From the preface to Physics for Mathematicians:
In addition to presenting the advanced physics, which mathematicians find so easy, I also want to explore the workings of elementary physics, and mysterious maneuvers — which physicists seem to find so natural — by which one reduces a complicated physical problem to a simple mathematical question, which I have always found so hard to fathom.
That’s exactly how I feel about physics. I’m comfortable with differential equations and manifolds. It’s blocks and pulleys that kick my butt.
12 thoughts on “How mathematicians see physics”
As an engineer with a mathematician girlfriend, I often can’t understand how simple problems, where we easily say “X quantity is conserved, plug equations together and solve a polynomial” can be hard for her to grasp. This book seems interesting, could you review it later?
Kim: I doubt I’ll have the time to read much of the book any time soon, so I wouldn’t expect a review.
What I found hard about elementary physics was the implicit assumptions. If you’re naive and take everything on faith it’s easy. And if you’re sophisticated enough to understand the justification of the assumptions, it’s easy too. But when you’re in the middle, it’s confusing.
I now understand some of the things that bothered me as a student, such as why some approximations are justified and others are not, but in college I only knew enough to be puzzled.
One of the major problems is caused BY mathematicians. The late 19th century program of “rigorizing” calculus took all the connection between physics problems out of the concepts of calculus and analysis.
Not until Abraham Robinson’s NSA (and later IST from Edward Nelson) did we put back the concept of an infinitesimal number that is so essential to the connection between physics and its mathematical models.
In physics, there IS NO LIMIT AT dx -> 0. Such a limit contains no atoms, no electrons, nothing. The point of “infinitesimal” is that it’s a model for “small enough” which brings up the point that really physicists ignore things that become non-dominant when size decreases to a small enough level.
As soon as you (the physicist) need to see more details than can be seen with such a simplified model, you have to change your model to re-incorporate the missing effects.
Questions that are popular with mathematicians, like whether a solution to a certain PDE is valid when the initial conditions are discontinuous are NON PHYSICAL. The world is entirely discrete atoms. The PDE doesn’t exist.
I had to laugh — the Amazon.com page for this book that you link to above asks “Looking for the audiobook version of this book?”. Somehow I don’t think that would be a good idea…
Dave: That’s pretty funny.
You made me laugh with this post, John.
Another quote from Spivak: “I could easily understand symplectic structures, it’s elementary mechanics that I don’t understand.”
I am so with you on this. The main reason why I’m not a physicist (apart from the three hour lab classes on Friday afternoon) is that I could never get the hang of the way that physicists use maths, even in basic stuff like Newtonian mechanics.
The way I remember it, though, was that physics lectures tended to derive all of the standard equations, and then the tutorial classes would give you a problem (e.g. the famous “drop a chain on a scale” problem) where some key assumption would be broken. At least some of the equations would, as a result, be useless, but you’d never know which ones, because it was never clearly articulated.
I think that the underlying problem is one that Warren Siegel best articulated: Most physics courses/textbooks are actually history courses/textbooks. You start with Archimedes and Newton because that’s where physics started, but you actually should start with Lagrange, Hamilton and Gauss.
I still remember the derivation of the the Rayleigh-Jeans law from my first year lectures. The Rayleigh-Jeans law is not part of the theory of physics. It is part of the history of physics. That alone makes it worth interesting, but does it really need to be covered in first year?
I compare it to the way that the theory of classical music in the European tradition is taught. Even though Palestrina lived before Bach, you study Bach (chorale harmonisation) before Palestrina (species counterpoint). By doing Bach first, you completely avoid the harmonic mistakes that Palestrina could only fix by trial and error.
If physicists taught music theory, you’d be forced to make harmonic mistakes before you got to learn any harmony.
In class I often heard this referred to as physical intuition. Often the math isn’t the hard part it is having an idea what should be happening so you know which combination of equations are relevant to the problem. Having an intuitive grasp of expected orders of magnitude help too so you know when you are roughly right, which factors can likely be ignored from your model etc.
I am a physicist. At least I try to be.
Much of learning physics is about getting a grasp of a series of different worlds. What happens in a crystal? What happens in a plasma? What happens in a supernova?
Each time, we take the same basic principles of physics – energy, mass, momentum, light, relativity, quantum physics – and get a feel for how they fit together in this scenario. A star holds up because the nuclear reaction in its core balances the crush of gravity. A table holds up because the electric fields of its atoms lock each other in place. By doing this over and over, we learn to judge the scales at which different rules apply, where different regimes exist.
