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While physics and astronomy sported mathematical models for centuries mathematical chemistry and biology appeared relatively recently. Most of the interaction seems to go one way, established mathematical theories (differential equations, combinatorics, graph theory, etc.) are applied to formalize and solve chemical and biological problems. I am interested in the reverse effect: development of new mathematical theories inspired by chemistry and biology, like trigonometry was inspired by astronomy, or calculus by physics.

One example I know of is genetic algebras of Etherington that describe the structure of genetic inheritance. They are structurally different from algebras that emerged from physical applications or inner workings of mathematics, for example non-associative in ways distinct from Lie, Jordan or alternative algebras, and rely on different intuitions. Are there other such examples?

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    $\begingroup$ Does "mathematics" here, include Computer Science? CS wasn't really a distinct field from Math until the late 70's, and in many ways it still is a type of mathematics. $\endgroup$ – RBarryYoung Jun 29 '15 at 14:59
  • $\begingroup$ @RBarryYoung Sure, especially where computational methods are concerned. $\endgroup$ – Conifold Jul 3 '15 at 0:18
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In 1959, Alfred Wigner presented a talk on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Many papers on the unreasonable effectiveness of mathematics in this field, that field, and some other field quickly followed suit. As an opposing point of view, the unreasonably effective (800+ papers, 30+ books) mathematician I.M.Gelfand noted that (emphasis mine)

Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.

I.M. Gelfand said this after developing an interest in biology due to the premature death of his son. He organized a weekly seminar that attracted the best minds in Russian biology and mathematics. Gelfand's interest in biology resulted in pioneering work in the field of biomathematics. Not only is mathematics applicable to some aspects of biology, biology has motivated many new developments in mathematics.

Just a few of the areas where biology has inspired new mathematics:

Mathematics inspired by population modeling

Modeling population dynamics has been a fruitful application of mathematics starting with Euler, who studied age distributions in stable populations (Euler 1760). What about unstable populations? One of the seminal papers (if not the seminal paper) in the development of chaos theory was written by Robert May in 1976. May was educated as a physicist (where mathematics is unreasonably effective), but then switched to biology (where mathematics is supposedly unreasonably ineffective). His paper on population dynamics (May 1976) discusses the logistic map, $x_{n+1} = \lambda x_n(1-x_n)$, which he used to model population dynamics. This paper marked the beginning of chaos theory.

Mathematical techniques inspired by evolution and animal behavior

The contributions of biology to optimization theory are immense. Evolution has motivated a number of techniques used in mathematics and artificial intelligence, including evolutionary programming (Fogel 1966), evolutionary strategy (Rechenberg 1973), genetic algorithms (Holland 1975).

Emulating animal behavior has provided a number of other optimization techniques. These include the critter of the month optimization techniques, starting with ant colony optimization (Dorigo 1996). Ant colony optimization has been used to attack a large number of problems from very diverse fields. Now there are bees algorithms, bacterial colony optimization algorithms, foraging algorithms, all based on the behaviors of simple creatures. I won't give references for all of these; there are now entire journals dedicated to this subject (e.g., the IEEE Transactions on Evolutionary Computation). Collectively, these fall into the category of swarm intelligence. One last technique that I will give a reference on is particle swarm optimization (Kennedy 1995, Eberhart 1996).

Mathematics inspired by DNA sequencing

The above areas are new areas of applied mathematics. The problem of how to sequence DNA produced not only new applied mathematics (e.g., the image below from Letunic 2007) but new theoretical mathematics as well.


All of these developments led Joel Cohen (Cohen 2004) to conjecture that mathematics is biology's next microscope, only better; biology is mathematics' next physics, only better.


References

Joel E. Cohen (2004), "Mathematics is biology's next microscope, only better; biology is mathematics' next physics, only better." PLoS Biology 2.12:e439.

Marco Dorigo, et al. (1996), "Ant system: optimization by a colony of cooperating agents," IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 26.1:29-41.

Leonard Euler (1760), "Recherches générales sur la mortalité et la multiplication," Mémoires de l’Académie Royal des Sciences et Belles Lettres 16:144–164.

Russell Eberhart and James Kennedy, "A new optimizer using particle swarm theory," Proceedings of the Sixth International Symposium on Micro Machine and Human Science.

Lawrence J. Fogel, et al. (1966), "Artificial intelligence through simulated evolution," Wiley.

John H. Holland, (1975), "Adaptation in Natural and Artificial Systems," University of Michigan Press (second edition, MIT Press, 1992).

James Kennedy and Russel Eberhart (1995), "Particle swarm optimization," Proc. IEEE International Conf. on Neural Networks.

