Have you ever observed the shape of a sunflower flower, the structure of a snowflake or the morphology of a fern? Beyond their fascinating beauty, we can also see mathematical objects in them, since the spirals of the sunflower flower follow a famous numerical sequence called the Fibonacci sequence , the snowflakes present particular hexagonal symmetries and the morphology of the fern describes a fractal geometry.
Many other examples illustrate the extent to which mathematical objects are present in nature. Conversely, mathematics is used to understand the phenomena that surround us: it is for example thanks to differential equations that we can precisely calculate the trajectories of the stars or predict the weather in a few days. Mathematical objects seem to apply to almost all sciences with remarkable performance. This efficiency is particularly intriguing in physics.
Eugene Wigner , Nobel Prize winner in 1963 for his contributions to the theory of elementary particles through the discovery and application of fundamental principles of symmetry, in 1960 posed the question of the "unreasonable efficiency" of mathematics in the natural sciences. This efficiency may seem superficial if we reduce mathematics to a simple toolbox used by other sciences. But the link between mathematics and nature is much deeper: mathematics is often essential to the understanding of phenomena and it makes it possible to make unexpected predictions that will only be observed much later. Here are two fascinating examples.
Cicadas singing to the rhythm of prime numbers
Magicicada Cassini cicadas have a very particular life cycle: these insects remain underground for years, and come out to reproduce sometimes every 13 years, sometimes every 17 years, and exclusively at these two time intervals.
Specialists have proposed to explain this phenomenon using evolutionary arguments: these insects tend to minimize their interactions with predators (who have a life cycle of n years). However, this explanation only becomes satisfactory with the help of the following purely mathematical result: if p is a prime number and n an integer strictly smaller than p , then the smallest common multiple of p and n is p × n . Indeed, imagine that the duration of the cycle of the predator is 4 years. If the cicadas (which have a 17-year cycle) are confronted with their predator one year, they will not be confronted the next time they appear, since this predator will appear in the 16th year and in the 20th year but not in the 17th year. In fact, they will only intersect after 68 years (because 17 × 4=68).
One then wonders if the Darwinian mechanism of natural selection would not in fact be a mathematician who would apply this theorem in his work. Whatever the answer, another question remains: is there another explanation for this phenomenon that does not appeal to abstract notions of mathematics?
“My equation was smarter than me”, says Paul Dirac, Nobel Prize
At the end of the 1920s, the two great physical theories, general relativity and quantum physics, were still young but already well developed and widely studied. However, it seemed at the time – and it is almost always valid today – that these two physics ignore each other: the first describes the large-scale behavior of the universe and the other is interested in the infinitely small.
Paul Dirac then decides to connect these two visions by formulating an equation which describes the quantum state of an electron while taking into account the principles of relativity. Armed with advanced mathematical concepts , hitherto confined to the world of abstract objects, Dirac established an equation that correctly described electrons and was consistent with Schrödinger's equation when particle velocities were very small compared to that of light.
But, alongside these successes, new problems appear since the equation also admits other solutions than the particles already known… In other words, Dirac's equation seemed to predict the existence of new particles with “negative energies”. A mathematical consequence of Dirac's equation, which was rejected by all physicists of the time, including by Dirac himself. It was only after a few years of debate and strenuous effort that Dirac gave in to his own equation and named the hypothetical negative energy particle an "anti-electron". This particle will be observed experimentally three years later, and will thus initiate the discovery of antimatter. Commenting on this episode, Dirac would later say , “My equation was smarter than me. »
Mathematics: language of nature?
The beginning of this fusional relationship between mathematics and physics is often attributed to the work of Kepler and Galileo in the 17th century. There is then an important turn in the history of science, expressed by the famous quote from Galileo:
“Philosophy is written in this vast book which constantly stands open before our eyes (I mean the Universe), and one cannot understand it if one does not first learn to know the language and the characters in which it is written. Now it is written in mathematical language, and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it, without which one really wanders in a dark labyrinth. (Galileo Galilei, "The Assayer", 1623)
The use of the adverb “humanely” gives this phrase an ambivalent meaning that has occupied, and continues to occupy, the greatest minds of the last three centuries. Does Galileo's assertion mean that mathematics is the true and only language of nature? Humans must then learn this language in order to understand the reality they observe. Or does it mean, on the contrary, that mathematics is a human invention which makes it possible to make the observed phenomena intelligible to study? There is of course no definitive answer that puts everyone in agreement on the nature of this mysterious link between mathematical objects (abstract objects) and physical reality (empirical objects).
Nevertheless, major philosophical schools have offered their theses on the subject. Some empiricists think that mathematical objects are the fruit of a process of purification (of an abstraction) of concrete and observed objects. Conversely, idealists , sometimes called Platonists, consider that mathematical objects exist as idealities separate from the observed world and these idealities apply to the phenomena of nature because they served as models for their constitution. The Kantians meanwhile think that we cannot apprehend the phenomena of nature outside of innate structures identical to all human beings: the "a priori" forms of sensitivity , which are space, time and the concepts of understanding . Space and time therefore constitute the conditions of any physical experience but also of any mathematical construction, it is inevitable that any natural science is a mathematical science . Finally, other more formal philosophers see no mystery in history since mathematics is just a set of symbols that scientists (physicists, biologists, economists, etc.) interpret at their leisure.
While scientists and philosophers continue to debate the issue, bees have been able to build nests for centuries whose hexagonal structure allows for the maximum number of cells for a minimum of wax. This mathematical property was conjectured in the 4th century but was only demonstrated in 1999 – the theorem is even called the honeycomb theorem .
Whether mathematics is just a representation that makes nature intelligible to study or whether it is its true language, this nature never ceases to fascinate, and it will certainly continue to do so for a long time. 
Athmane Bakhta , Engineer - researcher in applied mathematics, French Alternative Energies and Atomic Energy Commission (CEA)
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