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Mathematical Lives: Protagonists of the Twentieth Century from Hilbert to Wiles

Claudio Bartocci, Renato Betti, Angelo Guerraggio, and Roberto Lucchetti, editors
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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This idiosyncratic book “brings to the forefront some of the proponents of the mathematics of the twentieth century, who have put at our disposal new and powerful instruments for investigating the realities around us.” Additionally, “[s]pace has been made … also for literary texts, including contributions by two apparent mathematical interlopers, Robert Musil and Raymond Queneau, for whom mathematical concepts represented a valuable tool for resolving the struggle between ‘soul’ and ‘precision.’”

This blurb from the back cover of the book under review hits the nail on the head as far describing what is inside is concerned. The operative phrase, really, is “the realities around us,” surely a red flag to any mathematician: there’s going to be some armchair philosophizing in what follows, even if it be sub rosa or sotto voce. In any event, one has to approach the material in the pages that follow with a willingness to make allowances for the authors’ tastes and points of view, and this is particularly relevant when it comes to the choice of which figures to include and which figures to exclude in such a (quasi-) biographical effort as this one. We end up with something that is not going to be every one’s cup of tea.

Consider the following telling phrase from the Preface to the book: “Although many have heard of Russell, Gödel, von Neumann, or Nash, how many know about Emmy Nöther, Schwartz, Grothendieck, or Atiyah?” What is the authors’ audience, if they propose such a dichotomy? I stipulate that any one who knows a bit about the first four but not the second four is not a mathematical insider. All undergraduates in Mathematics know Emmy Nöther; all graduate students know the other three. Conclusion: the book’s audience in not the collection of professional mathematicians. Then who?

I’m afraid I’m not sure.

But leaving my confusion in the shadows for a while, where it presumably belongs, let’s try to get a gauge of what Mathematical Lives is about from its cast of characters. Who are included in this compendium of biographical sketches and essays? Well, to name the bigger shots, we have Hilbert (in the form of his 23 problems), Castelnuovo, Enriques, Severi, Poincaré, Bertrand Russell, G. H. Hardy, Emmy Nöther, P. A. M. Dirac, John von Neumann, Gödel, Turing, Kolmogorov, Bourbaki (Hah!), John Forbes Nash. Ennio de Giorgi, Laurent Schwartz, René Thom, Grothendieck, Gian-Carlo Rota, Smale, Atiyah, V. I. Arnol’d, Bombieri, Martin Gardner, F. W. Lawvere, and Andrew Wiles.

But what about (just off the top of my head) Hopf, Artin, Alexandroff, Pontryagin, Erdös, C. L. Siegel, André Weil, Hermann Weyl, Werner Heisenberg, Alfred Tarski, Jean-Pierre Serre, and Atle Selberg? Granted, p. 129 has a paragraph on André Weil, and p. 125 mentions Serre as well as Alain Connes (for instance). But where is Oscar Zariski? And, yes, Heisenberg is mentioned in the Dirac article, and, yes, Emil Artin is mentioned in the essay on Emmy Noether.

But this all attests, again, to the book’s idiosyncrasy: these mathematical lives are discussed in essays of various types, by various authors, with considerable variation in style, focus, and thrust. There are many very interesting essays in the book, some deep, some not so deep, others, well, bizarre, at least to my taste: I fear that, for example, “Carciopholus Romanus” on pp. 53-54, by L. Sinisgalli, is not my cup of tea, to return to an earlier metaphor, and the same can be said about the excerpt on pp. 221-222 from Le Corbusier’s The Modulor.

So, on the whole, while the tea served up in Mathematical Lives is ultimately not to my taste, it will please many another palate, and I guess that’s how it should be: after all, as the authors state on p. vi, quoting Marcel Schwob, “the art of biography consists precisely in choice,” and one man’s Van Gogh is another’s no go (sorry: couldn’t resist …).

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Preface.- Hilbert’s Problems. A Research Program for Future Generations: U.Bottazzini.-The Way We Were. The Protagonists of the “Italian Spring” in the First Decades of the Twentieth Century: G.Bolondi et al.- Intermezzo: Verlaine and Poincaré from Etudes littéraires : P.Valéry.- B.Russell. Paradoxes and Other Enigmas: G.Rigamonti.- G.H.Hardy. A Brilliant Mind: R.Lucchetti.- E.Noether. The Mother of Algebra: A.Brigaglia.- Intermezzo: Carciopholus romanus from Furor mathematicus: L.Sinisgalli.- P.A.M.Dirac. The Search for Mathematical Beauty: F.La Teana.- The Theoretical Intelligence and Pratic Vision of J. von Neumann: R.Lucchetti.- K.Gödel. Completeness and Incompleteness: P.Odifreddi.- Intermezzo: In Homage to Gödel: H.M.Enzensberger.- R.Musil. The Audacity of Intelligence: C.Bartocci.- The Life, Death and Miracles of Alan Mathison Turing: S.Termini.- R.Caccioppoli. Naples, Fascism and the Post-War Period: A.Guerraggio.- B. de Finetti. The Foundations of Probability: D.M.Cifarelli.- A Committed Mathematician: G.I.Bischi.- A.N.Kolmogorov. The Basis of Probability, and More...: G.Boffetta et al.- Bourbaki. A Mathematician from Poldavia: G.Bolondi.- Writing and Mathematics in the work of R.Queneau: A.Ferraro.- J.F.Nash, Jr.. The Myth of Icarus: R.Lucchetti.- E.De Giorgi. Intuition and Rigour: G.Dal Maso.- L.Schwartz. Political Commitment and Mathematical Rigour: A.Guerraggio.- R.Thom. Conflict and Genesis of Forms: R.Betti.- Intermezzo: The Dream: J.L.Borges.- A.Grothendick: Enthusiasm and Creativity. A New Language at the Service of Imagination: L.Barbieri Viale.- G.-C.Rota. Mathematician and Philosopher: D.Senato.- S.Smale. Mathematics and Civil Protest: A.Guerraggio.- M.F. Atiyah. The Deep Reasons for Mathematics: C.Bartocci.- V.I.Arnol’d: M.Pedroni.- Enrico Bombieri: C.Bartocci et al.- M.Gardner: E.Peres.- Intermezzo: The Door of Miracles from Le Modulor: Le Corbusier.- F.W.Lawvere: R.Betti.- An Interview with A.Wiles: C.Bartocci