Why Study Physics?
Physics seems to have a bunch of useful epistemic techniques which haven’t been made very legible yet.
The two big legible epistemic techniques in technical fields are Mathematical Proofs, and The Scientific Method. Either derive logically X from some widely-accepted axioms, or hypothesize X and then do a bunch of experiments which we’d expect to come out some other way if X were false. It seems pretty obvious that science requires a bunch of pieces besides those in order to actually work in practice, but those are the two which we’ve nailed down most thoroughly.
Then there’s less-legible methods. Things like fermi estimates, gears-level models, informal mathematical arguments, an aesthetic sense for kinds-of-models-which-tend-to-generalize-well, the habit of figuring out qualitative features of an answer before calculating it, back-of-the-envelope approximations, etc.
Take informal mathematical arguments, for example. We’re talking about things like the use of infinitesimals in early calculus, or delta functions, or Fourier methods, or renormalization. Physicists used each of these for decades or even centuries before their methods were rigorously proven correct. In each case, one could construct pathological examples in which the tool broke down, yet physicists in practice had a good sense for what kinds-of-things one could and could not do with the tools, based on rough informal arguments. And they worked! In every case, mathematicians eventually came along and set the tools on rigorous foundations, and the tools turned out to work in basically the cases a physicist would expect.
So there’s clearly some epistemic techniques here which aren’t captured by Mathematical Proof + The Scientific Method. Physicists were able to figure out correct techniques before the proofs were available. The Scientific Method played a role—physicists could check their results against real-world data—but that’s mostly just checking the answer. The hard part was to figure out which answer to check in the first place, and that involved informal mathematical arguments.
We don’t really have a legible Art of Informal Mathematical Arguments, the way we have a legible Art of Mathematical Proofs or Art of Scientific Method. Informal mathematics clearly played a key role historically in figuring out useful tools, and will likely continue to play a key role in the future, but we don’t have a step-by-step checklist to follow in order to use informal mathematical arguments (the way we do for The Scientific Method), or even a checklist to verify that we’ve applied informal mathematical arguments correctly (the way we can for Mathematical Proofs). If someone says that my informal mathematical argument is wrong, and I can’t either convert it to a formal proof or show some definitive experiment, then I don’t have a clear standard way to argue that it’s correct.
Yet there’s clearly something making informal mathematical arguments correlate quite highly with truth (if imperfectly), because they work well in practice!
The same applies to Fermi estimates. They work remarkably well in practice, yet there’s not a standard step-by-step checklist. There’s not some standard rules to check whether a Fermi estimate is correct. If someone disagrees with my Fermi estimate, there’s not a way for me to establish correctness other than putting in all the work to find more-rigorously-estimated numbers.
The same applies to gears-level models. Plenty of physicists (and engineers, and others in technical fields) have an intuitive sense that a gears-level model is useful and powerful in ways that a black-box model isn’t. But if someone comes along and says that their 50-variable linear regression gives more precise predictions than a model based on the internal structure of the system, and that we really don’t need those gears anyway, I expect most people would not have a strong explanation of why the gears matter. Such explanations do exist (see e.g. the Lucas Critique, or my own writing on the subject), but gears-level models are still in the early stages of becoming legible. As of today, we don’t even have a widely-accepted explanation/definition of what “gears-level” means! For most practitioners, it’s just a vague aesthetic sense, at most.
The really important point to notice here is not any one method, but that there seems to be a bunch of these. Enough that there’s probably more of them which we haven’t even given names to yet. And they seem to come disproportionately from physics. (Or from the kinds of applied mathematicians who are adjacent to physics, and not adjacent to people who write Definition-Theorem-Proof style textbooks and papers.)
So if we want to learn all these key illegible methods, use them ourselves, and maybe someday make them more legible, then physics (and physics-adjacent applied math) seems like the main subject to study.
One caveat to anyone following this advice: once you get past the basics, it is probably more important to study the work of physicists as opposed to physics per se. Even outside of physics itself, there are certain patterns of thought which will make it clear who the physicists are—for instance, I could guess that Uri Alon got his degree in physics, even though he’s known mainly for his introductory text on systems biology. It’s exactly those illegible epistemic tools which identify such people; once you have a little bit of a handle on the tools, you’ll hopefully be able to recognize other people using them. And by studying how those tools are used outside physics itself, we can get a wider view of their application.
