That’s not theory, it’s just… numbers

“What’s so great about him ? All he can do is score !”
(Soccer legend Edson Arantes do Nascimento, aka Pele’, speaking of Italian striker Gigi Riva)

In a recent brilliant post, condensed matter theorist and blogger Schlupp writes about the fascinating field of condensed matter physics (CMP), also outlining the differences between scientists engaged in theoretical and experimental research.
It is a relatively clear distinction, one which most people, including non-scientists, have really no trouble understanding. Experimentalists do the actual research, i.e., build instruments and make measurements, thereby discovering new phenomena; theorists try to understand what the measurements mean, develop theories based on the observations, ultimately attempting to “explain it all” and predict the outcome of yet-to-be-performed experiments.
Schlupp also draws a further distinction, namely among theorists who do “real theory”, and those (like herself and myself) who do “computation”. This is a much subtler one, and I do think it takes the mind of a physicist to appreciate it, because when I try to explain it to people outside physics (or even science), they have absolutely no idea of what I am talking about.

This is how goes a typical conversation between a Sensible Outsider (SO) and myself:
SO: Real theory… computation… I do not understand, I thought all theoretical physicists were engaged in some sort of “computation”. I mean, how can you make a theory without supporting it with calculations ?
Me: Well, yeah, but, you see, “real theorists” use nothing but pencil and paper. For the most part they work with fairly abstract concepts, eventually arriving at results cast in elegant closed-form mathematical expressions… on the other hand, those who do “computer simulation” (as we call it) put everything on a computer and just get a bunch of numbers out… it is not quite as elegant, as numbers have to be analyzed and interpreted.
SO: Yeah but… whose answer is right (elegant or not) ?
Me: Well, analytical (i.e., pencil-and-paper) calculations usually require approximations (often uncontrolled), because most condensed matter problems, involving as they do large numbers of interacting particles, are unsolvable with just pencil and paper… on a computer one can often get an essentially exact answer. But that’s a simplistic way to put it…
SO: How is it “simplistic” ? Who cares how one actually does the calculations, what matters is the final result, whether it is correct or not, and what you learn from it… isn’t it ?
Me: You would think so, wouldn’t you….
(Fellow physicists can point out our mistakes, but only SOs can really make us feel stupid…)

So, where is the distinction, exactly ? Should there be any, really ? Both analytical and numerical approaches have benefits and shortcomings; I see them as inseparable, both integral parts of the activity of a theoretical physicist, in CMP as well as in other fields. Obviously, one may be better versed at one than the other, but the distinction between “computational” and “theoretical” physicists is meaningless. The fact is, one should use whatever tool can afford progress, be that a computer, a pencil or else [0]. However, as Schlupp puts it:
“… pencil people sometimes say we cannot provide ‘real’ explanations”
Yup. That’s what they say, and not just “sometimes”. But, what exactly makes an explanation “real”, anyway ? Is working through pages of algebra, eventually obtaining an analytical expression of some sort, required in order for one to claim physical understanding ?
“All you computer people can do is calculate stuff”, or “a real theorist should use computers for e-mail only”, “clear thinking and computers do not go together”, are a few examples of pills of wisdom dispensed by otherwise seemingly intelligent colleagues. Particularly annoying to me has always been the relentless insistence on some supposed dichotomy between “thinking” and “calculating”, “insight” and “numbers”. Sure, the calculation is not the final outcome, but there is no final outcome without the calculation.
It is downright mind-boggling how, in the light of the tremendous progress afforded by computers in a wide variety of fields of inquiry, including CMP, a pervasively snotty attitude still exists among theoretical physicists toward those of us who are not afraid of getting our hands dirty with some C++ programming (not Fortran, please — that thing is for losers). In fact, while I would certainly hope that she’s right, I am afraid that Schlupp’s assertion that “this despicably ignorant opinion is dwindling” is more a reflection of her youthful optimism than of the actual state of affairs.
From conversations with colleagues at a number of prominent physics departments, whose search for a condensed matter theorist appears to be destined to yield a null outcome this year, I have learned that the problem is often precisely that the strongest candidates are “computationally inclined”, and that departments cannot overcome a “strong internal bias against numerics” [1].