Ivica Letunic and Peer Bork (2007), "Interactive Tree Of Life (iTOL): an online tool for phylogenetic tree display and annotation," Bioinformatics 23.1:127-128.

Robert M. May (1976), "Simple mathematical models with very complicated dynamics," Nature 261.5560:459-467.

Ingo Rechenberg (1973), "Evolutionsstrategie Optimierung technischer Systeme nach Prinzipien der biologischen Evolution," (PhD thesis), Friedrich Frommann Verlag, Struttgart-Bad Cannstatt.

Eugene P. Wigner (1960), "The unreasonable effectiveness of mathematics in the natural sciences," Communications on Pure and Applied Mathematics 13.1:1-14. Richard courant lecture in mathematical sciences delivered at New York University, May 11, 1959.

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    $\begingroup$ Vow, what a comprehensive answer! Frankly, I was always puzzled by Wigner's title, how is effectiveness of mathematics in physics unreasonable if so much of mathematics was explicitly developed to model physics. Biology makes a better case, although of course biological laws are still ultimately based on the laws of physics. $\endgroup$ – Conifold Feb 26 '15 at 18:43
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This is a correct observation. Chemistry and biology indeed contributed very little to mathematics itself.

One of the examples of chemistry contribution is the "Belousov-Zhabotinsky reaction". This was an experimental discovery whose explanation stimulated to some extent the development of the theory of dynamical systems (known as "chaos theory" in the popular literature). A lot of sophisticated mathematics was invented to explain the Periodic Table. But this was mostly applications of mathematics TO chemistry; I cannot say that chemistry brought new ideas to mathematics.

All examples from biology which come to my mind are about population genetics, the thing mentioned in the question. This is also related to dynamical systems and related algebra (Bernstein algebras, for example).

Volterra-Lottka systems (also from population biology) stimulated the qualitative theory of differential equations in the first half of 20-s century.

But most of it again works one-way: applications of mathematics TO biology etc. In the best case, chemistry and biology provide QUESTIONS to which mathematicians sometimes find answers. But no really new ideas in mathematics that originate in chemistry/biology.

It seems that there are no new mathematical theories (of importance to mathematics itself, with no regard to applications) which came from chemistry or biology.

This is indeed very different from physics which constantly feeds mathematics with new ideas. So I just confirm your observation.

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  • $\begingroup$ I wonder why. Because biology and chemistry are inherently "less mathematical" or physics already plucked all the low hanging fruit so what otherwise might have been inspired is already developed and ready to use. I thought molecular biology and quantum chemistry might have contributed because QM descriptions are too fundamental for what they study, and graph theory is too schematic. $\endgroup$ – Conifold Feb 18 '15 at 17:27
  • $\begingroup$ This is an interesting question, and the answer is not clear. Much of the existing mathematics was created under the influence of physics. Perhaps biology and even chemistry are "too young" for this? $\endgroup$ – Alexandre Eremenko Feb 18 '15 at 19:16
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How about: (1) the logistic equation of Pierre François Verhulst (describing the change in a population over time, published 1838) leads to (2) Feigenbaum's work on chaos.

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  • $\begingroup$ This is very interesting, I didn't realize that Feigenbaum universality was biology motivated. Could you expand and give a reference. $\endgroup$ – Conifold Feb 18 '15 at 21:59
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    $\begingroup$ Fiegenbaum was my undergrad research advisor, long ago (1977-1978). I had expressed an interest in combining software and statistical physics for my senior research project. I was assigned to him, and he asked me to investigate the behavior of $x_{n+1} = \lambda x_n (1-x_n)$. I thought what?!? That can't be a good research problem. It was. If search tools were anything like what they are now, I would have quickly run across Robert May's seminal 1976 paper. As it was, Feigenbaum showed me that paper only after I had spent a semester developing software to study that seemingly simple equation. $\endgroup$ – David Hammen Feb 21 '15 at 4:57
  • $\begingroup$ Plus one, by the way. $\endgroup$ – David Hammen Feb 21 '15 at 5:09
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  • The famous Polya's 1937 paper "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen" is considered, as far as I understand, as one of pillars of the modern theory of combinatorial enumeration. The title of the paper suggests that, at least to some degree, it was motivated by a question from chemistry.

  • Genetic algorithms.

  • Some algebro-geometric questions from algebraic statistics seem to be motivated by biological considerations.

My personal impression (based mainly on hearing people from a few IHES conferences devoted to interaction of mathematics and biology, and personal experience from working in a biotech industry) confirms Alexandre Eremenko's comment that "biology is too young": the structure of biological knowledge is mainly descriptive, as opposed to the physical one, organized in "theories".

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