In Folha, Viana talks about Gauss and the importance of ideas.
Imagem: Pxfuel
Column by Marcelo Viana, director-general of IMPA, in Folha de S.Paulo :
To the assertion that a certain theorem "will be very difficult to prove because there is no notation to represent prime numbers," Gauss responded with the famous ironic exclamation in Latin, "notationes versus notiones," meaning that what matters are the ideas (notions) and not the names we give them (notations). He was right, of course: in just one year, this theorem had already been proven by Lagrange.
Nevertheless, I am frequently asked about certain mathematical terminologies that spark passions. Such discussions are usually amusing, even if they have little scientific relevance. Let's look at three examples.
Natural numbers are integers excluding negative numbers. And zero, is it a natural number? "Yes and no. Including zero among the natural numbers is a matter of preference or, better yet, of coexistence. For an algebraist, it's natural to be in favor, but an analyst will probably be against it," replied the late Professor Elon Lima.
A rhombus is a polygon with four sides of equal length. Do we also require that the angles not be right angles, or is a square a special case of a rhombus? A triangle is said to be isosceles if it has two sides of equal length. And if the third side also has the same length, is the triangle still isosceles, or is that case (equilateral triangle) excluded? Even textbooks get confused with this.
A natural number is prime if it has only two divisors: itself and 1. But is 1 itself prime or not? Until the beginning of the 20th century, the affirmative opinion prevailed: for example, the "Exercises in Numerical Analysis," published by Lebesgue in 1859, explicitly states that 1 is prime. Nowadays, it is a consensus among mathematicians that 1 should not be considered prime (nor composite). Not because it would be wrong, but because in this way the statement of the fundamental theorem of arithmetic becomes simpler: "every integer greater than 1 can be written uniquely as a product of primes."