But, why ? What is it that makes many a prominent, successful and influential member of our community dismiss as “just numbers” the unquestionable advances made by a growing fraction of theoretical physicists who, grown tired of endless arguments over “whose approximation is better”, believe that the computing power available nowadays can be harnessed to generate insight, not merely push accuracy to the next digit ?
Which of our role models of the past would have turned away in disgust at the notion of using a computer to solve, say, the many-body problem ? Certainly not Enrico Fermi, in many respects the father of computational physics, who before everyone else saw the potential benefit of casting theoretical physics problems in a form suitable for fast computing machines.
Certainly not Richard P. Feynman, literally obsessed with computation. While he himself never formulated numerical methods (that I know of), almost his entire body of work was devoted to the developments of techniques capable of furnishing precise numerical answers to problems in various areas of condensed matter and particle physics; in fact, much of it underlies most modern methodologies to study on a computer analytically intractable problems in CMP. I cannot imagine him objecting to the use of computers to implement efficiently, for example, his famous diagrammatic technique. In a conversation I had with him about a decade ago, prominent russian physicist Igor Dzyaloshinkii said to me that in his view, his legendary teacher Lev Davidovic Landau, often known to rely more on his superb physical intuition than on rigorous calculations, would have enthusiastically embraced the use of computers to do theoretical physics. Honestly, I cannot think of anyone who would not have. The one thing that all of those great scientists had in common, was the desire for the truth, the eagerness to get to the bottom of problems, by whichever means available. They would not have had any time for ill-advised and misguided “theoretical purism” [2].

As Schlupp writes, sometimes “we may end up with just a bunch of numbers and no nearer understanding anything.
No question about it, that does happen. Sometimes scientists making use of numerical simulations are unable to see by themselves the physical implications of their results, and need help from more analytically inclined colleagues to do so. Other times numerics fail to provide any novel insight. So what ? Like in any human activity, there are successes and failures. The same can be said about pencil-and-paper theory as well, and even about experiments, some of which fail to generate anything useful in spite of all the money invested into them. A wholesale dismissal of the activity of a sizable fraction of the community based on the fact that it does not always work, or that its practitioners are not all uniformly brilliant, seems silly; it is akin to throwing the baby out with the bath water.
Frankly, I think that there are other reasons for the hostility toward computation, including a mechanism of self-defense by some who feel threatened by the emergence of methodologies which they themselves never did get around mastering, and which may put them “out of business”, so to speak. Interestingly enough, in other academic disciplines such as chemistry, materials science and engineering, as well as in different settings such as industry, where the need of obtaining reliable quantitative answers is more pressing, the widespread use of computers has gained acceptance much more rapidly.

[0] The repeated use on the part of many of us (and them) of the misnomer “computational physics”, implying somehow a different way of thinking, level of rigor and/or physics background of those who use computers, may not help. In my opinion, the sooner we abandon this terminology, the better.
[1] This bias has a way of suddenly disappearing as soon those “real theorists” need to find out whether their beautifully elaborate theories in the end make any useful predictions — in those cases, a collaborator well-versed in numerics, who can actually provide reliable results against which to assess their theories, is welcome. But as soon as the research project is finished and the beautiful theory has been proven accurate (all right, maybe to within a factor 4…) in some unphysical/irrelevant limit, the collaborator turns back into the “computing monkey” that (s)he initially was, and is metaphorically sent back to scientific purgatory. A simple exercise consists of going through the web pages of, say, the top 30 physics departments in the United States, and just counting how many of their CMP theory faculty engage in research heavily based on computer simulations.
[2] Just to avoid any misunderstanding — there is something beautiful and profoundly satisfying about arriving at a simple answer to a complex question, expressed in a compact mathematical form. Anyone capable of providing such a solution for a non-trivial outstanding problem deserves praise, and makes a valuable contribution to science. However, it is unfortunately the case that that happy state of affairs is usually not realized. Now, there is absolutely nothing beautiful about a wrong answer, no matter how intriguing its underlying idea and appealing the mathematical formula that expresses it.

