A Deeper Look on Mathematics

A Book Review

                                                             

What Is Mathematics, Really?

Reuben Hersh

Oxford University Press, 1999

ISBN 0-19513-087-1

Softcover, 368 pages, $16.95

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            The idea on the title of this book originally comes from the classical text of Richard Courant and Herbert Robbins, “What is Mathematics?” This contextual and unsatisfactory work leaves the author; Robert Hersh motivated to explore the nature of mathematics; where it comes from and what it is really. Dealing with that question leads him to a search for a new philosophy of mathematics later found in an approach he called humanism. His elucidating and thought-provoking philosophical investigation, explains the notion in such a way that it rejects the ideas of most philosophers of mathematics that treat the subject matter as isolated, timeless, ahistorical and inhuman as partly indicated in the three prevalent mathematical philosophies namely Platonism, formalism, and intuitionism or constructivism.

As I have read, Hersh argues that mathematics must be understood as a human activity, a social phenomenon, a part of human culture, historically evolved and intelligible only in a social context.

The table of contents gave me an idea that the book is divided into two main parts. Part One, Chapters 1 through 5, is programmatic and filled with different perspectives, while part two, Chapters 6 through 12, gives historical facts for a better foundation that would benefit the readers. The last chapter, 13, summarizes the book and cites some interesting mathematical examples and notes in the last part.

The beginning of the book started with an intriguing puzzle. This introduction is followed by a brief overview of modern mathematics, and then a presentation of the three mainstream mathematical philosophies---- Platonism, formalism, and intuitionism compared to humanism, which he believes as the most suitable, and superior of them all. Along the way, he considers some of the generally accepted properties of mathematics and attempts to discredit them.

As far as I have tried to read, I noticed that the heart of the book is the humanist philosophy of mathematics. None of the three mainstream philosophies of mathematics can account for the existence of its rivals so there goes the birth of humanism, a radically different view which accounts more closely resembles how mathematicians actually work. Hersh also insists that there is a strong link between philosophy and theology. Say for instance, some mathematicians and scientists did not believe in the existence of God, and others who did wanted to think of mathematics as independent of any divine intercession. This removed one of the main arguments in support of Platonism of the origin of math if it does not come from God. Many say that the answer was in the logical foundations of the subject or on the three mainstream philosophies but for Hersh, these philosophical alternatives fail to deal satisfactorily with the main questions of philosophy of mathematics, and this is what brings him to his humanist point of view.

Generalizations, Reflections and Conclusions

           Regardless of the book’s claim to be “easily comprehensible to anyone”, there are still some times when I got lost in some points while reading the book. Yet, I still think in some ways that Hersh has a point in his views discussed in this book. It’s so amazing that Hersh was able to figure out all of those ideas in just one book. What struck me most was the heart of Hersh for humans which lead him to his philosophy. Being a humanist as well as a God-fearing person, I heartily agree to his side even though me myself is not an expert in the fields such as mathematics and philosophy. Besides that, I honestly think that this book would be very valuable as a supplementary tool for enhancement of learning and accumulation of new knowledge for starters or even non-starters of the study of mathematical philosophies and ideas. It is not typical to find a book in any bookstores that holds a vast area of information regarding this subject matter. It takes a lot of expertise to compile a wide range of ideas in just one book. Yet, as I noticed some lack of information and carelessness, that’s why I recommend that readers will read other references and not depend on this for some contents were notably unreliable.

Generally, most of my criticisms were that Hersh was unable to use his social-historic-humanistic approach very effectively to directly respond to the problems of mathematics.  As a whole, his analyses and criticisms of other philosophies are very useful, but they don’t rely on the humanist point of view and could easily have been made without it. As this line of Shakespeare, Sonnet 18 was written in the first part of the book----. . . , which caught my attention, So long lives this, and this gives life to thee.", I still found the book very fascinating and enlightening (pertaining to mathematics) in many ways, yet, I am uncertain if Hersh succeeds in making his case for his humanist philosophy.