I Know the Math Says so, but Is It Really True?

[Dale]

Einstein is reported as having once said " If you can't explain it to a six year old, you don't understand it."

It is probably misattributed: http://en.wikiquote.org/wiki/Albert_Einstein#Misattributed

But even if it were correctly attributed it is wrong. Teaching a concept clearly and understanding it are two different skills

 

[strangerep]

The universe does not calculate, does not use mathematics, yet everything seems to work quite well without it.

Heh, the universe is one big analog computer.

 

[Klystron]

Mathematics is no more special than any other creation of the human imagination.

Arguably incorrect.

Tinker Bell is a special creation of the human imagination. As much as I like Tink and clapped my hands hard so she would get well, mathematics remains far more useful and special.

The universe does not calculate, does not use mathematics, yet everything seems to work quite well without it.

Of several logical fallacies inherent in this statement, an existential 'least' refutation may be simplest.

1. Human beings exist in (are part of) the universe.
2. Humans count, calculate and use mathematics.
3. Therefore, (part of) the universe calculates and uses mathematics.

 

[HAYAO]

Excellent article.

Yeah, I was on the both ends of this. When I was young and had no good mathematical intuition of a physical phenomenon, I was really frustrated. Now, as an assistant professor, I get frustrated when people don't understand the mathematical representation of physical phenomena.

I still do think that almost all, if not all, physical theories are based on certain postulates. These postulates are based merely on physical intuition, although a good one. For example, one of the postulates of special relativity is that speed of light is constant in vacuum. No one can really "prove" this, but it is based on a good reasoning and observation if anything else. And since the theory of special relativity reflects the real world and it works, we "assume" that the postulates are correct. Similar thing for wave representation of particles. We assume that the particle behaves like a probability (amplitude) wave function, but the theory based on this actually works and reflect what we'd expect to observe. The more we do experiments and the more we have stronger mathematics to dig deeper into this, the more it reinforces the theory or provide a more generalized theory that encompasses all the other theories (e.g. QM and QFT).

In many cases, mathematical representation of physics is a well-thought-out reasonable modeling/interpretation of the observed phenomena. If any student is doubting the math, it's not necessarily because they don't agree with math, but because they don't agree with the modeling/interpretation, due to lack of intuition (which can be trained to a certain extent; it's why people learn to accept it).

 

[Hornbein]

The math for general relativity was already known. The difficulty was discovering and showing that it related to the real world.

 

[bhobba]

The math for general relativity was already known. The difficulty was discovering and showing that it related to the real world.

As Hilbert, who discovered the GR equations slightly before Einstein (by five days), said: "Every boy in the streets of Gottingen understands more about four-dimensional geometry than Einstein. Yet, despite that, Einstein did the work, not the mathematicians."

Thanks
Bill

 

[weirdoguy]

If you can't explain it to a six year old, you don't understand it.

As a teacher, I think that is one of the stupidest things I have ever heard in my whole life.

 

[Hornbein]

As Hilbert, who discovered the GR equations slightly before Einstein (by five days), said: "Every boy in the streets of Gottingen understands more about four-dimensional geometry than Einstein. Yet, despite that, Einstein did the work, not the mathematicians."

Thanks
Bill

Einstein and his collaborator (Rosen?) were stuck so he went to Gottingen for help. He was quite loathe to do this. He said Gottingen had a reputation for stealing the results of others. Nevertheless out of desperation he made the trip. He was informed that he was trying to do the impossible, that one of the conditions he thought essential actually wasn't, and he already had the answer. Hilbert published his solution, which greatly angered Albert. He got Hilbert to back down and recognize AE's priority. I imagine it helped that Albert had the backing of Berlin.

 

[bhobba]

He got Hilbert to back down and recognize AE's priority.

In public at least, and likely in private, Hilbert always gave Einstein credit for GR. Of greater interest was when Einstein, Hilbert and others discovered solutions that violated energy conservation. Both were stumped. But they knew one person they thought could tackle it - the great Emmy Noether. And her famous theorem was borne. It may even be a more important discovery than GR. IMHO, it was just a precursor to modern science, which is very collaborative. The idea of the lone genius that could revolutionise science was fast fading.

