[Micaiah_Chang]

There are some skills in physics which are sort of foreign to the math method of problem solving. (Speaking as a physics major)

For example, picking a nice reference frame in simple mechanics problems is something that a physical intuition is good for. Same with spotting symmetries in an EM problem. Also, in physics you need to have a good grasp of what to ignore because they have only a small effect on the solution or because it operates on a different scale (fringing effects, transient solutions in ODEs), which often relies on a very hand-wavey type of reasoning.

Essentially, physical intuition often does not map to mathematical intuition

I have not taken enough math to actually speak for them; this is mostly gleaned from talking with math major friends and my own speculations.

[jules]

I majored in math and physics. The same kind of intuition that you use for physics you also use for math. In physics you may call it choosing a reference frame, in math you call it choosing a coordinate system. At least for me it's exactly the same kind of thinking. You hit the nail on the head here though:

> Also, in physics you need to have a good grasp of what to ignore because they have only a small effect on the solution or because it operates on a different scale (fringing effects, transient solutions in ODEs), which often relies on a very hand-wavey type of reasoning.

Physics is often taught in an exceedingly vague and hand-wavey way. Physics students have to learn to ignore that nagging feeling that something is not quite right. This is impossible for a mathematician. A mathematician wants to cleanly separate the math from the problem that is being solved using math. The problem specification consists of a list of assumptions, and the solution of the problem consists of 100% rock solid math. Physicists weave the two together, so that in the end it's often not clear what is actually being assumed. Furthermore, it's usually not explained based on which experiments those assumptions are justified. A counterexample is special relativity. There it's clearly assumed that the speed of light is constant, and the experiments on which that assumption is based are explained, and from there it's mostly logical deduction. In other topics that is sadly not the case. I would love a physics education where you start with the experiments and work from there, instead of saying "Bam! Here are Maxwell's differential equations. Now deduce things from that based on hand-wavey arguments". I don't mean having the students perform the experiments themselves, just describe what somebody else did and what the results were, and why that led people to believe that the laws of physics are as they are. To make time for that, we should remove the endless by hand solving of special cases of special cases. We live in the 21st century. Instead use numerical methods everywhere, which easily tackle the general case. Got n electrons with initial positions and initial velocities, and you want to see what happens? No problem.

/rant

[gsteinb88]

Thank you so much for this; you just described my undergraduate experience in physics perfectly, especially the endless nagging feeling that something isn't quite right -- I never managed to make that feeling go away. I ended up taking a bunch of maths courses towards the end of my undergraduate and can still remember one particular functional analysis lecture on dual spaces where I finally figured out what the hell those dual spaces in quantum were about. Sadly, this was after I had struggled through and finished the full quantum sequence.

Ironically though, I ended up drifting into the EECS department to do applied physics and I've found an environment much more similar to mathematics (and to the experiment-based approach you advocate) than to the physics department -- when you're trying to build systems rather than just solve problems, you can't just wave your hands. Instead, you have to pick apart your assumptions and figure out why you can ignore certain things and not others.

[jpmattia]

> "Bam! Here are Maxwell's differential equations. Now deduce things from that based on hand-wavey arguments". I don't mean having the students perform the experiments themselves, just describe what somebody else did and what the results were, and why that led people to believe that the laws of physics are as they are.

It's odd that you chose that example, because there is a rich history leading up to Maxwell doing his work, then Hertz verifying it, then Heaviside refactoring the notation.

Perhaps the deeper issue is that there are only so many lecture hours in a semester for adding these details, and you gotta start somewhere.

[pmoriarty]

Physicists seem very hand-wavey to mathematicians, and mathematicians seem very hand-wavey to logicians. If you want to learn what real formal rigor is like, study logic.