When I found this book on the shelves of the Henri Poincaré Institute’s library, some months ago, I knew immediately that it was the kind of provocative book I would like to read and review.
Editor Reuben Hersh has orchestrated a highly provocative book, particularly for mathematicians, but also for anyone interested in the philosophy and practice of mathematics. As Hersh states, the book came from oldies and original works from people spanning a wide range of backgrounds: mathematicians, computer scientists, cognitive scientists, and philosophers saying provocative things about math. It is a book about the objects, nature, purpose, practice, and justification of math; how math progresses or degenerates; who and what determines the directions, styles, good manners, and trends; and who certifies all of it and how.
Because I am interested in the concept of mathematical proof, I focus my attention more on the essays related to this topic. A provocative claim that has been around for a while is well summarized in two of the included essays: Thurston’s “On Proof and Progress in Mathematics,” and “The Locus of Mathematical Reality: An Anthropological Footnote,” by White. Thurston writes about mathematics as a human social activity, in which mathematical progress relies on the construction of better ways of thinking and understanding. Therefore, the matter has a lot more social content, including psychology and cognition (fields that are usually absent from the popular model), and discussions about mathematical practices, including ways in which mathematicians look at their field. Math, like any other field, has a strong social content of standards, such as validity and truth, as well as a choice of definitions and trends. If math, as a field with higher standards of clarity and convincing thinking, has social contents, one can imagine that physics, biology, and chemistry have even greater social contents, when favoring models and explanations and choosing the experiments to perform.
In math, the traditional definition/theorem/proof (DTP) model in Western mathematics relies on the concept of formal correctness. However, people are usually not as good at checking formal correctness as they are at detecting potential flaws. Some drawbacks of the DTP model are not explaining the source of motivation, and not being subject to tests. If, in practice, something goes against a well-established old theorem, mature mathematicians tend to favor the theorem over intuition from the new experience. The system is quite good at producing reliable theorems based on previous theorems, but not as good at producing new ideas. I favor the idea/program/test (IPT) model, which is strongly supported by authors such as Gregory Chaitin, Stephen Wolfram, and others who push the field of experimental mathematics.
White’s essay addresses the question of whether mathematical ideas are created or discovered: the age-old question related to empiricism, idealism, and realism. White seems to crudely consider math as nothing more than the product of primate behavior.
Two essays discuss the effects of mathematics on other disciplines. In one of them, “The Pernicious Influence of Mathematics on Science,” Schwartz claims that mathematics has focused so much on what to do, on precision and definitiveness, that its omissions are bypassed. In other words, mathematical activity blinds us not only to other possible “mathematicses” and other ways to do math, but also to the purpose of them all.
This is a very appealing collection of essays. Readers will not remain indifferent to any one of them, like it or hate it. I highly recommend this book to those wondering how math is carried out, both mathematicians and laymen. I also recommend it to educators interested in changing the dominant view of math and how to do math. Those thinking that math has become an exclusive club where people read what they wrote and then approve themselves, will probably enjoy some of the essays in the book, as will those looking for new ideas on how to break the chains. Readers comfortable with the ways in which math is carried out, or simply happy to be unaware of what math is, must avoid reading this book.
