Thursday, December 08, 2016

No, physicists have no fear of math. But they should have more respect.

Heart curve. [Img Src]
Even physicists are `afraid’ of mathematics,” a recent headline screamed at me. This, I thought, is ridiculous. You can accuse physicists of many stupidities, but being afraid of math isn’t one of them.

But the headline was supposedly based on scientific research. Someone, somewhere, had written a paper claiming that physicists are more likely to cite papers which are light on math. So, I put aside my confirmation bias and read the paper. It was more interesting than expected.

The paper in question, it turned out, didn’t show that physicists are afraid of math. Instead, it was a reply to a comment on an analysis of an earlier paper which had claimed that biologists are afraid of math.

The original paper, “Heavy use of equations impedes communication among biologists,” was published in 2012 by Tim Fawcett and Andrew Higginson, both at the Centre for Research in Animal Behaviour at the University of Exeter. They analyzed a sample of 649 papers published in the top journals in ecology and evolution and looked for a correlation between the density of equations (equations per text) and the number of citations. They found a statistically significant negative correlation: Papers with a higher density of equations were less cited.

Unexpectedly, a group of physicists came to the defense of biologists. In a paper published last year under the title “Are physicists afraid of mathematics?” Jonathan Kollmer, Thorsten Pöschel, and Jason Galla set out to demonstrate that the statistics underlying the conclusion that biologists are afraid of math were fundamentally flawed. With these methods, the authors claimed, you could show anything, even that physicists are afraid of math. Which is surely absurd. Right? They argued that Fawcett and Higginson had arrived at a wrong conclusion because they had sorted their data into peculiar and seemingly arbitrarily chosen bins.

It’s a good point to make. The chance that you find a correlation with any one binning is much higher than the chance that you find it with one particular binning. Therefore, you can easily screw over measures of statistical significance if you allow a search for a correlation with different binnings.

As example, Kollmer at al used a sample of papers from Physical Review Letters (PRL) and showed that, with the bins used by Fawcett and Higginson, physicists too could be said to be afraid of math. Alas, the correlation goes away with a finer binning and hence is meaningless.

PRL, for those not familiar with it, is one of the most highly ranked journals in physics generally. It publishes papers from all subfields that are of broad interest to the community. PRL also has a strictly enforced page limit: You have to squeeze everything on four pages – an imo completely idiotic policy that more often than not means the authors have to publish a longer, comprehensible, paper elsewhere.

The paper that now made headline is a reply by the authors of the original study to the physicists who criticized it. Fawcett and Higginson explain that the physicists’ data analysis is too naïve. They point out that the citation rates have a pronounced rich-get-richer trend which amplifies any initial differences. This leads to an `overdispersed’ data set in which the standard errors are misleading. In that case, a more complicated statistical analysis is necessary, which is the type of analysis they had done in the original paper. The arbitrarily seeming bins were just chosen to visualize the results, they write, but their finding is independent of that.

Fawcett and Higginson then repeated the same analysis on the physics papers and revealed a clear trend: Physicists too are more likely to cite papers with a smaller density of equations!

I have to admit this doesn’t surprise me much. A paper with fewer verbal explanations per equation assumes the reader is more familiar with the particular formalism being used, and this means the target audience shrinks. The consequence is fewer citations.

But this doesn’t mean physicists are afraid of math, it merely means they have to decide which calculations are worth their time. If it’s a topic they might never have an application for, making their way through a paper heavy on math might not be the so helpful to advance their research. On the other hand, reading a more general introduction or short survey with fewer equations might be useful also on topics farther from one’s own research. These citation habits therefore show mostly that the more specialized a paper, the fewer people will read it.

I had a brief exchange with Andrew Higginson, one of the authors of the paper that’s been headlined as “Physicists are afraid of math.” He emphasizes that their point was that “busy scientists might not have time to digest lots of equations without accompanying text.” But I don’t think that’s the right conclusion to draw. Busy scientists who are familiar with the equations might not have the time to digest much text, and busy scientists might not have the time to digest long papers, period. (The corresponding author of the physicists’ study did not respond to my question for comment.)

In their recent reply, the Fawcett and Higginson suggest that “an immediate, pragmatic solution to this apparent problem would be to reduce the density of equations and add explanatory text for non-specialised readers.”

I’m not sure, however, there is any problem here in need of being solved. Adding text for non-specialized readers might be cumbersome for the specialized readers. I understand the risk that the current practice exaggerates the already pronounced specialization, which can hinder communication. But this, I think, would be better taken care of by reviews and overview papers to be referenced in the, typically short, papers on recent research.

So, I don’t think physicists are afraid of math. Indeed, it sometimes worries me how much and how uncritically they love math.

