David Hilbert and the foundations of geometry
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The German mathematician David Hilbert is one of the first figures whose name is associated with abstraction in mathematics, particularly through his axiomatic approach. One of his best-known achievements was to provide geometry with rigorous foundations. Euclid had built geometry in his Elements, but it relied on implicit assumptions rooted in intuition and no longer met the standards of rigor demanded in the late 19th century.
Hilbert based elementary spatial geometry on a system of twenty axioms. It contains three primitive notions, which are therefore left undefined: point, line and plane. He said that the words "point," "line" and "plane" could be replaced with "table," "chair" and "beer glass" without preventing us from doing geometry, since only the relations between these terms mattered.
These axioms include, of course, Euclid's famous postulate: in a plane, through a point not lying on a given line, there is at most one line parallel to the given line.
Nicolas Bourbaki and the remaking of mathematics
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In France, when abstraction in mathematics is discussed, the name that probably comes up most often is Nicolas Bourbaki. Who is he?
Nicolas Bourbaki is a group of mathematicians formed beginning in December 1934 around Henri Cartan (1904–2008) and André Weil (1906–1998), brother of the philosopher Simone Weil. Rules were laid down for how it would operate: membership was to be renewed by co-option, members were to remain anonymous and leave the group when they reached the age of 50.
Dissatisfied with the books of the time dealing, among other things, with multiple integrals, they decided to write a major treatise on analysis. The project quickly expanded to cover all the mathematics that some might call "pure." The treatise was renamed Éléments de mathématique, with both the reference to the title of Euclid's principal work and the singular "mathématique" entirely deliberate. The current edition comprises some thirty fascicles. This treatise is the standard reference for what might be called "abstract mathematics," based on the axiomatic method and the concept of structure. Bourbaki nevertheless chose not to include the concept of a category (see the article Category theory: "abstract nonsense" ?), although several of its members were specialists in the field.
Despite a spoof announcement of its death in 1968 and a decline in activity, the group continues to organize seminars every year and to work on its major treatise. Its influence peaked in the 1960s and 1970s.
Laurent Lafforgue, from algebraic geometry to industry
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Laurent Lafforgue, born in 1966, was never a student of Grothendieck (see In Brief, "Alexandre Grothendieck's method"). Yet he is one of the mathematicians working in the tradition of Grothendieck's research—and to spectacular effect! Generalizing a method introduced by Vladimir Drinfeld, a Ukrainian mathematician born in 1954 and awarded the Fields Medal in 1990, he proved part of the Langlands conjectures in an important case, earning him the Fields Medal in 2002. Since then, he has been working, among other things, on topos theory, a concept introduced by Grothendieck.
The Langlands conjectures concern links between number theory and other areas of mathematics, such as group representations and automorphic forms (see our feature "Mathematics around the world: Canada" in Tangente 149, 2015). Still one of the great undertakings of the 21st century, they were formulated in 1967 by Robert Langlands, born in 1936 and awarded the Abel Prize in 2018, then later generalized within the framework introduced by Grothendieck. Andrew Wiles's 1994 proof of Fermat's Last Theorem uses techniques introduced in this context.
A professor at the Institut des hautes études scientifiques (IHES) from 2000 to 2021, as Grothendieck had been from 1958 to 1970, Laurent Lafforgue held the IHES chair in algebraic geometry established in 2019 by Huawei, the Chinese information and communications technology company. In 2021, he left the IHES to join Huawei's research center, where he continues his research into topoi and their possible applications. Today, even industry is taking an interest in "abstract nonsense"!
