Skip to content

Simplicity

by Michael Nielsen on May 2, 2004

Two very interesting posts (and discussion threads) at Uncertain Principles: here and here. Here’s the second post in full, with a few comments of my own thrown in.

In the very nice comment thread that’s sprung up around the last post (this is why I envy Teresa Nielsen Hayden), Mary Messall writes about Physics in general:

The thing is stories don’t give you numbers that can be checked by experiment. Equations do. Insofar as we demand that our science be experimentally verifiable, we’re demanding that it consist of equations. In that sense there’s no such thing as “a scientific explanation.” Explanations are inherently unscientific — unpredictive, unfalsifiable.

What’s more, I find (to my dismay) that a great many, perhaps even the majority, of the equations we’re given in class are used *without* interpretation. Sometimes I wander around demanding an interpretation for some specific expression from everyone in the department, and mostly I eventually come up with some story that satisfies me, but it’s amazing how many of the people I ask in the meantime don’t know and *don’t care*.

And they’re better at solving problems than I am.

I’m a little bit bitter about some of the professors who’ve had that attitude. “Interpretation is the same thing as popularization, as speculation. Frivolous. Unrigorous. Beneath us. Shut up and calculate.” They’re right, in a way. It can’t predict anything.

I guess I still think stories (and applications, which are usually disdained by the same people) are the [horse], and the equations are the [cart]. But the equations-for-their-own-sakes people may be better scientists than I am. I’m not sure.

It’s a big enough idea that it deserves a post of its own. I’ve written about something vaguely similar in the area of lecture prep– twice, in fact: one, two– so it should come as no surprise that I tend to think of stories as an integral part of physics.

Contrary to what Mary says, I’ve found that the very best physicists I know (and this includes a couple of Nobel laureates, if I may be permitted a JVP moment) are the ones who have the best grasp of the stories and interpretations.

I talked once with a distinguished colleague who’d had the chance to see many great scientists speak – people like Feynman, Weinberg, Gell-Mann, Yang, Wilczek, and others. He made a comment about Feynman that struck me very much. The comment was that whatever subject Feynman was talking about, it was obvious that he had thought about it deeply from every conceivable angle, and so had arrived at extremely simple conceptions of many things that were often thought to be rather difficult. As a result it simply appeared that he could see through to the core of things in a way that most of us usually don’t.

This isn’t quite the same as what Chad is saying, but I think it’s closely related. Almost without exception, the most outstanding scientists I’ve met have this common property of trying to see through to the core of things, arriving at an extremely simple conceptual understanding.

At least for the sort of physics that I do, it’s essential to ground your understanding of the physics in terms of the real motions of real atoms that are the basis of everything. If you can understand what’s going on in simple terms, and more importantly explain it that way to other people, that’s a big step toward being able to push experiments in new directions, and explore new phenomena.

I used to think that one of the big sources of differences in ability between different researchers is that some people are much better than others at keeping track of complex ideas.

I now think that’s a cuckoo way of looking at things, and that a more accurate picture is that some people take much more care than others to reduce things to very simple terms all the time. Such people, because they understand things in simple ways (often, in multiple simple ways), can cope with much greater apparent complexity.

To some degree, this is an issue of sub-fields. I work in atomic, molecular, and optical physics, where the problems we study generally involve a smallish number of atoms doing comprehensible things. Other fields rely much more heavily on sophisticated mathematical tricks to make their problems tractable, which makes it harder to tell stories about what’s going on. I took one class on Solid State, and after the first couple of weeks, I no longer had the foggiest idea what was going on in terms of actual electrons moving through solid materials– it was all “reciprocal lattice vectors,” which I still don’t understand– which made it a deeply unpleasant class all told.

I think this desire for simple conceptual pictures is not an issue of subfields. The mathematician Doron Zeilberger describes the great mathematician Gelfand – who worked in some exceedingly technical areas – as having a “unique approach to learning and teaching” that revolves around “giving the simplest possible example, and that Gelfand enforces in his famous seminar, by constantly interrupting speakers and making them explain clearly and simply”.

(The reciprocal lattice vector stuff simply sounds like it was badly taught.)

On the other hand, though, I think the link between success in physics and a good grasp of stories could be extended to many of the best and brightest regardless of research topic. Einstein’s real breakthrough with Special Relativity was a matter of storytelling– people knew before Einstein that Lorentz transformations would solve the problems with Maxwell’s Equations, but thought it was too weird. Einstein showed that not only was it the right solution, but it had to be that way, and he did it by providing stories to make it all make sense (again, see some earlier posts: one, two). Schroedinger’s equation is in some sense a story that makes Heisenberg’s matrix mechanics palatable (the theories are mathematically equivalent, but as I understand it, nobody could make heads or tails of Heisenberg’s stuff). And when you get down to it, what are Feynman diagrams but little stories about what happens to an electron as it moves from point A to point B?

Yes, in some sense, the equations are the main thing. But when you look at the history of physics, you find again and again that the real giants of the field are the people who matched an interpretation to the equations, who came up with stories to explain it all. Any fool with a computer can manipulate equations, but it takes real genius to explain what’s going on in a way that makes it make sense.

