During Richard Feynman’s Messenger Lecture Series on “The Relation of Mathematics & Physics” held at Cornell University in 1965, “The Great Explainer” addressed what he found to be the key differences between mathematics and physics. His thoughts are summarized below.

Differences in Epistemology

“Mathematicians prepare abstract reasoning that’s ready ‘to be used’ even though they don’t know what it’s being used for”

First, Feynman addresses the differences in the epistemological level of analysis between those studying mathematics, in particular singling out metamathematicians:

Mathematicians are only dealing with the structure of the reasoning and they do not really care about what they're talking about. They don't even need to know what they're talking about, as they themselves say, or whether what they say is true.

He next proceeds to describe the property of computability for formal systems and the theoretical possibility of human-made machines deducing theorems which the humans themselves are unable to understand:

Now, I explained that if you state the axioms to say "such and such is so" and "such and such is so", what then? Then the logic can be carried out without knowing what the "such and such" words mean.That is, if the statements about the axioms are true, i.e. carefully formulated and complete enough, it is not necessary for the man doing the reasoning to have any knowledge of the meaning of these words. He will be able to deduce, in the same language, new conclusions. If I use the word triangle in one of the axioms there might be some statement about triangles in the conclusion. Whereas, the man who is doing the reasoning, he might not even know what the triangle is! But, then he can read his thing back and say "oh, a triangle, that's just a three sided what-have-you and so and so", and so I know this new fact.In other words, mathematicians prepare abstract reasoning that's ready "to be used".

This in contrast with the epistemological level of analysis in physics:

The physicist has meaning to all the phrases and there's a very important thing that a lot of people who study physics, but don't come from mathematics don't appreciate: That physics is not mathematics and mathematics is not physics. One helps the other.But, you have to have some understanding of the connection of the words with the real world. If necessary, to at the end translate what you figured out into English, into the world of blocks of copper and glass that you're going to do the experiment with, to find out whether the consequences are true. This is a problem which is not a problem of mathematics at all.I've already mentioned the only other relationship that.. of course it obvious how the mathematical reasoning which have been developed are of great power and are in use for physics. On the other hand, sometimes the physicists' reasoning is useful for mathematicians.

Feynman stops there without further explanation, but a relevant example to include here is the work of Edward Witten on the positive energy theorem for which he was awarded the Fields Medal. In the paper On the Work of Edward Witten, mathematician Michael Atiyah later described its significance to mathematics:

“His ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems… [H]e has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics.” — Michael Atiyah

Differences in Applicability

“Mathematicians like to make their reasoning as general as possible”

Feynman moves on to humorously discuss the applicability of mathematics, contrasted by the interests of most physicists:

If you say "I have a three-dimensional space" [...] and you ask mathematicians about theorems then they say "now look, if you had a space of n dimensions" then here are the theorems". "Yeah, well I only want the case of three dimensions..." "Well, then substitute n = 3!". It turns out that very many of the complicated theorems they have are much simpler because they happen to be special cases.The physicist is always interested in the special case. He's never interested in the general case. He's talking about SOMETHING. He's not talking abstractly about anything. He knows what he's talking about, he wants to discuss the new gravity law, he doesn't want the arbitrary force case, he wants the gravity law!And so, there's a certain amount of reducing because the mathematicians have prepared these things for a wide range of problems which is very useful and later on it always turns out that the poor physicists has to come back and say "excuse me, you wanted to tell me about these four dimensions.."

On Intuition vs Rigor

“The poor mathematician has no guide but precise mathematical rigor and care in the argument”

Feynman next addresses the process of discovery in both subjects, emphasizing the advantage physicists have that their subject is, in some essential sense, applied rather than purely abstract:

When you know what it is you're talking about, that these things are forces, these are masses, this is inertia and so on, then you can use an awful lot of common-sense, seat-of-the-pants feeling about the world. You've seen various things, you know more or less how the phenomenon is going to behave.Well, the poor mathematician he translates it into equations and the symbols don't mean anything to him and he has no guide but precise mathematical rigor and care in the argument. Whereas, the physicist who knows more or less how the answer can go, is going to come out and sort of guess partway and go right along rather rapidly.The mathematical rigor of great precision is not very useful in physics, nor is the modern attitude in mathematics to look at axioms. Now, mathematicians can do what they want to do, one should not criticize them because they are not slaves to physics. It is not necessary that just because this would be useful to you, they have to do it that way. They can do what they will, it's their own job and if you want something else then you work it out yourself.

