Translated from french by Grok

Mental activity in mathematical research (as in any formalizable theoretical research) essentially consists of two parts: the technical and the heuristic. The first is demonstrative and powerful, the second is dynamic and elegant. Classically, these two parts are considered equally important. Our position is somewhat different, so we will explain our viewpoint.

In this context, one of the first generic examples to consider is proof by induction. Indeed, it is the tool par excellence for proving a known result. More precisely, it is a procedure that allows one to prove something without any additional information. Thus, from a cognitive standpoint, proof by induction is static. And in a certain way, it is this property that gives its demonstrative character its power. Nevertheless, we must note right away that if the result is not already known, proof by induction—despite its power—is completely useless.

Another generic example, this time more historical, is the discovery of the exact value of the Riemann zeta function at 2. The search for this value preoccupied many mathematicians (among them Bernoulli) until Euler, thanks to a surprising heuristic, managed to find it. Strictly speaking, from a technical point of view, this method is incorrect because it treats an infinite series as if it were a polynomial. However, its elegance and the genius of Euler’s intuition are justification enough for its existence on their own. Moreover, it was Euler himself who later provided the rigorous proof of this new result. So the classical technician or mathematician can accept this approach, viewing its first part as a minor lapse…

Our position, even if it may seem presumptuous, is more extreme in this regard. Heuristic needs no technical justification for its existence: its ontological beauty is more than sufficient. And we can even go further. Indeed, an idea—even an original one—is accepted by the community in a relatively consensual way once its truth has been proven. However, in pure mathematics this is not necessary; only consistency matters to us. We are not seeking the reproduction of phenomena but the creation of entities. Unlike physics, which is the science of truth, mathematics is the art of beauty.

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