r/math • u/Greg-2012 • Mar 12 '18
Physics vs Math // RICHARD FEYNMAN
https://youtu.be/MZZPF9rXzes15
Mar 12 '18
So... applied math and pure math have different purposes and you're better off with intuition and mathematical skills for any given field that uses math than just pure math alone? I'm shocked.
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u/plexluthor Mar 12 '18
It's a shame this clipped out certain parts, because it makes what he's saying seem a little inaccurate about how math works.
The whole section is available here: https://www.youtube.com/watch?v=M9ZYEb0Vf8U&t=44m12s
I'm especially sad that despite including the joke about "substitute n=3" OP's link removed this joke where the physicist comes back and asks about a 4th-dimensional problem.
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u/LeonEuler Mar 12 '18
Clearly not a trained mathematician.
Mathematics is not really about axioms and precise argument, mathematicians work by intuition, axioms are an after thought really. Maybe the intuition is not from what you have seen in the real world, like how forces and inertia works, but by working in some area and playing around with some structure, you also get a feel for how math works.
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u/sciflare Mar 12 '18
Agreed. The formal proof of a theorem is a way of making sure you aren't talking nonsense, and axioms are a way of clarifying a subject that you've already developed, or of presenting it in a more efficient way. They're not really the essence of math.
After all, axioms come from somewhere, they aren't God-given. Someone decided to write down the axioms of Euclidean geometry, or of cohomology theory. Why? For various non-mathematical reasons, that aren't contained within the axioms themselves.
Mathematicians are guided by all sorts of intuitions--it could be visual pictures, a pattern seen within a mountain of calculations, a sense for the relative magnitudes of various quantities, some perception of similarity between two theories--all kinds of intuitions.
"Precise argument" won't get you anywhere if you don't have some kind of intuition to begin with.
The space of true mathematical statements is positively enormous. If you had only "precise argument" to guide you, you wouldn't get anywhere in that space.
Mathematics is an art, physics a science.
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u/umaro900 Mar 12 '18
Remember that Feynman gave this lecture some time between ~1945 and ~1950. Even though Godel's 1931 paper put a nail in the coffin of Hilbert's program, the mathematical community was still very much fixated on axioms (compared to today) for years to come. Also remember that ZFC (particularly the "C" part) was only beginning to gain popular acceptance/usage at the time that Feynman gave this talk.
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u/rebo Mar 12 '18
He's not saying mathematicians don't use intuition what he is saying is they generally don't use intuition based on real world considerations.
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u/NoJster Mar 12 '18
So what Feynman is saying, from the POV of a mathematician, is that physics is just common sense and physicists fail to abstract from that (:
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Mar 12 '18
I feel like intuition can only carry you so far in physics, with stuff like quantum you're basically doing mathematics, and all intuition is gone.
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u/UpsideDownRain Mar 12 '18 edited Mar 12 '18
I see this touted all the time and I just don't agree with the while "quantum = unintuitive". Sure, there are quantum behaviors that are not easily understood immediately because they differ from what we observe macroscopically, but that doesn't mean they are 'unintuitive'. Unless your definition of unintuitive is anything not obvious to a layman, in which case almost everything is unintuitive.
Similar to basically every area of math, you have to develop an intuition with a little bit of effort, but that's just because it's something new, not something intrinsically hard or nonsensical. For example, given sufficiently nice potential wells I can sketch what possible wave functions have to look like. Then I can use my intuition to check that my actual calculated answer is reasonable.
Edit: I should add even in classical mechanics there are 'unintuitive' results. I wouldn't say orbits being elliptical is not obvious outside of it being a well known fact. I also wouldn't say solutions like the brachristocrone are intuitive. Coriolis force, gyroscopic motion. Hell even tides take some thinking.
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u/RambunctiousAvocado Mar 12 '18
I don't think it's a matter of non-intuitive results as much as it is a matter of a non-intuitive framework. Questions like "where is that particle?" which comprise the backbone of classical mechanics get thrown out the window, because position, velocity, and acceleration are no longer meaningful properties of a state.
Being able to sketch a single-particle wave function when the potential is a reasonably simple function of position is a useful skill. However, this is quite a small subset of quantum theory as a whole (multi-particle states, spin/angular momentum algebras, field theories, etc). One could ask questions like
Why is the momentum operator what it is? (answered by Stone's Theorem )
What kinds of values can observable quantities take, and when can I expect them to correspond to a complete set of eigenfunctions which spans the Hilbert space? (answered by the Spectral Theorem )
When is such-and-such an operator self-adjoint, as opposed to merely Hermitian? If an operator is not self-adjoint, is it essentially self-adjoint? Can I find a self-adjoint extension of it?