Feynman here argues that because physics regards natural phenomena, humans have a better propensity for intuition in this domain. This somewhat in opposition to how the process of discovery of certain mathematical theorems have been described, including John Forbes Nash Jr.’ discoveries about nonlinear partial differential equations:

Mathematicians in the 1950s had known about relatively trivial routines for solving ordinary differential equations (ODEs) using computers. There were however, no established methods for solving nonlinear partial differential equations, such as those that occur during the turbulent motions of a jet engine.

[…]

By the spring of 1958 however, Nash was able to obtain basic existence, uniqueness and continuity theorems using methods of his own invention. Astoundingly, the methods involved “transforming nonlinear equations into linear equations, and then attacking these by nonlinear means” — something nobody had thought of before, “a stroke of genius” according to Peter Lax, who followed his progress closely. About the technique, Lars Gårding, a Professor of Mathematics at the University of Lund and specialist in partial differential equations similarly later declared “You have to be a genius to do that”.

On the Usefulness of Models

Feynman next discusses the usefulness of models in physics, and their seeming lack of usefulness in the process of making new discoveries:

The next point is the question of, whether we should guess when we try to get a new law, whether we should use the seat-of-the-pants feeling and philosophical principles, i.e. "I don't like minimum principle, I do like minimum control" or "I don't like action at a distance or I do like action at a distance". The question is to what extent models help. It is very a interesting thing. Very often models help, and very often physics teachers try to teach how to use these models and get a good physical feel for how things are going to work out. But, the greatest discoveries, it always turns out, abstract away from the model. It never did any good. Maxwell's discovery of electrodynamics was first made with a lot of imaginary wheels on idlers and everything else in space. If you got rid of all the idlers and everything else in space, the thing was okay. Dirac discovered the correct laws of quantum mechanics for relativity simply by guessing the equations. The method of guessing the equations seems to be a pretty effective way of guessing new laws. This shows again that mathematics is a deep way of expressing nature, and attempts to express nature in philosophical principles or in seat-of-the-pants mechanical feelings is not an efficient way.

On the Applicability of Mathematical Physics

Why should it take an infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do?

Curiously, Feynman goes on to predict that at some point in the future, the nature of the world will not be expressed in the language of mathematics. Rather, there will be some other method of expressing how nature operates, which requires less computation:

I must say, I've often made a hypothesis that physics ultimately will not require a mathematical statement. That the machinery will ultimately will be revealed. It always bothers me that in spite of all this local business, what goes on in, no matter how tiny a region of space and no matter how tiny a region of time, according to the laws and how we understand them today, takes a computing machine an infinite number of logical operations to figure out.Now how can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do? And so, I made a hypothesis often that the laws are going to turn out to be, in the end, simple like the checkerboard and that all the complexity is from sizeBut, that is of the same nature as the other speculations that other people make. It says "I like it", "you don''t like it". It's not good to be too prejudiced about these things.

On the Need for Mathematics

Feynman next both quotes Sir James Jean and refers to novelist and physical chemist C. P. Show’s famous work “The Two Cultures” in his discussion of mathematics in physics:

To summarize, I would like to use the words of Sir James Jeans which says that said that "The great architect seems to be a mathematician and for you who don't know mathematics, it's really quite difficult to get a real feeling across and up to the deepest beauty of nature."C. P. Snow talked about two cultures. I really think that those two cultures are people who have had and who have not had this experience of understanding mathematics well enough to appreciate nature once.It's too bad that it has to be mathematics and that mathematics for some people is hard. When one of the kings were trying to learn geometry from Euclid he complained that it was difficult and Euclid said that "There's no royal road to geometry".

On Communication

«It’s perhaps that the horizons are limited that permits such people to imagine that the center of the universe of interest is man.»

Finally, Feynman addresses the need for physicists to master mathematics in order to be able to make new discoveries about nature, stating that mathematics is essential to our current understanding of how the world works:

We cannot, as people who have looked at these things, a physicist cannot convert this thing to any other language we have. If you want to discuss nature, to learn about nature, to appreciate nature, it's necessary to find out the language that she speaks in. She offers her information only in one form.We are not so un-humble as to "demand that she change" before we pay any attention. It seems to me that all the intellectual arguments that you can make would communicate very little to deaf ears. All the intellectual arguments of the world will not convince those of the "other culture".The philosophers who tried to teach you by telling you qualitatively about this thing. Me, who is trying to describe it but who is not getting it across because it's impossible. We're talking to deaf ears.It's perhaps that the horizons are limited that permits such people to imagine that the center of the universe of interest is man.

Video

Video of Feynman’s lecture is available at the link below: