More than half a century has passed since Eugene Wigner (1902-1995) published his famous article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, and yet it is still surprising today because of the depth of the problem he posed (see also the post Einstein, Wigner and the mystery of mathematics, in this same blog). Wigner was born in Budapest; in the early 1920s he went to Germany, first to Berlin and then to Göttingen, where he began his research in quantum mechanics and tried to collaborate with David Hilbert. After Hitler’s rise to power in 1933, Wigner, who was Jewish but a convert to Lutheranism, left Germany for good and settled in the United States (a life path almost identical to that followed by other of his illustrious compatriots, such as John von Neuman, Edward Teller and Gabor Szëgo). When uranium fission was discovered at the end of 1938, and fearful of the Nazis’ potential use in warfare, he was one of the group of scientists who, together with Albert Einstein, warned President Roosevelt of the danger; he later became actively involved in the manufacture of the first atomic bombs. For his contributions to the understanding of the structure of the atomic nucleus, Wigner was awarded the Nobel Prize in Physics in 1963. Three years earlier, he had published the paper to which I have referred above.
Wigner’s thesis, in his own words, states: “The enormous usefulness of mathematics in the natural sciences is a circumstance bordering on mystery; a fact for which there is no rational explanation”.
This dichotomy between the abstract and the useful is related to another one that also occurs in mathematics: the confrontation between the emotional and the rational. And both are probably nothing more than a reflection of similar tensions in the human brain.
Although the human brain is what distinguishes us as a species, it remains the most unknown organ. We have barely begun to scrape away the hard crust that prevents us from accessing its secrets, and we are even unaware of the reasons for its most basic uses or needs (for example, we don’t know the purpose of sleeping or dreaming). Our brain incorporates as standard a logical, rational structure, closely linked to language; clearly a very recent addition or improvement in the evolutionary scale. But it is also in the brain where the emotional, the passionate, the instinctive, filtered through the genes that we have inherited from the species that preceded us in the family tree, resides. The rational and the instinctive/emotive seem to oppose each other; thus, Piaget labelled as intelligence the most elaborate function of the brain, the act of understanding – as opposed to the instinctive: what we use when we don’t know what to do.
The instinctive is more linked to what we see, what we hear, what we smell, what we perceive via our skin, our touch, our senses in short; we relate to nature mostly through the senses. However, intelligence generates another type of knowledge that is intrinsically different, but with a power that allows us an incomparably superior understanding and sometimes mastery of what surrounds us: intelligence has allowed us to discover secrets of nature that are indiscernible to the senses. In a way this justifies why it is not irrational that physics gets it right when explaining nature. In Wigner’s words: “The physicist is interested in discovering the laws of inanimate nature”. The effectiveness of physical theories in explaining nature is therefore sought after, which is why it is more difficult to label it as unreasonable. The unreasonable happens because on many occasions the physicist ends up using mathematics and its concepts, which have not been created to explain any external physical reality, but have been created in the light of an inner and abstract reasoning, and on many occasions they are more the result of an aesthetic whim than anything else. Physicists, moreover, sometimes use certain mathematical theories because they have nothing better at their disposal, which makes the efficacy of mathematics even more irrational: “One possible explanation for the physicist’s use of mathematics,” wrote Wigner, “is that the physicist is an irresponsible person: when he finds a connection between two quantities which suggests to him a connection well known in mathematics, the physicist leaps to conclude that the connection is that already discussed in mathematics, simply because he knows of no other similar connection”.
Besides, in the cerebral dichotomy between emotion and reason, mathematics seems to fall on the side of the rational, and it is often claimed that it is even opposed to the emotional. This statement is not true, however, because mathematics has a strong emotional component, as anyone who has been involved in it or is familiar with its history knows well. Mathematics is split, like the brain: it is true that the rational is essential, but it also has a distinctly passionate component, as mathematics moves between the heat of discovery and the cold logic of proof (this enriching approach is the one we usually adopt in this blog).
With more or less importance throughout history, proof entered mathematics at the hands of the Greeks. A proof is an act of reflection, of meditation, which takes us away from reality and takes us into the depths of our thinking. A proof is a dialogue, a conversation with our own mind, which we try to convince of the logic of a reasoning. When we look for a proof, we move away from reality, we abstract from it; proofs are usually made by the mathematician in solitude, isolated, locked inside himself. It is not unusual for a mathematical demonstration to have little or no relation to the information conveyed to us by the senses and a great deal to the way in which thoughts are ordered and plotted inside the human mind. This is what Russell meant when he said that Pythagoreanism had left as its inheritance “the conception of an eternal world which reveals itself to the intellect and not to the senses”. And this is also what Wigner meant when he said: “Mathematics is the science of performing highly skillful operations with concepts and rules invented precisely for that purpose”.
That is why the usefulness of mathematics in the natural sciences is so irrational. Mathematics is full of abstract concepts that seem not to belong to this world, to the point that Karl Popper (the most influential philosopher of science in the 20th century) said that it is not even a science, or not in the sense that other sciences are, precisely because its main concern is not the study and knowledge of reality. Sometimes mathematics is more like theology than physics or chemistry, and perhaps that is why nobody has been able to find the reason for its irrational efficacy in explaining the phenomena of nature.
Las matemáticas son una invención humana, un conjunto de axiomas y reglas de la lógica que se combinan para desarrollarlas. Y, sin embargo, son de gran utilidad para describir el universo. Quizás sea porque los axiomas y las reglas que utilizan parten de nuestra experiencia o perspectiva de la realidad.
En geometría, al comienzo, se aceptó el quinto postulado de Euclides por su “evidencia”. Sin embargo, al eliminarlo, surgieron otras geometrías.
Tengo entendido que algunos investigadores trabajan eliminando o modificando los axiomas o las reglas lógicas que se utilizan de costumbre para descubrir nuevas matemáticas.