In 2019, I had the opportunity to spend seven months in Japan, under the JSPS long-term Fellowship Program. My primary affiliation was with the Department of Philosophy at Kyoto University, and my host was Dr. Jun Otsuka. I am a philosopher of mathematics and philosopher of science, and one of my main research interests concerns how mathematics is applied in science and what this can tell us about the nature of mathematics.
During my time in Japan, I worked on the more specific question of when, if ever, mathematics helps in the explanation of physical phenomena. Together with Dr. Otsuka, I looked at various case studies in which mathematics is used to help explain biological phenomena. Below I shall briefly summarize two of these examples.
Cicadas are large insects with a very loud call, and there are numerous cicada species found in Japan. However, my own research focuses on the North American periodical cicada. These cicadas spend a long time developing as nymphs underground before emerging as a single group in very large numbers. The two main varieties of periodical cicada in North America spend either 13 years or 17 years underground between emergences. The basic evolutionary strategy of periodical cicadas is called predator satiation. Cicadas are nutritious and have no natural defenses, so what they do is overwhelm potential predators
by appearing in such large numbers that there are too many for the predators to consume. In this way, some cicadas survive to reproduce and the cycle repeats itself.
But why these particular lengths of years? What, if anything, is special about 13 and 17? One theory that has been proposed by biologists is that it is advantageous for cicada periods to be prime numbers. Why? If we imagine that there are predators that are also periodical, we can see that prime numbers minimize the frequency of intersection with these predators. Compare a 12-year cicada with a 13-year cicada. If there are 3-year periodical predators, these will catch the 12-year cicada every time it emerges. But the 13-year cicada will be caught only every 39 years. This explanation relies on some mathematical principles from the area of mathematics known as number theory. In a way, number theory helps to explain why periodical cicadas have the life cycles that they in fact have.
Important work uncovering the evolutionary strategies of periodical cicadas has been done by Professor Jin Yoshimura and his research team based at Shizuoka University. From a philosophical perspective, my interest in this kind of example is that it seems to show that our core mathematical principles are true, because they best explain phenomena that we see in nature. In a similar way, we believe that scientific entities such as black holes and neutrinos really exist because they best explain the outcomes of our observations and experiments.
A second case study that I looked at during my JSPS-supported research in Japan concerns the flowering patterns of bamboo. I happened to be in Kyoto during the flowering of a species of bamboo that flowers only once every 60 years, and I was fortunate to be able to see examples of this bamboo flowering at Rakusai Bamboo Park in the western suburbs of Kyoto. Why is the flowering of bamboo important? Because when bamboos flower they also produce seeds, and these seeds are an important source of food for various kinds of animals and birds. It turns out that bamboos, like periodical cicadas, also follow the strategy of predator satiation. By having long intervals between flowering, and by flowering all at the same time, huge numbers of seeds are produced at once, which overwhelms the predators and results in plenty of seeds being left over.
This leaves one important puzzle still to be explained. As we have seen, cicadas have converged on periods that are prime numbers, in order to minimize intersection with periodical predators. However, most bamboo species have period lengths such as 32 years, 60 years, and even 120 years, none of which are prime numbers. Why is there this difference? This is an interesting, and still not fully resolved question. One hypothesis is that because plants have such long life cycles compared to animals, being prime simply doesn’t matter. Plants simply outlast their animal predators and can avoid them by having dramatically long intervals between successive flowerings. However, the situation is further complicated by the fact that bamboo life cycle lengths are not merely non-prime numbers, they are highly composite numbers. What does this mean? Basically, it means that they have a large number of prime factors. For example, 120 = 2 x 2 x 2 x 3 x 5. It is almost like these periods are the extreme opposite of prime numbers! One idea to explain this is that bamboo periods evolved as multiples of previous periods. For example, a 15-year bamboo might develop a mutation that leads it to change to a 30-year period. The advantage of this “evolution by discrete multiplication” is that the resulting flowerings (and seed production) are protected from predation by those 15-year bamboos which did not mutate, because both the 30-year bamboos and the 15-year bamboos will flower together at the same time.
As with periodical cicadas, important work on bamboo seed production and its timing has been published by research teams in Japan. One notable recent study is by Dr. Keito Kobayashi and his team at the Kansai Research of Forestry Research Institute in Kyoto.
During my seven months in Japan, my host Dr. Jun Otsuka, the Kyoto University Philosophy Department, and the Research Institute for Mathematical Sciences (RIMS) all provided me with a welcoming and intellectually stimulating environment. I was able to give several talks, and to attend a reading group on topics in the philosophy of science. Meanwhile, I lived with my wife and two daughters in a house near Ginkakuji, the Silver Temple, just a few steps away from the Philosopher’s Path. Since I am a philosopher by training, it seemed appropriate that my daily commute to my office at the Kyoto University main campus involved bicycling along the Philosopher’s Path!