Humanizing Mathematics and its Philosophy consists of essays collected and published in honor of the 90th birthday of Reuben Hersh, who asked that contributors address questions that arose from a remark to Hersh by Paul Cohen in the 1970s, predicting that at some point, mathematicians would be replaced by computers. Naturally, given Hersh’s conception of mathematics as a human social activity, he has taken exception to Cohen’s statement. For this book, Hersh asked contributors (selected by Sriraman and Hersh) to address two sets of questions:

Can practicing mathematicians, as such, contribute anything to philosophy of math? Can or should philosophers of math, as such, say anything to practicing mathematicians?

and

20 or 50 years from now, what will be similar, and what will, or could, or should be altogether different: About the philosophy of math? About math education? About math research institutions? About data processing and scientific computing?

The resulting essays are effectively described by Sriraman in his preface:

The colorful and eclectic essays in this Festschrift from numerous well-known mathematicians, philosophers, logicians, and linguists offer in part his [Reuben’s] colleagues’ attempts at answering Reuben’s questions and also in part glimpses into Reuben’s fertile mind and his influence on many generations and decades of mathematical life.

My own interest in Hersh’s writings on mathematics began with

*The Mathematical Experience* (1981, co-authored with Philip Davis), which I adopted as a textbook in a course for in-service middle grades mathematics teachers. My interest continued with

*Descartes’ Dream: The World According to Mathematics* (1986, again co-authored with Philip Davis),

*What is Mathematics Really?* (1997), and

*Loving and Hating Mathematics: Challenging the Myths of Mathematical Life* (2011, co-authored with Vera John-Steiner). Hersh’s thoughtful considerations on how mathematics is actually done have been a major influence on his philosophy of mathematics. For Hersh, mathematical entities are real objects that have three principal manifestations: social, mental, and neural. Further, “established mathematics undergoes steady revision and correction”; it is a naive notion that accepted theorems are absolutely certain.

Rather than summarizing and commenting individually on the 27 chapters in this book (five by Hersh and 22 by others; see the table of contents), here I will simply highlight a few that I found of particular interest or for which the title does not make entirely clear the topics considered in the entry. Then too, providing summaries of the philosophical essays would mangle the extensively qualified arguments and descriptions that the authors have so carefully crafted.

One of the ideas in Hersh’s “Pluralism as Modeling and as Confusion” that I find thought-provoking is that the various approaches to the philosophy of mathematics (such as formalism, Platonism, and intuitionism) can usefully be regarded as different models of mathematics — they need not compete to be the truth. D. A. Edwards, in “The Exact Sciences and Non-Euclidean Logic,” follows up on this, saying that “the attempt to enforce consensus carries grave risks.”

In “Xenomath!” Ian Stewart uses science fiction to make it clear that the mathematics we do is indeed human dependent. He contrasts human-developed mathematics with the mathematics that might be developed by extraterrestrials.

Several authors followed up on Hersh’s observation that most philosophies of mathematics focus on the “front” of mathematics, that is, how mathematics appears when published, rather than on the “back,” which is where the mathematics is done, with its leaps of intuition, discussions with colleagues, use of analogies, missteps, blind alleys, etc.

Michael Harris’s essay, “Do Mathematicians Have Responsibilities?” argues for the importance of the involvement of mathematicians in public discussions, pointing out that the certainty and objectivity that are mythically attributed to mathematics can and has damaged critical thinking in other disciplines. As one example, Harris uses a quotation from Thomas Pieketty’s *Capital in the Twenty-First Century* to illustrate this: “To put it bluntly, the discipline of economics has yet to get over its childish passion for mathematics and for purely theoretical and often highly ideological speculation.”

For anyone disturbed by the obvious differences between school mathematics (as usually taught) and mathematics (as practiced by research mathematicians), Bonnie Gold’s article “School Mathematics and ‘Real’ Mathematics” will provide an enlightening discussion.

Doron Zeilberger’s essay “What Is Mathematics and What Should It Be?” is charmingly provocative, calling for mathematics to stop being a religion/game/art form and encouraging it to become instead a science. I intend to assign this essay as the subject of reflective papers by students in my history of mathematics class at the end of the semester. It will help them consider the versions of mathematics that they have encountered throughout the course.

Chandler Davis took me aback with his title “Can Something Just Happen to Be True?” but I had to agree with his example: “I can’t help an immediate feeling that 5279 being prime is as accidental as the first snowdrop in my garden appearing on the left side of the walk rather than the right.” He continues to elucidate the implications of this idea through a discussion of dragon curves, involving interactions among himself, John Heighway, Donald Knuth, and Martin Gardner.

Carlo Cellucci’s essay “Varieties of Maverick Philosophies of Mathematics” discusses his agreements and disagreements with various aspects of Hersh’s philosophy. Jody Azzouni’s article “Does Reason Evolve? (Does Reasoning in Mathematics Evolve?)” includes a discussion of the changing acceptability of diagrammatic proofs within mathematics.

Alexandre Borovik makes the distinction indicated in the title of his essay, “Mathematics for Makers and Mathematics for Users.” Can a democracy practically teach rich mathematics to every child in order to allow all children to potentially become makers of mathematics? Borovik believes that to do so is ultimately very expensive. He explains a number of mental traits that should be developed in anyone who is to become a maker of mathematics:

the ability to engage the subconscious when doing mathematics; the ability to communicate intuition; the ability to learn by absorption; the ability to compress mathematical knowledge; [the] capacity for abstract thinking; and being in control of [his or her] mathematics.

I would encourage anyone interested to read the essay for Borovik’s explanations and development of these ideas.

Finally, Nel Nodding’s “A Gift to Teachers” builds on ideas presented by Hersh and John-Steiner: in their words,

We should discontinue ‘the use of mathematics as an academic filter: Instead, the goal is to treasure diversity in talent and interest; to provide advanced mathematics teaching/learning to motivated students, while decreasing the number who suffer from math phobia.

This book presents a rich variety of essays. I would encourage readers to dip into the book, picking and choosing the essays that fit their interests. They will be rewarded.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.