I think the title should be “why study physicists” not “why study physics”. Because what you are describing is a gift certain physicists have, and others do not. I had it in high school (often when the teacher would state a problem in class, the only thing that was obvious from the beginning was the answer, not how to get to it), and it saved my bacon in grad school a few times many many years later. Recently it took my friend and me about 5 min of idle chatting to estimate the max feasible velocity of a centrifugal launch system, and where the bottlenecks might be (surprisingly, it is actually not the air resistance, and not the overheating, but the centrifugal g forces before launch which make single-stage-to-orbit impossible). John Wheeler famously said something like “only do a calculation once you know the result.” Einstein knew what he wanted out of General Relativity almost from the start, and it took him years and years to make the math work. This pattern applies in general, as well. A physicist has a vague qualitative model that “feels right”, and then finds the mathematical tools to make it work quantitatively, whether or not those tools are applied with any rigor. I don’t know if this skill can be analyzed or taught, it seems more like artistic talent.
I’m not so sure. I think a lot of physicists get better at this through practice, maybe especially in undergrad. I have a PhD in physics, and at this point I think I’m really good at figuring out the appropriate level of abstraction to use on something (something I’d put in the same category as the things mentioned in the OP.) I don’t totally trust my own recollection, but I think I was worse at this freshman year, and much more likely to pick e.g. continuum vs. discrete models of things in mechanics inappropriately and make life hard for myself.
I’m sure one can train this skill, to some degree at least. I don’t think I got better at it, but I did use “the appropriate level of abstraction” to get the numerical part of my thesis done without needing a lot of compute,
By the way, I agree that finding the appropriate level of abstraction is probably the core of what the OP describes.
Yeah, I also seem to have a knack for that (as good as anyone in my cohort at a top physics grad school, I have reason to believe), but I have no idea if I got it / developed it by doing lots of physics, or if I would have had it anyway. It’s hard to judge the counterfactual!
Hmm, I do vaguely remember, in early college, going from a place where I couldn’t reliably construct my own differential-type arguments in arbitrary domains (“if we increase the charge on the plate by dQ …” blah blah, wam bam, and now we have a differential equation), to where I could easily do so. Maybe that’s weak evidence that I got something generalizable out of physics?
Could you write a list of physicists which have such “gift”? Might be useful for analyzing that specific skill.
Do you expect these less legible techniques to generalize well outside of physics, especially places where checking the answer is impossible or prohibitively expensive? Maybe the reason these techniques are widely and successfully used in physics is because the answers can eventually be checked, which limit the extent to which the techniques can be misused/abused?
That’s a natural hypothesis. A couple reasons to expect otherwise:
High dimensional world: to find something as useful as e.g. Fourier methods by brute-force guess-and-check would require an exponentially massive amount of search, and is unlikely to have ever happened at all. Therefore we should expect that it was produced by a method which systematically produces true/useful things more often than random chance, not just by guess-and-check with random guessing. (Einstein’s Arrogance is saying something similar.)
Physicists have a track record of successfully applying physics-like methods in other fields (biology, economics, psychology, etc). This is not symmetric—i.e. we don’t see lots of biologists applying biology methods to physics, the way we see physicists applying physics methods to biology. We also don’t see this sort of generalization between most other field-pairs—e.g. we don’t see lots of biologists in economics, or vice versa.
Relatedly: I once heard a biologist joke that physicists are like old western gunslingers. Every now and then, a gang of them rides into your field, shoots holes in all your theories, and then rides off into the sunset. Despite the biologist’s grousing, I would indeed call that sort of thing successful generalization of the methods of physics.
I thought the meme was that physicists think they can ride into town and make sweeping contributions with a mere glance at the problem, but reality doesn’t pan out that way.
Relevant XKCD.
That is indeed a meme. Though if the physicists’ attempts consistently failed, then biologists would not joke about physicists being like gunslingers.
I don’t think this contradict the hypothesis that “Physicists course-correct by regularly checking their answers”. After all, the reason Fourier methods and others tricks kept being used is because they somehow worked a lot of the time. Similarly, I expect (maybe wrongly) that there was a bunch of initial fiddling before they got the heuristics to work decently. If you can’t check your answer, the process of refinement that these ideas went through might be harder to replicate.
The second point sounds stronger than the first, because the first can be explained in the fact that biological systems (for example) are made of physical elements, but not the other way around. So you should expect that biology has not that much to say about physics. Still, one could say that it’s not obvious physics would have relevant things to say about biology because of the complexity and the abstraction involved.
This makes me wonder if the most important skills of physicists is to have strong enough generators to provide useful criticism in a wide range of fields?