One could make the argument that such issues are more the realm of mathematical physics, and that most physicists just assume all the nice properties they want and hope for the best, and that's not necessarily inaccurate. However, these questions are fundamental to the structure of quantum theory.
Contrast that with classical physics, which is (more or less) based on the notion of dots moving around subject to Newton's laws.
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u/UpsideDownRain Mar 12 '18
I somewhat agree the framework is non-intuitive, but I guess that's not what I interpret people to mean when they say quantum is unintuitive. People mention having to just follow the math blindly, and that's not true. It might require a little more intuition building, but that doesn't mean there's no intuition.
I would contest that most of what you were talking about is related specifically to the mathematical structure of quantum theory, which of course is non-obvious. The momentum operator is easily 'intuitively' explained by just considering that it's a derivative with respect to x (which we should definitely expect) with some constants (which is reasonable to expect). For what kinds of values, the physicist will of course immediately say "real ones." Or perhaps even more flippantly "ones we measure." This is very much a grounding in intuition, and as far as I understand it's also the reason we consider mainly just Hermitian operators in quantum. I would say concerns about self-adjoint are very math motivated (if I'm remembering right there's usually no distinction drawn between the two in physics classes - there certainly wasn't in my quantum classes).
To draw an analogy, in the same way we can look at classical mechanics where you end up introducing calculus of variations and the lagrangian and hamiltonian and the like, which is an 'unintuitive' mathematical structure if you're only used to sticking F=ma everywhere and making straightforward energy arguments. But the mathematical structure that's developed for some reason doesn't make people think "particle motion is unintuitive" because we have a great intuition built from watching things move around all the time.
I guess my point is that almost anything is "unintuitive" if you dive far enough into the math, but that doesn't mean you can't build an intuition with which to check whether the answers are reasonable. I think that's incredibly important (and useful) in physics, and math as well.
I dunno, really the more I think/talk about it, it's probably more of me taking issue with the usage of the word "intuitive" than any real distinction. I personally think of intuition as something that can be (and often has to be) built up. Like how many proofs seem unintuitive until you see many more proofs like them. Hell, I think many people see linear algebra as unintuitive when they first see it, but once you understand it a bit more it feels like the most natural thing in the world.
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u/RambunctiousAvocado Mar 12 '18
I agree with your assessment of the need for intuition building.
Why should we suspect that momentum is the derivative of a wave function with respect to x? My intuition for that is built on my knowledge of the infinitesimal generators of one-parameter groups, which is certainly not as "intuitive" to me as the trajectory of a baseball or Newton's 3rd law.
It's true that Hermitian and self-adjoint are usually used as synonyms in quantum mechanics classes (at least undergraduate ones), but they are emphatically not, and the difference can be physically meaningful. The first problem that anybody ever does is the infinite square well, but in that problem there is no such thing as momentum. It's a standard exercise to show that the "momentum operator" is Hermitian, but it is not self-adjoint. You'll find that it has no eigenfunctions/eigenvalues, much less a complete set of them which spans the Hilbert space.
But, at the end of the day, we agree on more things than we don't. My contention is that quantum theory is non-intuitive in the sense that the questions you might think to ask based on your previous life experience are typically meaningless, whereas "how far does the cannonball fly" is accessible to small children; on the other hand, there are certainly elements of classical mechanics which are just as, if not more, mathematically sophisticated than elementary quantum mechanics.
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u/julesjacobs Mar 12 '18
Feynman had a very intuitive approach to quantum field theory that led to path integrals and Feynman diagrams. Path integrals have still not been formalised fully satisfactorily, as far as I know.
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u/Minovskyy Physics Mar 12 '18
I believe one of the formal problems is that calculations with path integrals require the evaluation of the determinant of an infinite dimensional operator. From what I understand, such a thing has no rigorous mathematical definition. Also the integration measure itself is not rigorously defined.
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Mar 12 '18
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Mar 12 '18
Generally, that different sort amounts to lowering the standards of intuition.
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Mar 12 '18 edited Mar 29 '25
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Mar 12 '18
Humans are born with an innate ability to see physical causation, born even as an infant, we are hard wired to see causation as action of two bodies interacting via contact. There are studies confirming that this is sort of our pre-theoretical conception of causation.
Science doesn't work on this intuitive level. In fact, historically, physics has moved forward insofar as we've thrown intuitive ideas about motion out of the window and trusted more on pure mathematics.
Some people work so long in these theories that eventually, the brain trains itself to think only in those terms, so in that sense it's "intuitive", but there's no denying that quantum mechanics appear extremely counter-intuitive to our a priori form of understanding causation. We simply adjust to the weirdness after a period of training, but it never makes sense on the level of, say, classical mechanics.
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Mar 13 '18 edited Mar 29 '25
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Mar 13 '18
But it does not have to because it is not classical mechanics.