Math can do a lot of things for you, but in the end it’s merely a device to derive consequences from assumptions. Physics isn’t math, however, and physics papers don’t work by theorems and proofs. Theoretical physicists pride themselves on their intuition and frequently take the freedom to shortcut mathematical proofs by drawing on experience. This, however, amounts to making additional assumptions, for example that a certain relation holds or an expansion is well-defined.

That works well as long as these assumptions are used to arrive at testable predictions. In that case it matters only if the theory works, and the mathematical rigor can well be left to mathematical physicists for clean-up, which is how things went historically.

But today in the foundations of physics, theory-development proceeds largely without experimental feedback. In such cases, keeping track of assumptions is crucial – otherwise it becomes impossible to tell what really follows from what. Or, I should say, it would be crucial because theoretical physicists are bad at this.

The result is that some research areas can amass loosely connected arguments that follow from a set of assumptions that aren’t written down anywhere. This might result in an entirely self-consistent construction and yet not have anything to do with reality. If the underlying assumptions aren’t written down anywhere, the result is conceptual mud in which case we can’t tell philosophy from mathematics.

One such unwritten assumption that is widely used, for example, is the absence of finetuning or that a physical theory be “natural.” This assumption isn’t supported by evidence and it can’t be mathematically derived. Hence, it should be treated as a hypothesis - but that isn’t happening because the assumption itself isn’t recognized for what it is.

Another unwritten assumption is that more fundamental theories should somehow be simpler. This is reflected for example in the belief that the gauge couplings of the standard model should meet in one point. That’s an assumption; it isn’t supported by evidence. And yet it’s not treated as a hypothesis but as a guide to theory-development.

And all presently existing research on the quantization of gravity rests on the assumption that quantum theory itself remains unmodified at short distance scales. This is another assumption that isn’t written down anywhere. Should that turn out to be not true, decades of research will have been useless.

In lack of experimental guidance, what we need in the foundations of physics is conceptual clarity. We need rigorous math, not claims to experience, intuition, and aesthetic appeal. Don’t be afraid, but we need more math.


  1. Well ... philosophy is not a conceptual mud ...

  2. Yes you should never make assumptions! I think some physicists might be afraid of what the maths suggests (like the many world's theory)and reject it due to personal reasons.

  3. Lack of "fine tuning" is an assumption yes. Naturalness seems more of an aesthetic judgement to me unless by this you mean 'naturalism' as in "all explanations must be physical". Lastly, philosophers too should make their assumptions explicit..

  4. driod33,

    Every theory needs assumptions. The problem isn't the existence of assumptions, the problem is the lack of clarity about what exactly is assumed and what follows from what.

    1. Sorry,never jump to conclusions due to personal thoughts like that's impossible.

  5. an entirely self-consistent construction and yet not have anything to do with reality Sakharov conditions, matter in excess of antimatter, demand parity violation. Opposite shoes embed within mirror-asymmetric space (mount a left foot) with different energies. They vacuum free fall along divergent minimum action trajectories. Atom-scale emergent (unit cell) opposite shoes are visually and chemically identical, single crystal test masses in enantiomorphic space groups (doi:10.1107/S0108767303004161), P3(1)21 versus P3(2)21 or P3(1) versus P3(2). we need more math We need orthogonal observation. Measurably violate the Equivalence Principle.
    In lack of experimental guidance test space-time geometry with geometry. Look outside physics with chemistry.

  6. driod33,

    It's logically impossible, there's no jump in that argument. A theory without assumptions isn't a theory of anything.

  7. I can read a paragraph of English almost at a glance. I can understand a graph or a diagram with ease. But I find that to understand an equation, I have to copy it down on a piece of paper, so it has gone through my brain. My guess is that there are people who can absorb equations like I can absorb written English. Are there any studies of this? (When I was at last in Greece I looked forward to reading signs, since as a scientist I am thoroughly familiar with the Greek alphabet. But I was shocked to find that I almost had to sound out every single word on a billboard. That tells me a lot about my ability to read English so fast.)

  8. An ammonia molecular beam splits in an inhomogeneous electric field. Multipole selection of the population inversion into an ammonia maser is non-classically eerie. So what? Observation that only respects respected theory will be weak toward improvement and sterile toward discovery.

  9. Thank you for another interesting post, Dr. B. Just an observation -- I much prefer "physics math" to "math math," whose odd symbols and ever-present unintelligible proofs represent a completely different language to me. I was trained in both physics and engineering, and I can assure you that whatever fear physicists might have of math, it's far worse for engineers. Maybe it's my engineering brain that prevents me from truly comprehending the math of string theory and supersymmetry, but it's heartening to know that many physicists also have a hard time with these subjects.

