Iterative processes at the heart of mathematics | Tangente

Apply a computation to some data, then apply it again to the result: this is an age-old approach found throughout mathematics. Is multiplication itself not repeated addition? And exponentiation repeated multiplication? Soon after writing emerged, we already find evidence of approximate methods for solving problems, such as calculating a square root. The accuracy achieved without machines, and without a numeral system as convenient as our own, remains astonishing and suggests that effective iterative processes were already at work. Later, among Greek mathematicians, another leap in the art of repetition began to emerge: repetition was no longer confined to computation; reasoning and proofs themselves could also be repeated, yielding general truths about the infinite set of integers.

Poincaré: brilliant thinking within everyone's reach --------------------------------------------------

With his wide-ranging research in physics and astronomy, his standing as a philosopher, and his status as one of the last people to understand all the mathematics of his day, Henri Poincaré is often regarded as one of the last great polymaths.

As committed to the philosophical study of science as he was to popularizing it, Poincaré published several accessible works of seldom-matched clarity, including La Science et l’Hypothèse (1902), La Valeur de la Science (1905), and Science et Méthode (1908), all published by Éditions Flammarion. Dunod brought these three works together in 2013 (La Science selon Henri Poincaré; see Tangente 179, 2017). In the first of the three, he discusses at length the nature and place of mathematical induction. He describes it as follows:

"We see that the conclusion of each syllogism serves as the minor premise of the next. Moreover, the major premises of all our syllogisms can be reduced to a single formula. If the theorem is true for n – 1, it is true for n.

Thus, in proofs by induction, we merely state the minor premise of the first syllogism and the general formula containing all the major premises as particular cases.

This never-ending chain of syllogisms is thereby reduced to a sentence only a few lines long."

Poincaré also wrote for younger readers. One such contribution appeared in an edited volume republished in 2010 by Éditions Hermann under the editorship of Christian Gerini, Ce que disent les choses : quand Henri Poincaré écrit pour les enfants (see Tangente 156, 2014).

!Henri Poincaré

A way of thinking -----------------

Some historians of science, including Jean Itard (1902–1979), regard certain proofs by Euclid as the earliest examples of mathematical induction. Blaise Pascal would later give a formalized account of the method that was both explicit and lucid (see *Blaise Pascal takes on mathematical induction*). By the late 19^th century, mathematicians had explicitly linked mathematical induction to the fundamental nature of the integers and to their iterative axiomatic construction. Mathematical induction thus became the "pre-eminent form of mathematical reasoning", as Henri Poincaré explains in La Science et l’Hypothèse. The most famous scientist from Lorraine emphasizes in particular that "the essential feature of mathematical induction is that it contains, condensed, so to speak, into a single formula, infinitely many syllogisms […]; it is an instrument that enables us to pass from the finite to the infinite".