Science and Math
July 16, 2012
All
scientists are knowledgeable in
mathematics, although the type of math and the degree of study will vary considerably.
Biologists and
psychologists, for example, are especially skilled in
statistics, and
physicists are expected to have extensive knowledge of many mathematical fields.
There's even a
Journal of Mathematical Physics, a past editor of which wrote a popular book on group theory.[1] String theorists are also adept at bringing
mathematical physics to new heights, stripping away nearly all the
physics at the same time.
Pythagoras started the idea that mathematics is the route to our understanding of
nature when he wrote, "Mathematics is the way to understand the universe," and "Number is the measure of all things;"[2] but Pythagoras was a mathematician and
mystic, not a scientist. As I wrote in a
recent article (Planetary System Formation, March 21, 2012),
Galileo was the first to apply mathematics to science.
Pythagoras from the Nuremberg Chronicle, 1463.
Woodcut attributed to Michel Wolgemut and Wilhelm Pleydenwurff
(Via Wikimedia Commons).
Galileo, known as the father of modern physics, applied mathematics to his
experimental data, and he believed that nature obeys
mathematical laws. He expressed this belief in
The Assayer:
"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 in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth."[2]
In more recent times,
Physics Nobelist,
Eugene Wigner, wrote that it's a mystery why mathematics is so useful in science:
"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."[4]
Scientists other than physicists often use mathematics, although sometimes to an extent not appropriate to their discipline. Some think that the more mathematical their discipline, the more "scientific" it is, an idea that's sometimes called "
physics envy."
Chemistry Nobelist,
Ernest Rutherford, was dismissive of the non-mathematical sciences, saying that "all science is either physics or stamp collecting."[5]
Mathematics is surely useful, but at times it can also obfuscate. Even the great quantum mechanic, Niels Bohr, was interested in simple, verbal expressions of the fundamental principles that his mathematics tried to elucidate. Bohr always sought the truth behind the math through things such as the
Copenhagen interpretation of quantum mechanics. Ernest Rutherford, who was against stamp-collecting, still said that
"An alleged scientific discovery has no merit unless it can be explained to a barmaid."[5-6]
A recent paper in the
Proceedings of the National Academy of Sciences presents results of a study on the dark,
obfuscating side of mathematics in science.[7-8]
Tim Fawcett and
Andrew Higginson of the
School of Biological Sciences, the
University of Bristol (Bristol, UK), found that scientists will ignore
research papers in the fields of
ecology and
evolution if they have too many
equations.[7-8] Using as a study metric the number of equations per page, they found that the mathematical papers are referenced fifty per cent less often than those with little or no math.[7]
The Bristol biologists found that such papers received 28% fewer
citations for each additional equation per page in the main text. Not surprisingly, mathematical papers are cited most often by similar papers, but they are less likely to be cited by other papers.[8] Citations in these non-mathematical papers decreased 35% for each additional equation per page in the main text.[8] One encouraging result is that moving the equations to an appendix boosted a paper to a normal citation level.[8]
Says study author, Tim Fawcett,
"This is an important issue because nearly all areas of science rely on close links between mathematical theory and experimental work. If new theories are presented in a way that is off-putting to other scientists, then no one will perform the crucial experiments needed to test those theories. This presents a barrier to scientific progress."[7]
This Bristol study is apparently the first to examine the influence of mathematics on a paper's citation rate.[7] Although one long-term remedy would be to improve biological scientists' facility with mathematics, a viable near-term solution is to have a more explanatory text and move all the math to an appendix.[7] I think
physical scientists prefer to have the equations inline with the text.
One side of a random equation, pulled from the Mathematical Physics section of arXiv. I know all the operators, but I and my barmaid acquaintances avoid such papers)
References:
- Morton Hamermesh, "Group Theory and Its Application to Physical Problems," Dover Publications (December 1, 1989), 544 pp., via Amazon.
- R. W. Hamming, "The Unreasonable Effectiveness of Mathematics," The American Mathematical Monthly, vol. 87, no. 2 (February, 1980), via the Center for Cancer Research Nanobiology Program.
- Galileo quotations on Wikiquotes; Selections of a translation of The Assayer; Stillman Drake, "Discoveries and Opinions of Galileo," Doubleday & Co.(New York, 1957), pp. 231-280.
- 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, and here.
- Ernest Rutherford quotations on Wikiquote.
- George Gamow, "My World Line : An Informal Autobiography," Viking Adult, April 28, 1970, p. 66 (via Amazon).
- Scientists struggle with mathematical details, Bristol University Press Release, June 25, 2012.
- Tim W. Fawcett and Andrew D. Higginson, "Heavy use of equations impedes communication among biologists," Proceedings of the National Academy of Sciences, Published online before print June 25, 2012, doi: 10.1073/pnas.1205259109.
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