  10. Bea, this is totally off topic -- but as far as I can tell, you've never posted on what for many is THE number one scientific issue of our time, the question of anthropogenic climate change. I may have missed a post or two, so if I'm wrong, I'd appreciate it if you would correct me, and post the link. If I'm right, however, then it seems logical to conclude that you must be a skeptic on this issue, because just about every scientist in the "warmist" camp seems to be hammering away at the dangers of "climate change" and what we need to do about it NOW. And nowadays I'd imagine any scientist whose research is dependent on funding from any establishment source would be reluctant to risk being labeled a "denier" in the poisonous atmosphere in which we now live.

    I've engaged online with climate scientists, expressing doubts based on my own assessments of the data, and the responses I'm getting strike me as not only unsatisfactory from a scientific point of view, but also remarkably juvenile, filled with crude insults and ad hominem arguments.

    I've followed this blog for some time and I've always been impressed, both with your expertise and your ability to explain sometimes difficult topics clearly. And I've never known you to respond in an insulting manner to any comment, no matter how naive. I'm willing to admit that my take on all this could be wrong, but I have yet to receive a satisfactory answer as to why.

    I'm hoping, therefore, that you'd be willing to post your thoughts on this topic sometime soon, as I trust your ability and your judgement and would be more likely to accept your viewpoint than that of the obnoxious so-called "scientists" currently attempting to control everyone else's thinking by crudely dismissing anything that challenges their dogma.

    I am currently a skeptic, but if anyone can change my mind it is you -- so I'm hoping you'll be willing to share your thoughts.

  11. DocG,

    Yes, it's clearly off-topic, but I think I should answer this question. I have never written about climate change and I very rarely touch political questions. The reason is that on most matters I don't think I have anything to say that hasn't been said a million times by other people already.

    I also seem to be the theorist all through, meaning I care more about how humanity handles the questions (badly) than about the questions themselves. I know more about political theory and sociology and economics and the way we make decisions than I know about the particular issues that need to be decided today.

    Having said that, I think the evidence that the climate is changing is overwhelming. The question is just what, if anything, should we do about it. That partly depends on what you believe is causing this and that seems to be what most of the discussion circles about. Unfortunately, what gets drowned in this discussion is that we'll have to deal with it regardless of what's causing it and the longer we postpone coming up with action plans for this, the more economically costly it will get.

    Now here's the rub: I often write about what's going wrong in the scientific communities that I'm part of. In a nutshell, nobody's taking any measures to prevent social and cognitive biases in theory development. But these are systemic problems and are almost certainly present also in other disciplines. The consequence is, sad but true, that I am having more and more difficulties to trust scientists generally.

    Hence, for me the top one concern isn't climate change. Or immigration. Or Trump. The top one concern is to make sure we can trust science to work correctly.

    Having said that, I'll not start writing about climate change because it would eat up a lot of my time and I don't think anyone would benefit from it. I also have to say that since I'm very future oriented, climate change isn't among my top worries. It seems exceedingly unlike it'll eradicate mankind. We'll survive this somehow and adapt.

    We might however face some centuries of regress. And we will all be losers: There won't be some nations who will come out ahead and others behind - we'll all go down together. You don't need a degree in economics to see that if many people have to spend time fixing problems caused by climate change they have less time to do other things. So that's why we have to talk about this.

    But what worries me much more is that I'm not sure our present systems are resilient to phases of downturn. Meaning, we might run into a feedback circle in which even a short phase of regress leads to much more regress. The reason is that in some areas of our lives we only cope because we're constantly working full power. Take antibiotic resistance. Can we afford to slow down on that front even a little bit? Take our whole information-culture. If access becomes even a little more difficult (ie costly) it'll cut off lots of people quickly, leading to more downturn, which will cut off more people, etc.

    In summary, for me it all starts with the scientific system itself. We'll have to fix our own problems first. In a nutshell, that's what my upcoming book is about. (I'm done writing. I'm now waiting for my editor to get back to me.) Best,


  12. Well, you don't have to put everyone in the same sack, there are good theorists who are not afraid of math indeed and there are many bad ones who are.

    Besides that math and physics are two different worlds in terms of formalism and terminology, physicists often loosely talk about physical concepts without even knowing their mathematical counterparts and the surrounding mathematical formalism and terminology.

    I believe that right now only a handful of String theorists (Witten, Vafa and Moore are prime examples ) can follow mathematical formalism and read in productive way a mathematical paper; they are first rate physicists and Mathematicians effectively combining the two worlds.

    There are also great theorists with brilliant intuition in Physics but weak in backing up their ideas with Mathematical consistency, Susskind is an example I can think of on the top of my head.

