Memory plays tricks, but I think I remember this moment quite well. It was 2 years after my Baccalauréat. In the midst of a massive ingestion of math and physics in "prépa", I could not help thinking about it all after classes. My thoughts that night were on the laws governing electric and magnetic fields. The so-called Maxwell's equations. They involve a mathematical arsenal called "partial derivatives", discovered by Newton and Leibniz in the seventeenth century.

Lost in my Maxwellian dreams, I started to wonder: "How is it that abstract mathematical tools developed over millennia are suddenly perfectly adapted to describe something real?"

I soon learned that I was far from being the first to wonder about this. Had not Galileo written centuries ago that "the book of nature is written in the language of mathematics"?

Albert Einstein also asked [1]:
"How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?"
Finally, Eugène Wigner, physics Nobel in 1963, wrote in 1960 a text which title is as famous as explicit: The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

A few thoughts later, an idea dawned on me: math exist. They are not invented. They are discovered. And here also, I realized many had came to the same conclusion. Cédric Villani for example Fields Medalist 2010 [2]:
"I am among those who believe that there is a pre-existing harmony... [it is] an unexplained intuition; a personal and quasi-religious conviction."
Alain Connes, Fields Medalist 1982, is also worth quoting here [3]:
"Two extreme viewpoints are opposed in relation to mathematical activity. The first, to which I completely subscribe, is of Platonic inspiration: it postulates that there exists a mathematical reality, raw, primitive, which predates its discovery. A world which exploration requires the creation of tools, as it was necessary to invent vessels to cross the oceans. The second viewpoint is the one of the formalists; they deny any preexistence to mathematics, believing that they are a formal game, based on axioms and logical deductions, thus a pure human creation."
Then he adds,
"This viewpoint seems more natural to the non-mathematician, who refuses to postulate an unknown world of which he has no perception. People understand that mathematics is a language, but not that it is a reality external to the human spirit. The great discoveries of the twentieth century, especially the works of Gödel, have shown that the formalist viewpoint is not tenable. Whatever the exploratory medium, the formal system used, there will always be mathematical truths that will elude it, and mathematical reality cannot be reduced to the logical consequences of a formal system."
As Roger Penrose writes about math in general and the Mandelbrot set in particular [4],
"It is as though human thought is, instead, being guided towards some external truth - a truth which has a reality of its own.... The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there."
The Mandelbrot set, the Bernoulli numbers, the googolplex-th decimal of Pi [5], the non-trivial zeros of the Riemann zeta function, the Lorenz attractor... the list is endless, infinite indeed. All these things exist.

Connes, Villani, Penrose... I was finally in good company. This "personal and quasi-religious conviction," as Villani says, acquainted me with the possibility that something non-material might exist. It was probably, with the reading of Hermann Hesse, the beginning of my spiritual journey.

  1. Einstein, Geometry and Experience, 1921.
  2. Pierre Cartier, Jean Dhombres, Gerhard Heinzmann, Cédric Villani, Mathématiques en liberté, La Ville Brûle, 2012, page 60.
  3. Alain Connes interviewed by Sylvestre Huet, Libération, december 1, 2001.
  4. Roger Penrore, The Emperor's New Mind, Chapter 3 on Mathematics and Reality.
  5. A googolplex is 1 followed by 10100 zeroes. We'll probabbly never know what is this digit, but it exists.