This book is the result of a conference held in 2008 to celebrate the 150th anniversary of Giuseppe Peano's birth and the 100th anniversary of one of his most famous works, the Formulario Mathematico. The nine articles survey various aspects of Peano's work in mathematics and logic.
Most mathematicians today know Peano's name in association with his axioms for the natural numbers, but his work was much broader than that. In Skof's article on "Giuseppe Peano and Mathematical Analysis in Italy", for example, we learn that early in his career Peano put together a textbook based on the lectures of A. Genocchi, who was professor of Analysis at the University of Torino. Peano was Genocchi's assistant, and the book was presented as being by Genocchi "con aggiunte da G. Peano" (with additions by G. Peano). Many of these additions, marked as "annotazioni", dealt with the need for hypotheses in "well known" theorems, often including counterexamples. Many of these results are still mentioned today, though we do not always remember their origin.
Peano was Professor of Analysis (or Calculus) from 1890 to 1931, making several contributions along the way. His interest in precise hypotheses and counterexamples continued, leading, for example, to his famous space-filling curve. Skof includes an impressive list of results of this kind found by Peano, from a closed formula for Dirichlet's function to work on integration.
Other fields in which Peano did substantial new work include numerical analysis and the foundations of geometry, discussed here in the articles by Allasia, Freguglia, and Marchisotto. But Peano's work in logic and the foundations of mathematics was of fundamental importance, and several articles focus on that. Of particular note is Grattan-Guinness's contribution, which places Peano's work in the context of the history of logic, calling attention in particular to the shift from part-whole logic (in which, to use an anachronistic description, "being an element of" and "being a subset of" are not distinguished) to Cantorian set theory.
Peano was a great inventor of symbols. He felt that ordinary language was too ambiguous for mathematics, and argued that mathematical papers should include symbolic statements of theorems as well as descriptions in words. The ultimate result of this was the Formulario Mathematico, in which he attempted to include "collections of all the theorems now known referring to given branches of the mathematical sciences." Were the theorems written out in words, such a collection would have to be enormous, but Peano's dream was to use such a concise and efficient logical notation that allowed everything to be expressed in a small space: "the collection of the theorems on a given subject [might become] less long than its bibliography." A famous quote from a letter to G. Vitali captures the idea: "Of Lebesgue's book may result a line or half a page." Roero's article about the Formulario describes the project, traces its origins, and explores its influence.
As in all publications of this kind, the articles vary in quality and interest, but overall this small book provides a great way to begin exploring the work of Giuseppe Peano.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and the editor of MAA Reviews.