Abstract

Topology uses simple geometric and algebraic ideas, but its huge success and vast ramifications make it a tough nut for historians of twentieth‐century mathematics. Two books have addressed it well: Dieudonné chronicles about one thousand key definitions and theorems, and essays in James focus on forty central themes. Both assume considerable mathematics, but neither offers a historical synthesis of the simplest core ideas. Now, Alain Herreman uses semiotics to watch these leading ideas develop through the founding works of Henri Poincaré, Oswald Veblen, James Alexander, and Solomon Lefschetz. Herreman states outright that semiotics will not exhaust these meanings, but he makes it a revealing tool.The method is especially suited to Poincaré, who will define one technical term repeatedly in a single work, each time differently, as if it was the first, and perhaps no definition will match any use of the term in proofs. For Poincaré no term gets meaning from a definition. Each functions in relation to the others—that is, specifically in relation to other terms in Poincaré's work. It is no use invoking standards of rigor current in Poincaré's time and place or then‐current definitions. Poincaré was well known at the time for using neither: Poincaré's meanings must be derived from his writing, as Herreman does.Herreman bases his semiotics on Hjelmslev yet refutes Hjelmslev's concern that mathematics may be “monoplanar,” with no content beyond the signs themselves . The book depicts four levels of content at work in these authors: algebraic, geometric, arithmetic, and set theoretic. Herreman says a sign has algebraic content if its use depends on its written expression, the way polynomials are formal expressions added and multiplied by formal rules. Early topologists—here especially Alexander—sought purely combinatorial methods. Thus a “cube” is a set of six “faces,” twelve “edges,” and eight “vertices,” each taken as primitive and described only by a short table showing which ones meet which. Combinatorics typifies arithmetic content for Herreman. Yet a cube is also an infinite set of points. Herreman speaks of geometric content when a sign indicates both a set of parts and a set of points. Today we might use different notations for the set of points and the set of parts—Poincaré et al. did not. Herreman will not reconstruct their works or restate them in other words; rather, he uses these levels of content to organize extensive quotations and analyze the relations in each text as they move toward deeper union of the algebraic and geometric.The chapter on Lefschetz makes a great finale. Lefschetz is arguably Poincaré's closest and greatest student, though the two never met. Like Poincaré's, his work is at once compelling and baffling, decisive for the future of mathematics yet brutally difficult to absorb. Semiotics serves well in presenting this mathematician who “never stated a false theorem or gave a correct proof,” as his friends joked.The book does not go far into theorems. Yet it requires some background. A beginner might enjoy it with Alexandroff , a gem itself, written with unusually strong historic sense. Specialists will enjoy reading it with the original works for its fresh viewpoints and novel connections. It is a fine way to analyze the works, to see how they create their own meanings