Defining the Physics Mindset

Going beyond just the math

In my July recap post, I linked to a post by Freddie de Boer with the title “You Can't Understand Physics Without Understanding Its Math and I'm Certainly Too Dumb to Understand the Math.” In it, he talks about an encounter with a mathematical physicist:

After doing the requisite amount of admitting that I knew nothing, I told him that I read physics books and watched YouTubes and got the standard popular science accounts of where the field was. I expected him to dutifully say “yes yes, quite right,” or whatever, to be condescending but vaguely in favor of learning about physics even as we both understood that I knew nothing. But this was not his reaction. On the contrary, he was utterly adamant: all of those popularizations, in his view, were so inherently distorted compared to the actual physics - compared to the math - that they were not just insufficient, but actively wrong. He was very, very convinced that all of the ways that physics is typically distilled down for a popular audience result only in greater ignorance. It wasn’t that the way the math was conveyed in math-free terms was wrong; it was that the very project of trying to explain any physics beyond the most basic Newtonian mechanics, without math, was inherently wrongheaded.

When he had told me about his focus, I had said that it was surprising that he was in a math program, not a physics program. But as he inveighed against the concept of a non-math approach to relativity or quantum mechanics, it became clear to me that there was no conflict because physics simply was math. It was like saying that you were surprised that someone was a Literature major instead of a Words major. And anything that attempted to explain physics without math was wrong, inelegant, and fundamentally impure.

As someone who writes pop-physics, I encounter a fair bit of this kind of thing, though this is a particularly extreme version. Most physicists will grudgingly admit that at least some popularizations are good and useful, though pretty much everyone in the field has at least one hot-button topic that they’re not willing to accept any simplification of. Then again, de Boer also says this guy had “idiosyncratic politics and a deep Euro-chauvinism that was both obnoxious and charming,” so maybe he was just kind of a dick in general…

Anyway, this stood out to me at the time, and I was reminded of it again when writing my big Two Cultures thing the other week. In that, I offered one of my usual comments about the justification of liberal arts education:

I think in the end it’s mostly about norms and practices, and habits of mind. That is, different disciplines go about things in different ways— literary scholars seek to describe every story in terms of power relationships between identity groups, physicists seek to describe everything in the universe as a simple harmonic oscillator. Being exposed to a wide range of those approaches helps build a kind of flexibility of mind that comes in handy, a recognition that people from different backgrounds may approach similar situations in different ways, and that they’re not necessarily wrong to do so. We have physics students study poetry not because English faculty can teach them something about writing that physics faculty can’t, but because seeing how English faculty approach thinking about poetry tells you something about the full range of ways of engaging with the world, and that can come in handy down the road.

I’m obviously being flippant in that passage, but I stand by this as a general idea, and it’s one of the things I say to students in the introductory engineering courses when I teach them. We have engineering students take physics classes from the physics department not because the engineering faculty can’t teach them what they need to know, but because there’s a difference in approach that’s useful for them to see. There’s a mindset to doing physics as a physicist that’s slightly different than the engineering approach to similar problems, and it’s helpful for them to experience both sides. If nothing else, some of them may end up deciding that they find the physics approach more congenial and switch majors, making everyone involved happier.

A big part of that physics mindset is, to my mind, stuff that stands in direct contrast to the attitude de Boer’s acquaintance displayed. That is, while you can’t do physics without being able to do the math, there’s more to being a physicist than just the ability to do the math. And I figure it’s worth a post trying to tease out a bit of that.

Substack’s built-in generator with the prompt “enrico fermi milking a spherical cow”

The ability to do math is obviously one of the prerequisites for being a working physicist, but to be truly successful and even celebrated as a physicist involves more than simply carrying through formal calculations. In fact, there are any number of faintly disparaging terms within physics for that sort of operation: physicists will speak of “plug and chug” problems, “grinding through” calculations, “brute force” approaches, and so on. These just-the-math-ma’am approaches are valid as far as it goes, and in many cases even necessary, but as a matter of disciplinary culture, we prefer methods that have a kind of elegance and insight that goes beyond just doing the math.

I would tend to divide the key elements of this into two aspects, though they’re not cleanly separable. For the purposes of this piece, I’ll call them “Useful Reductionism” and “Cultivated Intuition.”

“Useful Reductionism” is the tendency that is both core to the physics approach and the root of a lot of (self-)mockery: the extreme version is the old “assume a spherical cow…” joke. Nobody loves to tell that more than physicists, because it’s solidly in the “Funny because it’s true” category: when a physicist sits down to attack a problem, the very first step in the process is to abstract away most of the complexity. As Tom Swanson often points out, the spherical cow is actually a step up from the real starting point, namely the point-cow approximation.

