lhe Francophonie Summit in Villers-Cotterêts (Aisne), which took place on October 4 and 5, revived an old question: that of the relationship between languages and sciences. I am often told that this does not affect mathematics, because its texts are nothing more than long sequences of formulas, supposedly universal and independent of language. An essential fact is thus ignored: an article consisting solely of formulas would be incomprehensible to a human being and, therefore, useless.
Indeed, in 1910, Alfred North Whitehead and Bertrand Russell published Mathematical principlesa monumental work where, with some exceptions, only mathematical formulas occur. The proof of 1 + 1 = 2 only appears on page 89 of the second volume, the first already having 696 pages.
One of the rare sentences in natural language, in English, is the one that follows this demonstration: “The previous proposal is sometimes useful” (“The above proposition is sometimes useful”). A touch of British humor. But I don’t know anyone who has read this book. Mathematical texts, to be interesting, must remain readable. Which implies an element of implicitness, including the use of indefinite or polysemous terms, which the reader would be able to understand in the context.
Visual representations
In 1623, Galileo described mathematics as a true language, essential to understanding the world: “Philosophy is written in this immense book always open before our eyes, the Universe. But we cannot understand it without first learning its language and knowing its characters. It is written in mathematical language, with triangles, circles and other geometric figures, without which it is humanly impossible to grasp a single word. Without it, it is a vain wander through a dark labyrinth. »
I like the idea that to understand something we can rely on visual representations. When we read a proof of geometry, even elementary, are the figures illustrations of the text or does the text serve to explain the figures? It would be necessary to develop a rigorous use of figures as objects of reasoning in their own right, endowed with their own grammar, to turn it into a true language. This was a wish expressed by the German mathematician David Hilbert in 1900, and one that has largely yet to be fulfilled.
You have 36.47% of this article left to read. The rest is reserved for subscribers.
