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Math Anxiety

November 21, 2016

My training as a physical scientist included a lot of mathematics, the apotheosis of which was a course in continuum mechanics, all of which I have long forgotten since I never had a practical use for it. While I did well in mathematics (see figure), I never really liked mathematics until the advent of experimental mathematics. I'm a dyed-in-the-wool experimentalist.

Figure captionThis document yellows with age as my hair grays.

A good way to look at this is that gray hair is better than no hair at all.

(Click for larger image.)

Mathematics is an inextricable part of the sciences, and I would be remiss if I didn't mention the famous paper, "The unreasonable effectiveness of mathematics in the natural sciences,"[1] by Physics Nobelist, Eugene Wigner. However, mathematics has become a part of science only in recent times. There are no equations in the works of Aristotle (384 BC-322 BC), who didn't examine a large enough statistical sample when he wrote that the "lesser sex" had fewer teeth.

We can place most of the "blame" of the introduction of mathematics into science on Galileo. Galileo was arguably the first physicist in the modern sense, as evidenced by his design, performance, and analysis of experiments. His analysis involved mathematics; and, as he stated in his 1623 book, The Assayer (Il Saggiatore),[2]
Philosophy is written in this grand book - I mean the universe - which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.[3]

La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l'universo), ma non si può intendere se prima non s'impara a intender la lingua, e conoscer i caratteri, ne' quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto.[4]

Galileo, sketch by Ottavio Leoni (1578–1630)Galileo Galilei (1564-1642), sketch by Ottavio Leoni (1578–1630).

My favorite among his experiments is his determination of gravitational acceleration by rolling balls down inclined planes.

Galileo also proposed an experiment to determine the speed of light.

(Portrait at the Biblioteca Marucelliana, via Wikimedia Commons.)

As I wrote in a previous article (Science and Math, July 16, 2012), some scientists from disciplines other than physics have what's called "physics envy." They think that the more mathematical their discipline, the more "scientific" it is, but there's a downside to physics envy. A 2012 paper in the Proceedings of the National Academy of Sciences by Tim Fawcett and Andrew Higginson of the School of Biological Sciences, the University of Bristol (Bristol, UK), showed that papers in biology containing too many equations are generally ignored.[6]

Purity, xkcd comic no. 435
Physics envy, or math superiority? A cartoon from Randall Munroe's xkcd Comics, licensed under a Creative Commons Attribution-NonCommercial 2.5 License. (xkcd comic 435.)

Fawcett and Higginson found that, based on the number of equations per page, mathematically dense papers are referenced half as often as those with little or no math.[6] It's been said that every equation reduces book sales by half, and such a marginal effect was seen in the mathematical biology papers, where each equation in the body of the article reduced the number of citations by 28%.[6] However, moving equations to an appendix results in normal citation levels.[6]

An equation from a quantitative biology paper in population dynamics
An equation from a random paper in the biological sciences. Such equations cannot be avoided in this paper's particular subject area, population dynamics. (Onur Gün and Atilla Yilmaz, "The stochastic encounter-mating model," arXiv, November 4, 2016.[7])

Fawcett and Higginson's statistical analysis was rebutted in another paper that looked at not only biology papers, but also physics papers, containing equations. This reanalysis showed that citations for the biology papers were not dependent on the number of equations.[8] Since physicists are supposedly more mathematically adept than biologists, the authors of this new study performed a systematic analysis of the citation counts of a thousand papers published in Physical Review Letters, volumes 94 and 104.

Their analysis was aided by the contribution of extensible markup language (XML) source files for these articles by the publisher, The American Physical Society.[8] For physics papers, equations were not found to have an influence on citations.[8] While equations in that analysis were not found to be a detriment to physics citations, the authors did identify the factors that lead to citation; namely, have "a dozen coauthors, cite several dozens of papers; and, on the average, use 1 figure per page."[8]

Fawcett and Higginson have defended their thesis of more math leading to less readership in a recent paper in which, they too, looked at physics papers.[9-11] They examined the same dataset of 2000 articles in Physical Review Letters. Their analysis shows that citation counts in physics, as in biology, are lower when articles have a high density of equations, and they speculate that this trend is true among all sciences, and it diminishes the dialogue between theory and experiment.[9,11]

Since physicists are well trained in mathematics, this finding suggests that extra math courses in the other sciences won't encourage access to the equation-laden articles. The solution might be found in a different approach to exposition in math-heavy papers.[11] Says Fawcett,
"Physicists need to think more carefully about how they present the mathematical details of their work, to explain the theory in a way that their colleagues can quickly understand. It takes time to scrutinize the details of a technical article--even for the most distinguished physics professors--so with many competing demands on their time scientists may be choosing to skip over articles that take too much effort to digest."[11]

References:

  1. E.P. Wigner, "The unreasonable effectiveness of mathematics in the natural sciences," Communications on Pure and Applied Mathematics, vol. 13, no. 1 (February, 1960). pp. 1-14. A PDF file is available here.
  2. Galileo Galilei, "Il_Saggiatore."
  3. Nature is "written in the language of mathematics," Galileo Galilei, The Assayer, English text.
  4. Nature is "written in the language of mathematics," Galileo Galilei, Il Saggiatore, Italian text.
  5. Scientists struggle with mathematical details, Bristol University Press Release, June 25, 2012.
  6. Tim W. Fawcett and Andrew D. Higginson, "Heavy use of equations impedes communication among biologists," Proceedings of the National Academy of Sciences, vol.109, no. 29 (July 17, 2012), pp. 11735-11739, doi: 10.1073/pnas.1205259109.
  7. Onur Gün and Atilla Yilmaz, "The stochastic encounter-mating model," arXiv, November 4, 2016.
  8. Jonathan E. Kollmer, Thorsten Pöschel1, and Jason A. C. Gallas, "Are physicists afraid of mathematics?" New Journal of Physics, Volume 17 (January, 2015), http://dx.doi.org/10.1088/1367-2630/17/1/013036. This is an open access publication with a PDF file available here.
  9. Andrew D. Higginson and Tim W. Fawcett, "Comment on 'Are physicists afraid of mathematics?'," New Journal of Physics, vol. 18 (November, 2016), http://dx.doi.org/10.1088/1367-2630/18/11/118003. This is an open access publication with a PDF file available here.
  10. A. D. Higginson and T. W. Fawcett, "Statistical Analysis of the Effect of Equations on Citations," zenodo.org (PDF File).
  11. Even physicists are 'afraid' of mathematics, University of Exeter Press Release, November 11, 2016.

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