What is Mathematics, Really

The book written by Reuben Hersh published in the year 1997 under Oxford University Press, Inc. titled “What is mathematics, really?” is a book that proposes a philosophy of mathematics that the author called “humanism” and uses that philosophy to analyse timeworn questions of proof, certainty, and invention versus discovery. It is a philosophical idea pushed by the author himself in contrast to what most of the philosophers of mathematics viewed it- being “inhuman”. And on that note, it certainly can upsurge numerous ardent debates.

According to Hersh, the author, his main objective of writing the book is to “show that from the viewpoint of philosophy, mathematics must be understood as a human activity, a social phenomenon, part of culture, historically evolved, and intelligible only in a social context he called, “humanist”, a viewpoint “. He used the word “humanism” to include all philosophies that see mathematics as a human activity, a product, and a characteristic of human culture and society.

The author himself claimed that his book is a subversive attack on traditional philosophies of mathematics particularly the philosophies: Platonism, Formalism and Neo-Fregeanism. Hersh said that, “I am defending our right to do mathematics as we do. To be frank, this book is written out of love for mathematics and gratitude to its creators”. Those words of the author clearly stresses that he indeed wanted to put up a thought-provoking investigation into the philosophy of mathematics and to awaken and change people’s intuition towards mathematics – to know and treat mathematics as what he does.

The book has two main parts: Part 1 for chapter 1-chapter 5 is programmatic. It is a paradigm for the main problem in philosophy of mathematics. Within this part, a quick overview of modern mathematics, presentation of mathematical Platonism, and the heart of the book: the social-historic philosophy of mathematics that the author call humanism were expounded. The Part 2, from chapter 6-12, is historical.

Among the topics discussed in the chapters covered under this part, I am most absorbed with the topic: “Criteria for a Philosophy of Mathematics” which is in the second chapter of the book. In this this chapter, Hersh provided a list of 13 criteria for a philosophy of mathematics. The author said that “the list is a vantage point from which we can evaluate theories, including our own”. He designates recognizing the scope and variety of mathematics, fitting into general epistemology and philosophy of science and being compatible with mathematical practice, research, application and teaching as the essential criteria for a philosophy of mathematics. While elegance, economy, comprehensibility, precision and simplicity as the desirable. He also noted that “consistency” is essential, but not as hard to attain as the three mentioned essential criteria. Also, he said that we should reject novelty and originality as inessential and unattainable as well as certainty and indubitability, as false and misleading. The twelfth criteria, “applicability”, does not refer to mathematical applications, but to philosophical ones. The author stresses that,

“Your philosophy may increase your feeling of being at home in the universe, or your

ability to sleep with a clear conscience. But it should also be helpful in analysing

philosophical problems, perhaps even in solving one or two. If it's useless, who needs it?”

          Indeed, a philosophical idea without spirit, is definitely useless! It is like how most of the philosophers of mathematics viewed mathematics as “inhuman”, no life, and no spirit.

The last criteria on the list is Acceptability. This criterion is never explicitly demanded. Yet in practice it's the most important.

            On the contrary, among the topics discussed in the book, I find the topic: Mainstream Before the Crisis in the sixth chapter of the book as the most somewhat mind-bugging. The idea of the two parallel streams: Mainstream and; Humanist and Mavericks seem so vague except for ideas that Mainstream sees mathematics as “superhuman”, while Humanist sees mathematics as a “human activity, a human creation”. All of the other discussions and arguments in the chapter are way too vague and too rational.

            A defect of this book is neglecting non-Western mathematics. And the author clearly said that. The book focuses only on “Western Mathematics. What about the “non-Western mathematics”? India and China also sent important contributions to the world of mathematics. “But compared with Greece, we hardly know the history of the philosophy of mathematics in Indo-America, Africa, or the near and Far East. The literature on non-Western mathematics is valuable, but it's not philosophical.” according to Hersh.

            As a whole, the book was interesting. Topics were really relevant in providing a backbone to what Hersh, the author, tried to argue - that mathematics must be understood as a human activity, humanistic.