed into the sail, and indeed, the car moved forward (this part he demoed to the class as he was telling the story and just before he did it, he took a poll to see how many people thought it would move forward, backward, or stay still - "stay still" won the poll)
The student reportedly got 100% on the test and the professor threw out that question on future exams.
Given that the equations involve continuous functions of some bulk properties of a fluid, while the fluid itself is made up of discrete particles, is there anything fundamentally or metaphysically problematical if these blow-ups are unavoidable? Or is it about finding practical ways to avoid them when modeling real-world scenarios?
I guess that it's just that it would be super convenient if the equation that work for a fluid would work for any part of the fluid, however small it is.
As a non-mathematician, I found the Numberphile video on YouTube very helpful for breaking down the basics of Navier-Stokes equations in a very easy to comprehend way:
https://www.youtube.com/watch?v=ERBVFcutl3M
From the perspective of trying to get to the actual physical behavior of such systems, the exercise is purely theoretical since we know there are other mechanisms at work to prevent the “ blow up” ...
(I'm not a fluid flow person and this is probably not a very good comment.)
Yeah, that seems likely. But sometimes it's difficult to draw the line between issues that are "purely theoretical" and issues that make a practical difference. Conceivably at some scale the Navier-Stokes equations match up with molecular dynamics simulations in a way that is illuminating. Then in that situation maybe the problems with the N-S equations become nice indicators of what's going on.
Because we're drilling down into finer and finer descriptions. If there was a black hole in earth orbit where it could be closely observed I'm sure we'd find small details of its behavior and the behavior of dust crossing its event horizon that require difficult math to explain.
I wonder what a good textbook on theoretical physics would look like if it assumed all the math involved as a prerequisite. (Say, a 10-volume set "compressed" into, what, 300 pages still containing a complete treatment of the subject?)
Jackson's electrodynamics is probably the closest thing to this I've seen. He often mentions offhand that it's trivial to show one thing or another, but when you actually do the derivation it's many pages.
Doesn't this point to some renormalization strategy being necessary? Or at least some correction to how renormalization is being applied if its leading to subspace infinities. Not sure how mathematicians view renormalization when you are trying to ask questions about indefinitely far distances into the future but it seems like there needs to be some necessary coarseness which suppresses the infinities, not sure why those terms wouldn't naturally arise.
Possibly, but I either interpreted, or heard this somewhere before, that turbulence is way harder to make sense of and predict (?). Not a physicist, but essentially that both are nuts, but that turbulence is even nuttier
From a laymens POV, it seems like half of the equation would represent the energy used in the bonds of each molecule representing smoothness (almost moving as a solid), with the other half laying out a dynamic list of finite variables that disrupt or exceed the energy forming the bonds.
Its almost like they would have to define a theoretic limit similar to 9.808175174 m/s^2 to limit the finite possibilities of outside forces acting on the molecules before the molecules themselves become the outside force, with those that exceed the limits be classified with a custom equation, and those below the threshold fitting nicely in a bow wrapped package.
And just to add more SWAG, is it going to be easier to produce these finding/understandings in space, where theoretic limits are more well defined and we'll be forced to use custom equations within our atmosphere looking for an answer from a perspective that is environmentally more complex?
OK. I was thinking more of a Newtons apple moment in space, where someone with a syringe and fluid may make an elegant observation on blow-up or collapse.
Eddys in a river with more eddys creating eddys is a dizzying theory though.
If you think the Navier-Stokes equations are hard, try the version that plasma physicists have to use, instead.
Rather than confront those, cosmologists have chosen to pretend that, while every single thing they can see is plasma (excepting, uniquely, planets), none of it does anything plasma-ish.
Since plasma physics is scale-invariant, freaky phenomena seen in labs should be playing out at stellar, galactic, and super-cluster scale. If they don't, it needs explanation why not.
Huge props to solar physicists, who confront plasma physics, face-to-face, daily.
I picked up a copy of Chandrasekar's "Hydrodynamic and Magnetodynamic Stability" a while back and started working through it. The basics are relatively approachable, but dang the differential equations can get gnarly. It feels like a large portion of the art is in asking the right question that can aviod toppling into a mess of non-linearities.
Really good book, as far as I got through it. The beginning has a nice derivation of Navier-Stokes, which was mew to me.
"Butterfly effect" refers to chaotic systems, i.e. those where the outcome is highly sensitive to the starting position. They are deterministic, not random, but they are hard to predict because this sensitivity means any small errors in simulation can give completely wrong results.
"From... elementary theories we build up descriptions of more and more complex systems. But in all these efforts we take for granted that we may use any language we wish and as many [languages] as necessary. That is, we choose whatever mathematical formalism is most useful and then interpret the symbols and measurement operations in very highly developed natural language. To a large degree, the simplicity of natural laws arises through the complexities of the languages we use for their expression."
– H. H. Pattee