What mathematical techniques came after calculus?

April 21, 2021 12:10 PM

I was reading this article about Nobel prize-winning physicist Richard Feynman, and I came upon an interesting quote. I hope that the quote can be explained to me by someone with a background in math and/or physics.

The article was written by Stephen Wolfram, who was a PhD student in the physics department at CalTech, where Feynman was based at the time. Wolfram writes:

It’s kind of interesting to look at [Feynman's handwritten note with calculations related to a Feynman diagram]. His style was always very much the same. He always just used regular calculus and things. Essentially nineteenth-century mathematics. He never trusted much else. But wherever one could go with that, Feynman could go. Like no one else.

So this quote is interesting to me for a couple of reasons. I'm wondering what sorts of more-modern mathematical techniques were available to Feynman to use. What's more advanced than calculus? I'm also curious why Feynman didn't trust the newer techniques. Is there something that's unreliable about those methods? Feynman was a genius, so I think he probably understood the newer techniques very well, but he consciously chose not to use them. He must have had his reasons.

posted by alex1965 to Science & Nature (8 answers total)

I don't know if this covers everything the author is talking about, but Feynman was famous for using differentiation under the integral sign (an advanced calculus technique) to solve integrals that were traditionally solved using residue integration (which is a technique from Complex Analysis, a course that does come after Calculus.).
posted by wittgenstein at 12:21 PM on April 21, 2021 [2 favorites]

He taught himself, famously, calculus. Not in he read it in a book, but *rederived it* from first principles when he was a kid, developing his own notation, as the ideas he was playing around with in physics required it. He didn't realize what he had done until much later, when he encountered calculus in school.

With that story in mind, he knew calculus inside and out, and everything you could do with it, so perhaps trusted it more. He didn't stumble upon these other methods in solving problems, so probably didn't think they were necessary.
posted by chiefthe at 12:48 PM on April 21, 2021 [3 favorites]

General relativity requires concepts from linear algebra and topology, which were developed in the second half of the 19th century. Topology was a very live topic of mathematical research in the early 20th century, and when Einstein began applying it, he was using some very new mathematical ideas.

Quantum electrodynamics, which Feynman pioneered, is often formulated in terms of group theory. I guess group theory originated in the first half of the 19th century, but it's been under pretty intense development ever since and certainly represents "modern" mathematics.
posted by mr_roboto at 2:06 PM on April 21, 2021 [2 favorites]

In addition to complex analysis (Feynman in his memoirs mentions that contour integration is something that he particularly hated doing and would use lots of older techniques to avoid), tensor calculus is an example of something that is (a) not "regular calculus", (b) used constantly in physics, (c) from more recently than the 19th century.
posted by goingonit at 2:06 PM on April 21, 2021 [4 favorites]

It's probably worth noting (if you are not aware already) that Wolfram is famous for using computers to solve things. His work in cellular automata, where simple rules can create complex patterns, relies on computers, rather than doing calculus by hand. Things like self-organisation, such as the patterns given as examples in the talk, do not lend themselves to "classical tools".

There is a recurrent theme in the talk: the reluctance of Feynman to use a computer - and wanting to actually understand why and how these patterns form, rather than having a computer just run the numbers and create them. I think this is what Wolfram is implying with his 19th Century comment.

As a final note: Wolfram is good at marketing (see his book: "A New Kind of Science") and one of the strongest proponents of computer-assisted physic (as a developer of the Mathematica tool).
posted by swordfishtrombones at 10:44 PM on April 21, 2021 [2 favorites]

Lots of things are more advanced mathematically than calculus, particularly in the sense of being used by physicists. For example, although group theory was invented in the early 19th century it wasn't used by physicists until the 20th century, and isn't regularly taught to undergraduate physicists. In fact the application of 'advanced' mathematical concepts to domains like physics has transformed some specific areas of 20th and 21st century research. Those topics aren't intrinsically harder* than calculus to understand and use IMO, they were just developed later and are taught later in most places.

*For example, group theory can be approached intuitively using symmetries without requiring more than intermediate high school level mathematics, and many people can use calculus tools effectively without a full mathematical understanding of them.
posted by plonkee at 1:30 AM on April 22, 2021

On re-read, I agree that Wolfram is talking about Feynman's reluctance to use a computer to understand and develop physics.

But it might be helpful to note that in modern mathematics, the way that new research often works is that say some tools exist in sub-area A, someone works out a mathematical relationship between sub-area A and sub-area B, everyone uses that relationship to apply the tools in sub-area A to sub-area B and new discoveries are made in sub-area B. Since theoretical physics has mathematics as its language, if you can write some physics into sub-area B language those new mathematical tools and discoveries can help you learn more about that physics.

Feynman preferred to use a particular set of mathematical tools to understand and develop physics. In general, most people seem to find it easier to use a wider variety of mathematical tools, to understand and develop physics. Many new tools have been developed since calculus was applied to physics, and many more will be developed in the future.
posted by plonkee at 2:26 AM on April 22, 2021

Response by poster: I'm marking this question as resolved, but I was hoping to get some sense of why, exactly, Feynman didn't like the more-advanced tools that were available to him (complex analysis, tensor calculus, computer-based approaches, etc.).
posted by alex1965 at 7:59 AM on May 5, 2021