This is an absolutely crucial step, though, and a process that gives the physics approach a lot of its power. Any real-world scenario will involve an enormous number of fiddly details— the exact size and weight of the cow, the color and patterning of its coat, the color of the barn, etc.— and you would drive yourself nuts trying to keep track of them all. The key realization embodied in the physics approach is that while any of these factors could matter, many of them won’t, or at least won’t make enough of a difference to care about at the level that you can measure in a practical experiment.

So, the starting point for physics is to strip the problem down to the absolute simplest, stupidest case you can imagine— point cows— and work out what happens there. If that tracks with experiment, great; if not, then you make the smallest addition to your model that you can— cows as spheres— and see if that works. And you keep doing that until you reach the level of precision you need.

This then feeds very neatly into the other aspect, what I called “Cultivated Intuition.” That is, physicists prize the ability to have a sense of what the result of a calculation “should” look like without doing complicated math. The goal is to build up a the kind of intuitive expectation that we have for everyday actions: as I sometimes put it to intro physics students, you may not yet know how to predict the exact trajectory of a ball launched at a certain angle at a certain speed, but if I lob a ball in your direction, you’ll probably be able to make a decent attempt at catching it.

I say “Cultivated Intuition” here because this isn’t purely intuitive, because a lot of the situations physicists care about are not, in fact, things we encounter in everyday life, for which we have any useful pure intuition. They’re more abstract or artificial, but through enough experience dealing with simple cases, you can build up a sense of how things ought to work, and make a rough prediction of the general trend.

This process is aided immeasurably by the reductionist approach, which often simplifies things to a point where analytical solutions— equations you can write down with a pencil and paper— can be found and analyzed. This lets you get a sense of how things scale in various limits, and you can combine them in ways that let you make reasonable predictions. If considering a cow as a point predicts that output should increase as some quantity increases, but considering the cow as a sphere predicts it should decrease, you can play those expectations off against each other to predict that there will be an optimum value for a given size of cow. If you’re clever enough, you can even make a reasonable guess as to where that optimum value might be.

Same prompt, this one from Dall-E

This kind of cultivated intuition is extremely useful in two ways: first, if you’ve got a good sense of how things work, you can save yourself a lot of work doing complicated calculations that would be a waste of time. If your intuitive model lets you estimate that whatever effect you want to study will only show up at energies a million times greater than you can generate experimentally, you don’t need to expend time and resources working out whether he limit is really two million times greater or only half a million. You move on to something else that might produce more plausibly testable results.

It’s also helpful to have an intuitive sense of the physics involved as a check against something going wrong. If you only know the math, the only thing you can do is run the calculation and report the result, which might very well be wrong, particularly in cases where the calculation is very demanding or resource-intensive. If you’ve got some intuition to bring to bear, though, you can sometimes look at the output and say “Well, that can’t be right…” and go hunting for the error.

Both of these draw very heavily on processes that are very close to the simplifications disparaged by de Boer’s fellow party guest: a lot of analogies relating complex systems to simpler ones for which we have solid intuition. This kind of thinking is central to one of the most practically impactful calculational techniques in history, the Feynman approach to fundamental physics, with all the little diagrams telling stories about ways an interaction might happen. As a certain type of physicist will take pains to explain, these interactions are not literally happening, but it’s a way of thinking about the physics that incorporates both useful reductionism— you only calculate the number of Feynman diagrams that you need in order to reach the precision you need— and cultivated intuition— the whole process of representing interactions with particle exchanges constrains the possible outcomes in ways that let people say “Even though we haven’t calculated this in detail, we expect the net effect to shift things in this way,” which is useful information to have.

The other figure, along with Feynman, who exemplifies the beyond-just-equations approach would be Enrico Fermi, who was famous for estimating real-world outcomes through simple approximations and basic intuition. One of the disappointing (to me) omissions from Oppenheimer was the famous story where Fermi got a good and fast estimate of the Trinity test’s blast strength by dropping small pieces of paper and measuring how far the blast wave blew them. This kind of thing is widely celebrated in physics, and helps define us in a disciplinary-culture sense.

To be sure, there are times when intuition fails— a detailed calculation yields results that go against what you would expect from the simplified picture(s)— but those are both rare and very exciting. And there are situations where there’s just no recourse to intuition— when competing effects are similar in size, and you just have to throw up your hands and grind through all the details. But in the best case, those end up serving as the basis for developing new forms of intuition, and moving the field forward.

So that’s my pitch for physics as more than just the math, and also a sneaky defense of popularization. While the detailed mathematics will be the ultimate arbiter of what is and isn’t true, being a successful physicist requires the cultivation of skills that go well beyond what we think of as pure mathematical ability.

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That was way more fun to write than a lot of the heavier stuff I’ve been droning on about recently; I’ll try to do more in this vein. If you want to see if I succeed, here’s a button:

And if you want to dispute my characterization or share a different favorite physics joke, the comments will be open: