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Infinity and Truth

Chitat Chong, Qi Feng, Theordore A. Slaman, and W. Hugh Woodin, editors
World Scientific
Publication Date: 
Number of Pages: 
Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore 25
[Reviewed by
Tom Schulte
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Based on the talks given at the Workshop on Infinity and Truth (Institute for Mathematical Sciences, National University of Singapore, 2011), this book brings together nine papers on the foundations of mathematics. It is mostly thought-provoking and engaging, tackling questions formally unsolvable within ZFC, such as Projective Determinacy and the Continuum Hypothesis, and pondering if the foundations of mathematics lean to any one philosophical view.

Of the 234 pages that make up the book, 110 are given over to “Hilbert, Bourbaki and the Scorning of Logic” (by A. D. R. Mathias). As a result, his essay overshadows much else here. As the author says, “…I am writing eight-nine years later, and benefitting from the enormous development of logic and set theory that has taken place in that time. But a modern account of foundational questions must allow for that development, and Bourbaki shut their eyes to it.” Reading the essay is a bit like seeing an effective prosecutor dismantling Bourbaki on the witness stand. It is difficult to watch this thorough exposure of the Bourbaki group (and they have no defense attorney!), which goes as far as to blame them for deficiencies in present-day French instruction in mathematics (“…fostered by the errors and obscurities of a well-known undergraduate textbook.”). This feels much more like history of mathematical pedagogy and does not wax philosophical, inquisitive, and even awed like the other, shorter papers. Napoleon is brought out to explain the foundation of the modern French university system and the emperor’s 1808 decree is quoted in such detail that we know it bans women from universities. Mathias does not always provide translations of the French sources. When the accusation is that Dieudonné asserts one is a mathematician or logician, but not both, having the original text and the translation would be best. The author’s summary assertions are that “There is a collapse of intellectual level in French schools… the cause of the collapse of mathematical understanding is the suppression of the teaching of logic … the consequence of Bourbaki’s disastrous treatment of logic.” And then, “Following the dropping of logic in the curriculum, schoolchildren in France are no longer taught to prove theorems.” Surely, this article deserves a rebuttal.

“Toward Objectivity in Mathematics” (Stephen G. Simpson) is, like the rest, on target for the title of this collection, exploring set-theoretic realism, Platonism, and the philosophy of mathematics from Penelope Maddy. Also a pleasure to read is “Reasoning about Constructive Concepts” (Nik Weaver), presenting concisely a formal system for reasoning about concepts. The final entry is a list of questions proposed by the participants of the Workshop to a mythical Oracle of Mathematics that can answer truthfully any yes-no question. The result is an overview of where researchers in this area see the boundaries of current knowledge as it may abut philosophy; for instance: “Does there exist an objective justification for the concept of actual infinity?”

Tom Schulte uses four tests and a final to provide objective justification for many mathematical concepts to students of Oakland Community College. 

  • Invited Lectures:
    • Absoluteness, Truth, and Quotients (Ilijas Farah)
    • A Multiverse Perspective on the Axiom of Constructiblity (Joel David Hamkins)
    • Hilbert, Bourbaki and the Scorning of Logic (A R D Mathias)
    • Toward Objectivity in Mathematics (Stephen G Simpson)
    • Sort Logic and Foundations of Mathematics (Jouko Väänänen)
    • Reasoning about Constructive Concepts (Nik Weaver)
    • Perfect Infinites and Finite Approximation (Boris Zilber)
  • Special Session:
    • An Objective Justification for Actual Infinity? (Stephen G Simpson)
    • Oracle Questions (Theodore A Slaman and W Hugh Woodin)