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18 Responses to “That’s not theory, it’s just… numbers”

  1. Massimo (formerly known as Okham) Says:

    If you prefer to do analytical work, then by all means stick to it. Nothing wrong with that preference, and I was not trying to convince anyone that “computational is better than analytical”. My point is that “analytical is not better than computational ” either, and I do see a bias in this direction, not in the opposite.

    One important difference between analytic and computational works is that in the former case I can (in principle) recheck all the derivations myself. With comp work on the other hand this is not possible.

    Why is it “not possible” ? Of course you can. You look for bugs in the code, go through all procedures and make sure that they reproduce known limiting results. In fact, in my experience you end up learning a lot of physics in this way, because you have to think of relevant physical limits to explore.

    I also respectfully submit that, unless you manage to derive an exact solution (in which case no more work is needed, neither analytical nor numerical, and you deserve to get a prize), your analytical expression is approximate, and in most cases you’ll have no idea of how far off you are from the exact answer (that is, until someone does the same calculation numerically). I think both approaches are needed. for different reasons.

  2. Anonymous Says:

    SO is, of course, right

    And so are you. (I am right as a matter of course, so this needs not be mentioned.) The last sentence of footnote 2 pretty much sums it up.

    I once said to another theorist – not himself of the numerical persuasion – that a certain wrong mechanism was nevertheless “a neat idea”. He answered very much like the Queen to Alice: “I can give you five nice but wrong mechanism before lunch.”

    I am not sure it is all “youthful optimism” with me.* I tried to figure out where I got the optimistic idea from, and I think it has more to do with the places I’ve been to: I was educated at ProvincialTECH, after all, where the most computer hostile faculty member just retired with a lifetime achievement of Hirsch index h=5. And my German postdoc place also had a numerical bend to it. Sure, some people there, e.g. the Über-boss, are more into analytic work,** but they are never afraid to use computers if appropriate and usually make fun of the attitude you describe. Hm, actually, I would consider the whole of Germany to be rather friendly to numerics.

    The more interesting question is where the hostility comes from. BTW, another puzzling (to me) aspect is that mathematics itself does not appear to suffer that much from it. (But I may have the same bias as in my assessment of CMT. Most mathematicians I know are from ProvincialTech and ProvincialUni.) One aspect is probably the feeling of being threatened you mention, whether by your looks or by your computer skills. Another one might be the persistent fear and inferiority complex that CMT might after all not be “fundamental” enough. Better make ourselves seem more important by keeping these pragmatic computers far away. Hell, material scientists use computers and we sure want to be WAAAYYY more “fundamental” than they are.***

    A third reason might be the large importance attached to being the first to have had an idea as opposed to being the one to figure out which one of competing hypotheses is actually true. Democritus is sometimes credited with having “discovered” the atoms. No, he didn’t. Ideally, you should state contradicting hypotheses: Whichever it will turn out to be, you will have said it first, while the inconvenient fact that you ALSO stated the opposite will hopefully be forgotten.

    Oh, and I think the previous commenter meant that the READER of a paper might more easily check an analytic paper by someone else than a numerical one. I am actually not sure this is true unless the involved math is very simple and the paper contains enough detail. Also, the reader can even less check experimental results and this does not make experiment invalid.


    *Hey, I am a POSTDOC! Since when are postdocs optimistic? We are pessimist cynics, and if I do not show this sufficiently, I can certainly write some more pessimistic and cynic posts.
    ** Über-boss is excellent at accepting approaches and research topics he does not work with/on himself. This gift is rare, not only in Germany.
    *** Another handicap from my engineering degree: I have to work HARD at looking down on applied scientists and engineers, while others can do it easily and without thinking about it.

    • Massimo (formerly known as Okham) Says:

      Re: SO is, of course, right

      I am right as a matter of course, so this needs not be mentioned.

      You know, Schlupp, you are so right it’s almost scary…

      One aspect is probably the feeling of being threatened you mention, whether by your looks or by your computer skills

      Bingo ! You hit the nail right on the head, as usual….

      Oh, and I think the previous commenter meant that the READER of a paper might more easily check an analytic paper by someone else than a numerical one. I am actually not sure this is true

      It’s not. If the calculation is non-trivial, the effort required to check it is comparable to that of writing your own code. And actually, those who are interested in checking someone else’s calculation (analytical or numerical) are typically equipped to do it with little effort, either because they are themselves well-versed in the specific analytical methodology utilized, or because they already have a comparable code (or can easily write one).