If you suppose that physics tries to deal with the whole of reality in one gulp, then its ability to come up with simple general rules would be remarkable. But actually , physics deals with extremely simplified and idealised situations … frictionless planes, free fall in a vacuum, and so on. Even experiments strive to simplify the natural messiness of reality into something where only one parameter changes at a time
(This is partly a response to the comment above, but I got kind of carried away.)
The Standard Model of particle physics accounts for everyday life (except gravity) in ridiculous detail, including all the “natural messiness” you have in mind (except gravity). It consists of some simple and unique (but mathematically tricky) assumptions called “quantum field theory” and “relativity”, plus the following details, which completely specify the theory:
* the gauge group is SU(3) x SU(2) x U(1) (or “the product of the three simplest things you could write down”)
* the matter particles break parity symmetry, using the simplest set of charges that works
* there are three copies of each matter particle
* there is also a scalar doublet
* the 20ish real-valued parameters implied by the above list have values which you can find by doing 20ish experiments.
I dare anybody to give a specification of, say, all of known organic chemistry or geology with a list that short. You don’t need to spell out any mathematical details, so long as a mathematician could plausibly have invented it without being inspired by physical reality (which are the rules I’m playing by in this comment—I think QFT, relativity, and concepts like “gauge group” and “parity symmetry” that I assume knowledge of are all things math could/would have produced eventually).
In some sense I’m handwaving past the hard part, but I think the remarkable thing about physics is that the hard part is entirely math; if you did enough math in a cave without observing anything about the physical world, you would emerge with the kind of perspective from which the known laws of physics (except gravity) seem extremely parsimonious. (Gravity is also parsimonious but sort of stands alone for now.) On the other hand, if you go do a lot of experiments instead, the laws of physics will seem bizarre and complicated. Which I admit is kind of a strange fact! It’s not clear that “math parsimony” is the same concept as, say, Turing-machine-based Kolmogorov complexity, and it definitely isn’t anybody’s intuitive notion of “simplicity”.
And of course, quite a lot of the “natural messiness” of the world is captured by even simpler Newtonian-mechanics models, although chemistry becomes a kind of nasty black box from a Newtonian perspective.
The SM is itself a kludge. It’s not a satisfactory TOE for a bunch of reasons besides gravity.
It is definitely not a TOE, but it is a successful EFT that accounts for everything except gravity/cosmology.
You are responding as though I said something like “physics doesn’t work at all”, when I actually said it works via idealisations and approximations. To talk of Effective Field Theories concedes my point, since EFTs are by definition approximations .
You said “extremely simplified and idealised situations … frictionless planes, free fall in a vacuum, and so on”. That’s a pretty different ballpark than, say, every phenomenon any human before the 1990s had any knowledge of, in more detail than you can see under any microscope (except gravity).
Do you consider everything you’ve experienced in your entire life to have happened in “extremely simplified and idealised situations”?
This is true of the physics most people learn in secondary school, before calculus is introduced. But I don’t think it’s true of anyone you might call a physicist. I’m confused by the chip you seem to have on your shoulder re physics.
Calculus isn’t a magic trick that allows you to dispense with idealisations and approximations. You can start dealing with friction and air resistance, but you don’t get one equation that is completely precise and applicable to anything.
I don’t have a chip on my shoulder about physics: everyone else has a halo effect
I think the use of symmetries for causal inference is an important insight in physics. Without constraints, there are an enormous amount of possible causal theories that can explain any given set of observations. But adding symmetries can often pin down the theory uniquely, and in a testable way (because the symmetry would fail for almost all the ways the theory could be false).
It also turns out that there’s an obscure method in the causal inference literature that I think can be analogized to this. If you’ve got two variables A and B, then the causal relationship A→B or B→A can be hard to figure out, just from their distribution P(A,B). However, if you can embed them into a family Ai and Bi which have sufficiently different distributions, but which all share the mechanisms, then likely precisely one of P(Ai|Bi) and P(Bi|Ai) will be constant as a function of i, with the constant one being the one that encodes the direction of causality. I think this is a special-case of the physics method of requiring theories to respect symmetries, where in this case the symmetry group is the group of permutations on i.
I’m wondering about the different types of intuitions in physics and mathematics.
What I remember from prepa (two years after high school where we did the full undergraduate program of maths and physics) was that some people had maths intuition (like me) and some had physics intuition (not me). That’s how I recall it, but thinking back on it, there were different types of maths intuitions, which correlated very differently with physics intuition. I had algebra intuition, which means I could often see the way to go about algebraic problems, whereas I didn’t have analysis intuition, which was about variations and measures and dynamics. And analysis intuition correlated strongly with physical intuition.
It’s also interesting that all your examples of physicist using informal mathematical reasoning successfully ended up being formalized through analysis.
