This is probably one of the oldest videos with Feynman as a teacher:
Messenger Lecture 2: a playlistHe talks about the relationship between mathematics and physics. Recall that Feynman wasn't really happy at Cornell. But I think that the lecture is very charming, isn't it?
Update: I was an idiot. The Messenger Lecture was from 1964 when Feynman was already happy at Caltech - for more than a decade. ;-)
In the first ten minutes, he explains e.g. how the "1/R^2" scaling in Newton's law becomes less mysterious with Le Sage's theory of gravity (with the unbalanced inflow of particles from various directions). Except that he can also easily falsify the theory (because it predicts a new type of friction).
He emphasizes that it's impossible to understand the real meaning of the laws without the maths and talks about the laymen who search through a book after book, trying to find an explanation without any maths that even the best expositors were unable to previously offer. ;-)
Not much has changed since the late 1940s. It's like he was specifically designing these sentences for the typical dumb lay readers of math-phobic populists such as Lee Smolin who are eager to convince everyone that she doesn't need to master the maths well because the key things are about philosophy, anyway. Well, they're definitely not. The maths underlying physics is not just a language, he says: it is also a method to think.
Feynman is a bit sloppy about the infinitesimal vectors needed to prove that a central force keeps a constant area per time (angular momentum conservation). But with the analytic approach replacing the geometric one, one can afford to be more stupid. :-)
Feynman divides the schools of teaching to the Babylonian and the Greek attitude. The Babylonians would learn a lot of examples while the Greeks would build on a few axioms, an approach that modern mathematics follows. You can still start at different places; no axioms are at the "ultimate" bottom because of the interconnections. It's helpful to remember many results on par with the axioms, in the Babylonian way, he argues.
He also gets to a cool feature of physics: we often derive principles that are more important and universally valid than the derivations! Mathematics doesn't have that: theorems usually don't appear where they're not supposed to be. :-) For example, the angular momentum conservation law started from "wrong" Newton's laws but has survived all the subsequent revolutions.
Feynman shows that laws in physics often have qualitatively different yet equivalent descriptions: Newton's force acting at a distance vs a field-theoretical description with a gravitational potential vs the principle of the minimal action. In the cases when the different philosophies are inequivalent, we have to leave the decision to Nature because all of our experience proves that philosophy never works.
The state-of-the-art theories of his time combined the minimum action principle with a local description - he meant quantum field theory in a path integral approach. ;-) That's still true today except that many equivalent descriptions of systems, including quantum gravity, are also local in the world sheet, the boundary of spacetime, or other spaces.
Feynman was amazed (5/6, 4:00) that the laws of physics can always be expressed in so many ways. With the dualities in string theory, he would surely be much more stunned than ever before. Well, this clearly seems to be a feature of the mathematical structures that are relevant for Nature. They can be studied from many different angles.
Physics and mathematics help each other but differ: physics needs a connection of the concepts with objects in reality. A related fact is that physicists are interested in specific cases, not abstract things - for example, physicists want to talk about the gravitational force, not an arbitrary one. He ends 5/6 with a joke about 4 dimensions.
Extreme rigor or focus on axioms is not too useful in physics but there's no reason to criticize mathematicians because they're free. Finally, in the last part, Feynman revealed that he was troubled that even physics of finite regions needed an infinite number of degrees of freedom. Well, yes, in quantum gravity, it's guaranteed that the entropy is bounded by the area, so there is a sense in which the number of degrees of freedom must be finite. Feynman would prefer the philosophy about a checkerboard but he still realizes that one shouldn't have prejudices.
Feynman says something that we recently ran into (during a discussion of a review of McMahon's book): without the listeners who actually mastered the maths, any qualitative talk about the philosophy and intuition behind the laws of physics is talking to deaf ears.
See also: Mathematics and physics: boundaries and interactions