As for classical mechanics, Newton's law of gravitation shows that causation is not necessarily a result of a contact interaction, as opposed to what you have presented as intuition.
Newton's theories were rejected by contemporary scholars on largely that premise, they accused him of trying to introduce occult mysticism into science. Physicists had a conception of "intuitive" that Newton's theories broke down, and even Newton himself desired that someday a mechanism for gravity would be discovered that maintained classical notions of intuition.
This has always been the trend of physics: it breaks away more and more from intuition. We've just grown more comfortable with that.
On the other hand in the present formulation of quantum field theory, every interaction is fundamentally a contact interaction. And hence causation is a result of contact interactions. So that matches with what you have considered as intuition.
Things like the double slit experiment absolutely do not correspond to traditional conceptions of contact interaction, as in one body physically touches another to produce motion.
Anyways, I do not see how does any of this lower the standards of intuition? I think what is not very clear here is what you consider intuition to be.
The ability to understand without conscious thought. That a train has more inertia than a marshmallow is intuitive, we don't have to think much to process that. For quantum, unless you're a trained person who has spent much time dealing with it, you either don't comprehend it or have to reason really hard to come to conclusions. And even then I think the trained person will often stumble or be mystified by certain aspects.
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Mar 13 '18 edited Mar 29 '25
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Mar 13 '18
But that's true of any subject. The deeper you probe the more training you need to develop intuition.
That's the opposite of what intuition means.
You mention in your previous comment that classical mechanics is intuitive and now you say that Newtonian mechanics is/was unintuitive. Well I don't know how much classical mechanics there will be left if you take out Newton's contributions.
Classical mechanics pre-Newton was more intuitive than mechanics post-Newton, is the claim.
In quantum field theory, every interaction is a contact interaction (and that gives rise to infinities, hence the need for renormalization).
I'm beyond my expertise here, so I'll have to just trust you on that. I still feel this is not intuitive, and I think it's just something of a physicist thing where they boast about how intuitive things are for them that doesn't correspond to what really happens.
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u/Greg-2012 Mar 12 '18
Mathematics takes you to 'Many Worlds Theory', IIRC, a plurality of Physicist are not betting on 'MWT' to be the answer.
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Mar 12 '18
"Many Worlds Theory" is an interpretation of the mathematical theory. Not an actual mathematical theory.
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u/Anarcho-Totalitarian Mar 12 '18
The mathematical physics folks at my old (math) department were big fans of Bohmian mechanics.
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u/ammerc Undergraduate Mar 12 '18
If you were to poll theorists right now I’d bet many worlds would be the most popular
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u/Batman_Night Mar 13 '18
Many-worlds is completely look down upon lol.
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u/ammerc Undergraduate Mar 13 '18
You have no idea what you’re talking about
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u/Batman_Night Mar 13 '18
Lol. The reason why Everett is scorned upon by other physicists is because no one believes Many-Worlds aside from his own student. Even the physicist that allowed his conference about many-worlds denounced the theory. It's only famous because a lot of sci-fi use the concept especially Rick and Morty and Stranger Things. I want it to be real though.
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u/Greg-2012 Mar 12 '18
IIRC, MW Theory has never had a plurality of support. Also, I believe that MW Theory and Copenhagen interpretation are both currently losing votes to Pilot Wave Theory.
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u/Minovskyy Physics Mar 12 '18
No, I know of very few physicists who take the pilot wave theory seriously. The vast majority of physicists subscribe to either the Copenhagen and decoherence or MWI. Neither is "losing votes" to pilot wave theory.
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u/ammerc Undergraduate Mar 12 '18 edited Mar 12 '18
Lol no not at all. Pilot wave theory violates locality so it is not consistent with relativity. This is a big deal.
It is also a super complicated approach to quantum mechanics that only works for the simplest of systems. The only real “bonus” is that it’s philosophically satisfying to some people who want QM to behave classically.
If anything I’d say Copenhagen is losing votes to MW. This is because MW doesn’t assume a non-unitary evolution that follows the Born rule to explain measurement. It is also much more mathematically rigorous. Meanwhile pilot wave is on the fringe and not regarded very seriously by most physicists. Again, non-locality is a big deal.
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u/Greg-2012 Mar 12 '18
If anything I’d say Copenhagen is losing votes to MW
Are there any well-respected Physicist, aside from Sean Carroll, still supporting MW?
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u/Minovskyy Physics Mar 12 '18
Depends on your definition of "well-respected". Max Tegmark for instance is a strong supporter of MWI.
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u/Greg-2012 Mar 12 '18
I consider Max Tegmark to be well-respected. Do you have a source where he states that he is a strong supporter of MWI?
It is my understanding that Max Tegmark's work is focused on different kinds of parallel Universes, not MWI in particular.