  3. Anonymous Says:

    “Why is it “not possible” ? Of course you can. You look for bugs in the code, go through all procedures and make sure that they reproduce known limiting results.”

    I (as the reader of your computational paper) can NOT look for bugs in YOUR code (only YOU can do it!), and so I am left with no means of verifying what YOU did. There is no such problem with analytical papers.

    • Massimo (formerly known as Okham) Says:

      I (as the reader of your computational paper) can NOT look for bugs in YOUR code (only YOU can do it!), and so I am left with no means of verifying what YOU did. There is no such problem with analytical papers.

      This is a weak line of argument. You can write your own code and see whether it reproduces my results. This is a well-defined procedure. Sure, you have to know how to do it, but I also have to learn how to do a diagrammatic calculation, for example. Which one is easier is a matter of personal taste.

  4. Anonymous Says:

    “You can write your own code and see whether it reproduces my results. This is a well-defined procedure.”

    Of course. But wouldn’t you admit it, that it will take me a slightly bigger time investment to convince myself that you are not playing Schoen-Batlogg?.. (nothing against YOU personally of course)

    • Massimo (formerly known as Okham) Says:

      In my experience, it depends on how hard the analytical calculation is. With numerical results you may be able to get a code from someone else independently, or download one. If the analytical calculation is involved, or requires knowledge of topics that are not treated in the average graduate program, checking it independently is far from trivial. And because numerical calculations aspire to be exact, the comparison with a relatively simple-minded analytical approach is not appropriate, in my opinion.

  5. Anonymous Says:

    And what if my code gives a different result? It would be slightly different to point out where exactly you are making an error, right? I bit more difficult than just stating that Eq.(xx) does not follow from Eq.(yy) because of so and so… How would I then argue with you?

    • Massimo (formerly known as Okham) Says:

      If your code gives a different result, and you are reasonably sure that you have done everything correctly, you need to contact me and we need to go through a somewhat tedious procedure of checking intermediate results. I have done it many a time. Keep in mind that the comparison is only fair if the analytical calculation is sufficiently involved. If we are talking, say, a straightforward mean-field type calculation, then I agree that checking is simple but one is comparing apples and oranges, to some extent…

      • Cherish Says:

        Hmmm…I thought you were supposed to publish your correct results and make jabs at people whose paper you were checking, describing how their results are obviously invalidated by your findings…

  6. Anonymous Says:

    Well, we are talking about a sufficiently involved calculation that you can do within a couple of days and then post on the arxiv rejoicing in finding a mistake in your competitor’s work. As you see, some innocent pleasure that your scheme of doing things deprives me of. And also, yours only works with well-meaning competitors, while mine works all the time (provided I did calculations correctly).

    • Massimo (formerly known as Okham) Says:

      Well, we are talking about a sufficiently involved calculation that you can do within a couple of days and then post on the arxiv rejoicing in finding a mistake in your competitor’s work.

      It has happened to me, and I have done it to others. Keep in mind that in our field we tend to have similar general-purpose codes, and it is really not a big deal to reproduce someone else calculation quickly.

      And also, yours only works with well-meaning competitors

      I am sorry, it has occurred to me only now that you may be worrying specifically about possible fraud. In that sense I have to agree — whereas one will always be able to prove unequivocally and unambiguously wrong an analytical calculation, a numerical calculation will often only be considered “wrong” simply because nobody else can reproduce the results.
      In practice though, at least in my experience numerical errors are spotted rather quickly… I don’t have the sense that this is really that big a problem.

  7. Anonymous Says:

    I mentioned Schoen 🙂 so I assumed you got the message (I hope you know who he was). But I see your point — having similar codes might indeed help, I did not know about that part.

  8. Anonymous Says:

    Oh, well… He was rather an instructor in a driving school car who got some ride from his student driver.
    In other words I still think he got off ther hook too easily. Old boys network…

  9. Filipe Guimarães Says:

    Some people think that Fortran is for C++ what others think
    that “computational physicists” are for “real theorists”.

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