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Opinion – Marcelo Viana: How mathematics reaches infinity


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In common parlance, it’s called the “domino effect”: if we line up upright dominoes in such a way that the fall of any one causes the fall of the next, and we drop the first piece, then they will all fall. Mathematicians call it the principle of induction: if a certain statement is true for the number 1, and if the fact that it is true for an integer N implies that it is also true for N+1, then it is true for all integers positive.

The first to use the word “induction” in this sense was the Englishman John Wallis (1616–1703), in the work “Arithmetica Infinitorum”, published in 1656. His compatriot Augustus de Morgan (1806-1871) coined the expression “induction mathematics” in 1838, as the title of an encyclopedia entry, but it used more “continuous induction”.

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The German Dedekind (1831–1916) preferred to speak of “complete induction”, to emphasize that with it we can rigorously prove statements for all numbers, an infinite set, while the experimental sciences are limited to verifying their laws only in a finite number of examples.

But the principle of induction was discovered much earlier: it was already used, albeit informally, in the works of the Persian Abū Bakr al-Karajī (‎953–1029), around the year 1000, and the Indian Bhāskara (1114–1185) . The first to use it rigorously was the Jewish mathematician Levi ben Gershon (1288–1344), better known as the Gersonides. In 1575, the Italian Francesco Maurolico (1494–1575) used induction to prove that the sum of the first N odd numbers is equal to N2.

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The first to formulate the principle of induction explicitly was the Frenchman Blaise Pascal (1623–1662), in his “Treatise on the Arithmetic Triangle”, published in 1565. Jacob Bernoulli (1655–1705), dean of the remarkable Bernoulli family, used it regularly in his works, making him widely known.

Pierre de Fermat (1607 – 1665), on the other hand, used quite a related principle, called the infinite descent method: it consists in showing that if an equation has some positive integer solution then there is another smaller solution; in this way it is shown that there is actually no solution. This kind of reasoning goes back to Euclid’s “Elements”, where it was used, for example, to prove that every integer is divisible by some cousin.


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