This observation makes me wonder if there are different forms of “informal mathematical reasoning” underlying these intuitions, and how relevant each one is to alignment.
An algebra/discrete maths intuition which is about how to combine parts into bigger stuff and reversely how to split stuff into parts, as well as the underlying structure and stuffs like generators. (Note that “the deep theory of addition” discussed recently is probably there)
An analysis/physics intuition which is about movement and how a system reacts to different changes.
Also the distinction becomes fuzzy because there’s a lot of tricks which allow one to use a type of intuition to study the objects of the other type (things like analytic methods and inequalities in discrete maths, let’s say, or algebraic geometry). Although maybe this is just evidence that people tend to have one sort of intuition, and want to find way of applying it at everything.
Interestingly, I have better algebra intuition than analysis intuition, within math, and my physics intuition almost feels more closely related to algebra (especially “how to split stuff into parts”) than analysis to me.
Although there’s another thing which is sort of both algebra and analysis, which is looking at a structure in some limit and figuring out what other structure it looks like. (Lie groups/algebras come to mind.)
Have you read What Should A Professional Mathematician Know? The relevant bits are in the last two sections.
Physics also seems to help with clear philosophical thinking, and has lots of unintuitive stuff that trains the skill of looking past your models and into the Real Thing. Of Deep Fundamental Principles, I also think physics has some of the easier ones to see, like the conservation principles (once you get why a physicist can be so confident of the non-existence of a perpetual motion machine, you can start to imagine how there could be other Deep Principles which could justify seemingly excessive confidence about other complicated domains).
On the other hand, mediocre physicists tend to be too arrogant, and the current generation of physicists seems to have lost the way in some important (but hard to pin down) sense.
the current generation of physicists seems to have lost the way in some important (but hard to pin down) sense
My impression of physics (1) post-1970-or-so is that it’s lost the balance between theory and experiment that makes science productive. Hypotheses like “superstring theory” or “dark matter” are extremely difficult to test by experiment (through no fault of the physicists’ own). Physicists have tried to to make up for it with improvements in theory, but without experiments bringing discipline to the process it doesn’t quite work.
In one sense, this is good news. Physicists have reached the point where it is extremely difficult to observe a physical phenomenon they can’t predict, which is very similar to saying the project is almost complete.
(1) Here I’m speaking mostly of particle physics. Condensed-matter physics has been much more successful over the past 50 years or so. Other disciplines may vary.
Related work:
Armstrong on informal probabilistic proofs of the Goldbach hypothesis and Fermat’s last theorem
Arguments against P = NP in Arbital (disclaimer: I wrote these)
Gwern discussing mathematical error, specifically about the puzzle of why most math theorems are correct even if errors in proofs are relatively common
Also, while this post focuses a lot on Physics, my experience is that top level math people are quite comfortable with informal math reasoning.
we don’t have a step-by-step checklist to follow in order to use informal mathematical arguments
If we did, the checklist would define a form and the mathematical arguments would become formal.
Terrence Tao uses the term post-rigorous to describe the sort of argument you’re talking about. It’s one of three stages. In the pre-rigorous stage, concepts are fuzzy and expressed inexactly. In the rigorous stage, concepts are precisely defined in a formal manner. In the post-rigorous stage, concepts are expressed in a fuzzy and inexact way for the sake of efficiency by people who understand them on a rigorous level; key details can be expressed as rigorously as necessary but the irrelevant details of a full proof are omitted.
This feels correct; my experience of physics+calculus in high school was that it was harder than any other class I took but made me [feel on the inside] a lot smarter, like the insights were important and unlocked new patterns of thought, even though everyone else in the class had a much easier time with the material (high-level elective with calculus as prerequisite screened out all but the most committed).
I have fond memories. I would read three paragraphs in the text book, ask my physics-gifted friend, “wtf?” and ze would say, “Oh, that just means x,” and I would say, “Oh, it seems obvious now—why didn’t they just say x??”
My friend was VERY patient with me.
One major pattern of thought I picked up from (undergraduate) physics is respect for approximation. I worry that those who have this take it for granted, but the idea of a rigorous approximation that’s provably accurate in certain limits, as opposed to a casual guess, isn’t obvious until you’ve encountered it.
Yes. It’s empiricism. If you use informal arguments in an apriori way, you get philosophy.
Well, it’s got lots of unintuitive stuff. Much more so than armchair metaphysics ever came up with.
Having a broad selection of theories choose from is, in a way, helpful to the process of honing in on the one true theory … But it puts a lot more stress on the honing-in part of the process.