As Wigner said of Einstein:

'I have known a great many intelligent people in my life. I knew Planck, von Laue and Heisenberg. Paul Dirac was my brother-in-law; Leo Szilard and Edward Teller were among my closest friends; and Albert Einstein was a good friend. But none of them had a mind as quick and acute as von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me. But Einstein's understanding was deeper even than von Neumann's. His mind was both more penetrating and more original than von Neumann's. And that is a very remarkable statement. Einstein took extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity, and for all of von Neumann's brilliance, he never produced anything as original.'

Even the great Feynman, himself like Einstein beyond genius at the level of a magician, said knowing what Einstein did, he could not have invented Relativity. Ohanian has said such people are sleepwalkers. They did not know where they were going but were unerringly led there. Einstein was perhaps the greatest sleepwalker there ever was, except maybe for Newton.

Thanks
Bill

 

[Dale]

they knew one person they thought could tackle it - the great Emmy Noether. And her famous theorem was borne. It may even be a more important discovery than GR.

I agree 100%. IMO it is the single most important theorem in all of physics.

 

[PeterDonis]

Einstein and his collaborator (Rosen?)

Marcel Grossmann.

 

[fresh_42]

I agree 100%. IMO it is the single most important theorem in all of physics.

For anyone who is interested in the original paper (which can be translated if you use google's chrome, right-click + "Translate to English"):
https://de.wikisource.org/wiki/Invariante_Variationsprobleme

and here are the facsimile (pages 37 ff. and 235 ff.)
https://gdz.sub.uni-goettingen.de/id/PPN252457811_1918

 

[strangerep]

IMO [Noether's theorem] is the single most important theorem in all of physics.

Surely that can't be true.

(Try deriving Kepler's 3rd law using Noetherian symmetry/conservation techniques... )

 

[strangerep]

Marcel Grossmann.

Hmm. I thought it was just Einstein+Grossman who put together GR using Riemannian geometry (but I haven't read a full history). What exactly did they need help with from Gottingen?

 

[PeterDonis]

I thought it was just Einstein+Grossman who put together GR using Riemannian geometry

Einstein came up with the field equation (at least by one possible route--see below for another route that Hilbert took). Grossmann helped him to learn Riemannian geometry.

What exactly did they need help with from Gottingen?

Einstein was stuck regarding a particular aspect of the field equation. (It's been a while since I read up about this so I can't say off the top of my head exactly what aspect it was.) I don't know that his primary purpose in visiting Hilbert in Gottingen was to see if Hilbert could help him get unstuck, but it probably was at least in the back of his mind. I also don't know that the talks with Hilbert were the primary thing that got Einstein unstuck, although they might well have helped.

However, the greater impact of the Gottingen visit was not on Einstein but on Hilbert. After mulling over his talks with Einstein, Hilbert came up with a simple, quick route to the field equation for gravity using the principle of least action. He came up with what is now called the Einstein-Hilbert Lagrangian based on obvious and simple considerations, and then it was a simple matter to derive the Euler-Lagrange Equation for it, and boom! he had the field equation. As MTW say in their discussion of six different routes to the field equation, "no route to the field equation is quicker".

 

[Dale]

Surely that can't be true. (Try deriving Kepler's 3rd law using Noetherian symmetry/conservation techniques... )

I am not sure why you think that is a necessary feature for a theorem to be classified as the single most important theorem in physics. I stand my my opinion, but being an opinion you are welcome to your own.

 

[fresh_42]

Try deriving Kepler's 3rd law using Noetherian symmetry/conservation techniques...

Is that really hard? Kepler's third is basically Newton's gravitation approximated, and Newton's gravitation is energy conservation in an $r^{-2}$ potential.

This would be at least my plan to approach it. Where is my mistake?

 

[PeterDonis]

Newton's gravitation is energy conservation in an $r^{-2}$ potential.

I think you mean an $r^{-1}$ potential.

 

[fresh_42]

I think you mean an $r^{-1}$ potential.

Sure. But is the plan ok?

 

[PeterDonis]

is the plan ok?

Newtonian gravitation is not the only theory that has energy conservation in an $r^{-1}$ potential (the obvious other such theory is electromagnetism). So that alone is not enough to get you to Newtonian gravitation.

 

[fresh_42]

I have read a comment on stackexchange that the only reason that Newton's mechanics can't be completely derived from Noether would be its incompleteness. I'm not sure how to take this since my physics knowledge is far too basic.

 

[fresh_42]

Newtonian gravitation is not the only theory that has energy conservation in an $r^{-1}$ potential (the obvious other such theory is electromagnetism). So that alone is not enough to get you to Newtonian gravitation.

As far as I understood the challenge, uniqueness wasn't required. The gravitational potential for masses can be measured and from there on only math is necessary.

 

[strangerep]

Kepler's third is basically Newton's gravitation approximated, [...]

Huh? Kepler's third law is an exact result from the theory of Newtonian gravitation. (A symmetry underlies it, but that symmetry is not associated with a conserved quantity.)

[Btw, I leave it to Mentors to decide if/when to fork this into a new thread. ]

 

[strangerep]

I have read a comment on stackexchange that the only reason that Newton's mechanics can't be completely derived from Noether would be its incompleteness.

? The Kepler problem is overcomplete in that it has more first integrals (constants of the motion) than are needed to solve the equations of motion, i.e., energy, angular momentum, and the LRL vector.

But none of these yield Kepler's 3rd law.

 

[bhobba]

I have read a comment on stackexchange that the only reason that Newton's mechanics can't be completely derived from Noether would be its incompleteness. I'm not sure how to take this since my physics knowledge is far too basic.

Well, one reason is you need to assume the Principle of Least Action (PLA). That is derivable from QM. So at rock bottom, classical mechanics is a limiting case of QM. Even assuming the PLA and Noether, a bit more is required as detailed in Landau - Mechanics eg why does mass appear in the free particle Langrangian.

Thanks
Bill

 

[strangerep]

To tie off this little detour about Kepler's 3rd law (K3L hereafter), a short answer is that K3L can be derived from a property of the Lagrangian known as Mechanical Similarity. (L&L vol1 p22 has more detail.)

In essence, if some of the variables in a Lagrangian are homogeneous (meaning that, e.g., $L(\alpha r) = a^k L(r)$ for some constant $k$ (where $k$ might be different for different variables), there are cases where the homogeneities in the different variables can combine to result in merely multiplying the Lagrangian by a constant factor. This doesn't change the equations of motion, but does correspond to the existence of similarly shaped orbits of different size and energy. This symmetry doesn't commute with the Hamiltonian, hence is not associated with a conserved quantity.

There are other cases where mechanical similarity is useful. L&L give examples.

 

[droogiefret]

I think what happens is that lay people (like myself) have these flashes of awe and there is this need to try to comprehend. Then we talk to physicists and everyone gets frustrated..lol!

My most recent example is magnetism. I'm reading that one of the objections to Newtonian explanations of gravity is that it is force acting at a distance - and I realize that that is exactly what magnetism is. So then I'm googling and Youtubing in a fairly haphazard way.

And I get to the point of realising that there is very little discussion of what a magnetic field could be made of. How can something exist that is not made of anything? And, understandably I guess, the physicists and mathematicians tend not to be particularly long suffering. It's not a scientific question. Or, if you studied the maths and understood it you'd gradually find you're more comfortable with the existence of non-physical fields. Or, the maths works if we assume it's made of photons, and although it's not really photons, it's virtual photons, because that's how the maths works.

I get it that physics and maths need to be practical. Measurement and prediction are real - wanting to intuit directly some kind of basic reality is a kind of wrong headedness. But still ... trying to comprehend a magnetic field is irresistible.

 

[fresh_42]

My most recent example is magnetism. I'm reading that one of the objections to Newtonian explanations of gravity is that it is force acting at a distance - and I realize that that is exactly what magnetism is. So then I'm googling and Youtubing in a fairly haphazard way.

Here is a nice interview with Richard Feynman about such questions in general, and magnetism in particular:

How can something exist that is not made of anything?

How can it fail to exist if we can measure it and see what it does?

 

[droogiefret]

Thank you - yes that sums up the problem nicely. Feynman is being as patient as he can be. Yet I'm sure the interviewer is thinking that Feynman has avoided the core of his question. I guess it's a language problem.

 

[phinds]

I guess it's a language problem.

Depends on what you mean. It is NOT a problem w/ the English language, it is only a "language" problem in the sense that math is not English.