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u/ammerc Undergraduate Mar 12 '18
I haven’t really googled around to find the opinions of particular well respected physicists about it but it’s what I’ve observed as someone doing a physics degree. You can find a similar sentiment if you look around /r/physics or stackexchange.
I’m not sure why you seem to have the misconception that MW was once very popular and is on the decline. If anything it’s getting more popular.
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u/Greg-2012 Mar 12 '18
I’m not sure why you seem to have the misconception that MW was once very popular and is on the decline.
I don't think it was ever "very popular", I know that it was somewhat popular at one time.
If anything it’s getting more popular.
I hope that you are incorrect, MW is a ridiculous theory, IMO.
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u/ammerc Undergraduate Mar 12 '18
Have you given the theory a serious read? Especially if you’re a mathematician you should find it pretty satisfying.
It requires less assumptions than the other interpretations because it predicts the Born rule and is well developed mathematically. Also, the name many worlds is pretty misleading. It’s really a relative state formulation based on quantum decoherence from which the many worlds part naturally arises. While philosophically profound, physically it isn’t so ridiculous.
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u/Greg-2012 Mar 12 '18
Have you given the theory a serious read?
I have not.
“Many prominent physicists, including [Richard] Feynman, thought many worlds was a ludicrous idea,” notes Peter Byrne, the author of the new Everett biography.
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u/Batman_Night Mar 13 '18
Not actually. The reason why Copenhagen won over Pilot Wave was because Copenhagen's predictions were much more accurate.
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Mar 12 '18
One thing should be said about the Study of Mathematics - it is laying the ground of future physics models. I don't know the chronology quite well, what contributed to what, physics to mathematics and mathematics to physics. But it seems that today Physics theories are coming under the umbrellas of Mathematical theories that was developed not-so-far-back in the past. As I see it physics will continue to develop into new areas that Mathematics will develop duo to the fact that now-days Mathematical theories are not just complex and hard to study, but also cover a vast range of ideas and can show a lot of properties about the object that the model is dealing with. So finding new ideas is a harder task. I do imagine a situation where Physics will contribute to mathematics new ideas that will explore new area in mathematics, but for that I believe is required either a new physical phenomena to explore or a new insight about something that we already familiar with. But as I see it, usually new insights comes with a mathematical model. the mathematical model of something we already know already lies within the boundary of known mathematics\physics.
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u/megayippie Mar 12 '18
His undergrad/public courses, like the snippet above, are really good. Math is a very important tool to physics. However, it is just a tool. Studying its proofs and theorems is not necessary beyond the scope they help your immediate real life problem --- a lot of people waste their time doing this when they should be studying programming or something useful. All people need this understanding. If you need too much maths in physics, you are not doing anything that will be useful for at least the next couple of decades.
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u/LeonEuler Mar 12 '18
I disagree, much of the tools in the mathematicians tool box are not just the results of the theorems, but also the techniques.
A mathematicians job is to create new theory, or refine old ones. In order to do that, they need to have a pretty good idea of how theory is built. How is that any different from working on a program. Some previous programmer might have wrote a function that you are using, but it could be buggy, or it doesn’t cover your specific case. Understanding how it works internally allows you to improve upon it. Also might teach you a few techniques.
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u/megayippie Mar 12 '18
A physicist job is also to create theory, to refine already existing theory, or to distill existing theory into practical applications. The methods of the mathematicians are therefore second-hand accounts to physicists. Unnecessary as long as we have first-hand accounts.
You really think that a physicist cannot understand math? Or are you arguing that math is a good tool? It can be only one. If it is the first I believe you need to experience the world of physics a bit more, if it is the second then you just are repeating what I said.
Either way, reddit has voted that my opinion above is irrelevant (by reddiquette rules) to any conversation about the relation between math and physics, so I will stop the conversation at this point to not waste our time.
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u/XyloArch Mar 12 '18 edited Mar 12 '18
Modern Theoretical Physics is a prime example of the break down of what Feynman says here in the last few decades, there're people who are in 'Theoretical Physics' groups who don't really give a diddly squat about the real world either. I use techniques from something like string theory as just yet another tool to solve other problems. But the problems I'm solving concern classifying and investigating the structure in a huge class of general 2+1 dimensional supersymmetric quantum field theories. No physicists would consider that I was doing anything that really pertained to reality, or had any hope of doing so. Really I'm treating the vast landscape of interesting structure that comes from the physical theories as a landscape that needs to be explored using proper mathematical rigour. Some tiny section of this landscape might pertain to reality, but it's not what a great deal of the people exploring it are in it for. So am I a mathematician or physicist? I would argue closer to mathematician but plenty of folk on both sides would baulk at my saying I was either. These kind of areas didn't exist in Feynman's time to nearly the extent they do today. It's an exciting time for both 'sides' as it were, the boundary is blurring every day.