I don’t have a good answer to “What’s a photon?”, but at least I can say this: If you feel that interpretations and stories are an important part of physics, you’re in good company.

From → General

10 Comments
  1. Mary Messall permalink

    You can see that I was regretting the much stronger condemnation that I had expressed in your own comments, of the “shut up and calculate” attitude…

    Part of this is just me trying to convince myself that I can be a good physicist in spite of the fact that I seem to approach the subject differently than my professors and peers (or even because of that.) But I do think they’re important questions, even if you’re not a neurotic grad student. It’s nice that you and Chad Orzel take the subject seriously too.

  2. This may be somewhat tangential, but I’m reminded of Kevin Smith’s Chasing Amy: I can’t help but think of the opening scene, with their buddy asking “What’s a Nubian?” and being told to shut the f* up-

    I envision an early quantum optics conference with some gadfly in the back row continually asking “by the way…what’s a photon?” just to annoy the experimentalists…

  3. Mary Messall permalink

    I haven’t seen Chasing Amy (I think I’ve seen all of Kevin Smith’s other movies.) But the scene sounds recognizable anyway…

    Today at lunch I rehearsed some of the mathematical questions that used to bother me in high school, with a friend. I was curious about whether I was badly taught, or whether everyone had to figure out their own answers to these:

    Why does differentiation undo integration? What do imaginary exponents have to do with sines and cosines? What do hyperbolic sines and cosines have to do with hyperbolas? What do tangent lines have to do with tangent functions? Why does the small angle approximation work? What’s the difference between statistical error and measurement error? What’s a standard deviation anyway, apart from a slightly intimidating formula?

    Took me years to get answers to some of those. I listen in class, then accost the professor, then look in the book, then think really hard and draw a lot of pictures, then look in other books, then draw some more pictures, then accost random people who may or may not know any physics or math. If I haven’t worked it out by then, I get mad.

    I have a story from today, actually, but I’ll resist…

  4. Michael Nielsen permalink

    Chasing Amy was a great movie. Haven’t seen Clerks or any of his others, though.

    Mary’s comments remind me of a High School maths teacher of mine who insisted that the connection between integration and differentiation was just a mysterious fact that had to be accepted as a given, but not questioned. Fortunately, my University lecturers were a great deal better.

  5. mitch permalink

    Here’s a tough question: what’s the analogue for mathematics of “understanding … in terms of the real motions of real atoms”? Somehow I don’t think it’s set theory.

  6. Mary Messall permalink

    I was going to say that the question that I asked in that spirit most often was “What exactly are we adding up?” Because it tends to be the innumerable infinite series expansions and the mysterious integrals that really lose me.

    But then I thought about how long it took me to understand curl and divergence, or even how bad my intuition was for a long time on sines and cosines and exponentials. I eventually caught on to those by drawing triangles (well, vectors, in the case of curl and div, but when you resolve vectors, you draw triangles). And I realized that it was by filling in little triangles on top of those rectangles you draw when you’re learning to integrate numerically, that I finally understood the relationship between derivatives and integrals. And it was by thinking in terms of slopes (always involves triangles, rise over run) that I finally saw why a Taylor series works.

    So I think what I want is “understanding, in terms of the real triangles.”

    My freshman geometry teacher would be pleased to know I’ve come to such an appreciation…

  7. I’ve always liked von Neumann’s possibly apocryphal response to a young colleague: “Young man, in mathematics you don’t understand things, you just get used to them.”

    I tend to believe, in this spirit, that understanding is related to the ability to connect one thing to another. This means having different ways of representing the “same” knowledge, and knowing of all sorts of unexpected connections between things.

    To pick up on the curl example, for me I felt I understood curl a lot better when I saw Stokes’ theorem used to simplify the taking of line integrals. And I felt I understood curl even better when I realized it was just a generalization of the fundamental theorem of calculus. Indeed, at that point, I could see exactly _why_ you’d be led to define the curl in the first place.

  8. Michael: Indeed I’ve always found it more interesting and gratifying to solve problems by trying different representations until I found one that made the problem very straightforward to solve. However I haven’t found that most people were like me.

    Re: “Shut up and calculate”, I just found out that while that phrase is often attributed to Feynman it may actually be Mermin’s – see http://physicstoday.org/vol-57/iss-5/p10.html

  9. A Scott Crawford permalink

    Mary,

    The problem with stories and interpretation is that there’s a tendency to use analogy and metaphor in place of reference to actual experimental apparatus and data. Terms become vague and mistranslated (a la the “uncertainty” principle), and eventually devolve away from their material and observable scientific basis into theological Pythagorean weirdness.

    At the same time it’s worth noting that equations are a descriptive tool, and are only as useful as a given mathematical systems ability to weather advances in technology and measurement while remaining generally predictive. Blame engineers if you will, but we’re never going to come up with an empirical way for you to physically measure mathematical singularities;) Or instantaneous type stuff. Or ideal anything.

    Oh!… To the question, “what’s a photon?”. I’m partial to the answer, “a tiny weeny smidgen of light!!”.

Trackbacks and Pingbacks

  1. More On Stories In Science « Steps & Leaps

Comments are closed.