The skill of a physicist is to look at the system and have an idea of what can be discarded in order to make the problem tractable. Maybe the chicken might as well be spherical. Maybe not. This gives us a way to start testing these expectations, by deriving some result or relation and seeing if that holds up.
The universe turns out to be doing some pretty impressive calculations continuously, femtosecond by femtosecond, using every particle as its own computer. So very few things, even individual molecules, are hard to solve for from first principles. Instead we have to try to approach the problem from a number of angles, looking for a schema that discards enough but no more, and leaves the essence of the problem intact.
I confess this is not always easy. I am an intuitive thinker, and sometimes my intuition starts far too complex and after failing to solve a problem I have to go back a few steps and start from an even simpler standpoint. When I later add complexity back in, I have the simpler version to compare to, a specific case as a reference in the generalisation. Often the simpler case has more mathematical rigour, but is less realistic, so we use the tools and language of the more rigorous case to explore the more complex one.
A final note on the historical angle; I agree that the historical approach often fails to teach the abstractions clearly enough, but often the language used in a field is determined by its history. Astronomy is particularly rife with standard candles and parsecs. So if you do end up in a field, you have to learn the history anyway, just to understand what’s going on. Maths loses most of this because once you have a generalisation it can completely supercede the previous special cases, and also because it is a field that generally builds up from the simple to the complex. Physics begins with whatever is human scale, and excavates from there, usually from the complex, murky and chaotic down to the elegant.
Physics has a frightening tendency to throw nonlinearity into systems that seem linear. So you take a reasonable model and apply it to a different material, or a different temperature regime, and suddenly you’re being asked at a conference how you deal with phenomenon X or the Y effect which you hadn’t even heard of. The limiting factor of the success of a line of research, then, becomes its capacity to expand to allow for these oddities of different applications and continue to be useful and accurate.
As a person who went to school for math the farther physics classes got from direct sense experience the more easy it was for me to deal with. i.e. E & M way more easy than Classical Mechanics (at least newtonian mechanics anyway). Newtonian mechanics, specifically Newton’s third law: Impossible. Used only to wiggle out of argument like a B movie plot twist. Or at least that’s what it felt like. Actually all physics explanations felt backward to me: you’d start in a random place wander in seemingly arbitrary directions, then arrive at a result suddenly and without motivation. Whereas in math, you’d get Theorem, proof that ends with statement of theorem. Not to say every step in a math proof was obvious, but each step was pulled from something that we’d already proved, discussed, or was clearly derivable from the former.
Why does water pressure only depend height of the water? Something to do with the column of water pressing down on it? Folks go on about inserting and removing straws as if that’s supposed to prove anything…
When I think really hard about it on my own, this very slowly makes sense because the water is being acted on by vector field where all the vectors are pointing in the same direction all with roughly the same magnitude. Consider water being acted on by a general vector field: how would water shape itself then? What happen to the density of the water if I increased the “gravity” at certain points? The reason why water is more dense at the bottom isn’t just because of the water on top being pulled down by gravity, it’s also because the container below is pushing up on the water farthest down.
For me this problem becomes extremely clear when we consider how the water pressure should behave by considering different vector fields to a body of water, and containers.
So generally a major reason I get so confused by concrete physical phenomena is because the assumptions are rarely pointed out or questioned.
Another gripe about physics pedagogy is that everything is supposed to be “intuitive” which is really frustrating to me on two levels:
1) If stuff like Newtonian Mechanics is soooo intuitive why did it take well over 1600 years to develop eh?
2) If this was intuitive, I wouldn’t be sitting in your class. duh.
When I took Real Analysis nothing was intuitive. Everything was weird and hard. Over time with much practice manipulating these weird hard definitions I developed intuition about what things like Epsilon Delta proofs were doing, and now they’re perfectly intuitive. But that’s now, not when I first learned it. Which is why, I wish physics instructors took an anti-intuition approach, and relied on developing “intuition” later.
(My apologies for the rant: I have many feels related to physics education vs math education)
I guess I am in the physics camp, though I have studied a fair amount of pure maths. I cruised through physics until I hit the later QM classes. I did this through “intuition”, I guess I naturally tried to explain everything around me so I had built this intuition over many years. I could visualize the problem and sort of let that direct the maths for me.
However pure maths took me a long time to get used to. The way in which I did physics, derive it all from what made sense to me, just didn’t apply here because nothing made sense. I would try and visualize the problem, however I just couldn’t see it in action (I did very well in complex analysis, I guess because it was easy to “see” it). For me, those hand wavy calculus derivations in classical mechanics worked so well because I thought like that anyway, whereas I felt as though real analysis was trying to break my mind, my hand wavy arguments no longer held up nor really helped me to understand what was going on.