    On the other hand there are (primary Mathematical) physicists lost in the Mathematical formalism who simply can't see the Physical picture or essence behind it i.e. what is physical important beneath this formalism.

    So things are not black and white as you present them.

  13. @Sabine,

    I know, a blog is not a paper, but I miss an aspect about mathematics (in relation to physics). Nearly all the mathematical tools are designed to simplify reality. That’s why mathematics is extremely successful when applied to phenomenological physics (the relations between phenomena). However, if we want to describe the foundations of physics – reality at the most fundamental level – nearly all the mathematical tools are useless.

    So I am a bit curious about your contribution to the new contest: “How can mindless mathematical laws give rise to aims and intention?”

  14. SE Grimm,

    Not all mathematical tools are designed to simplify reality. Some of them weren't designed for anything in particular, they were just found to be useful. And it's more about describing reality, though in practice that often implies simplification.

    I don't think I'll contribute to that essay, I don't find the question very interesting. Best,


  15. Respect: I find it useful to look at the existing equations of physics - say a solution of the Einstein Equation - and ask myself what are the consequences of letting the solution taking you where it wants to go, rather than limiting it with community beliefs and unproven conjecture.

  16. Thanks so much, Bea, for your very thoughtful and reasonable response to my query re climate change. I understand your reluctance to go into detail on this topic, but I have a question regarding an essentially technical issue that comes up when I attempt to analyze certain types of data. As you know, there are various graphs now available that track global temperatures during various time periods, and when I study these graphs it seems that, as far as I can tell, there is no long term correlation between CO2 emission levels and temperature.

    While CO2 levels have been steadily rising, according to virtually all the graphic displays, temperatures have been both rising AND falling AND remaining more or less level since roughly 1910, with no clear overall trend (though it is now warmer than it was 100 years ago). When I raise this issue on certain climate science blogs, I am very rudely dismissed as someone with no understanding of statistics. I am informed that if I understood how statistics worked I would realize that "eyeballing" the data means little, and the only "scientific" approach to understanding is via some sort of statistical analysis based on methods too complex for a non-scientist to understand. My response has always been that if there is a discrepancy between the data itself and some statistical analysis of that data, then something must be wrong. At that point my posts are greeted with howls of derision.

    What do you think, Bea? When a graphic representation of the data itself is simple and straightforward enough to evaluate by eye, is it possible for a statistical analysis to reveal some truth hidden beneath the surface that only a mathematician can understand?

  17. DocG,

    I am afraid I have to tell you that it is entirely correct what you have been told. Data can contain correlations that don't reveal themselves to the eyes and only a statistical analysis will pin them down.

    Incidentally, my previous post discussed exactly such a case. You can study the data by eye and it will look like no correlation (look at the paper by the physicists). But do an appropriate (standard) analysis and you find a statistically significant trend. Incidentally, anybody who has done statistical data analysis can tell you that this happens frequenty. The opposite is also true: you might look at a data set and believe it has a correlation and yet the analysis will show it hasn't.

    You ask "When a graphic representation of the data itself is simple and straightforward enough to evaluate by eye, is it possible for a statistical analysis to reveal some truth hidden beneath the surface that only a mathematician can understand?"

    Well, you don't need to be a mathematician to understand that the human eye isn't a device to perform quantitative statistical analysis. Also, statistical analysis isn't all that complicated and it's much more used by scientists of various disciplines than by mathematicians themselves. Having said that however, graphic representation don't always reveal correlations to the eye.

    I will not approve further comments on this. As I told you above I don't discuss these topics on my blog because they tend to become huge time sinks and I'm already running behind on to many things. I hope you understand.



  18. Thanks so much for your response to my correlation question, Bea. What you've written is helpful. I'd love to get more specific on this issue, however, because I'm really curious regarding your thoughts on the manner in which statistical analyses can be used to produce artifacts driven by confirmation bias as well as legitimate correlations. But I do understand your situation and won't pursue this issue here. In my experience this sort of controversy never reaches a satisfactory conclusion in any case, and it can be a huge eater of time and energy.

  19. When you write:

    In lack of experimental guidance, what we need in the foundations of physics is conceptual clarity. We need rigorous math, not claims to experience, intuition, and aesthetic appeal. Don’t be afraid, but we need more math.

    I could not agree with you more. But here I second P. Dirac who was pointing out conceptual problems within the existing physical theories and insisted on searching for better Hamiltonians in order to have sensible math applied. Yes, I am speaking of renormalizations and soft diagram summation. Here we have conceptual physical problems, the simplest of which is the self-induction of the electron. Math is right when it yields an infinite self-induction effect (self-mass or self-energy). It is we who impose a wrong physical construction.


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