Book Report: What is Mathematics, Really?

The title of this book is What is Mathematics, Really? by Reuben Hersh. The author got its title referring to the classic he discovered which is What is Mathematics? by Richard Courant and Herbert Robbins. The authors of that book explained a great deal of the content of mathematics while Hersh deals mathematics by exploring its nature, where it comes from and what it is, really. The book presents the philosophy of mathematics in real contact with mathematical practice and teaching.

                In this book, the author explores and declines the three main streams of mathematical philosophy: Platonism, formalism and intuitionism. He shows what he calls “humanism” which means that “mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context”. He notes that mathematics is a social enterprise. He tries to show that his philosophy of mathematics deals with the issues, such as existence of finite and infinite mathematical entities, intuition, proof and truth, better than the three philosophies he rejected.

                The book is divided into two parts. For the first part, it discusses the mainstream philosophies of mathematics and the said issues which the philosophy of math deals with. The introductory sample discusses the problem about counting the various parts of a 4-dimensional cube and reflects on what kind of sense the calculations could make. Hersh counts the vertices, edges and faces of a 3-cube and does the same for 2-cube and 1-cube. The three sets of formulas show a clear pattern that is easily generalized to 4 dimensions and this leads to a list of questions about the existence of a 4-cube. Hersh uses possible answers to the questions to help explain different philosophies of mathematics including his own humanism. After this, he then turns to the main point of his book which is to explain why he rejects the three mainstream philosophies and why he believes his humanism is superior.

                Hersh explains Platonism as the idea that “mathematical entities exist outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social”. He rejects Platonism for reasons that it does not relate to material reality, it violates the empiricism of modern science, and it insists on acceptance of a “strange parallel existence of two realities, physical and mathematical” but it does not explain how he two interact with each other.

                Formalism says that mathematics is an otherwise meaningless game played by explicit, but arbitrary rules. Hersh argues that the rules are not arbitrary but rather, are “historically determined by the workings of society that evolve under pressure of the inner workings and interactions of social groups and the physiological and biological environment of earth”.

                Intuitionism accepts the set of natural numbers as the fundamental datum of mathematics from which all meaningful mathematics must be obtained through a process of finite construction that does not make use of the law of the excluded middle. Hersh says that the intuition of the natural numbers is simply not universal.

                Hersh offers humanist or the social-historical point of view as an alternative to the mainstream philosophies of mathematics he rejects. He says that there is no need to look for a definition of mathematics beyond its social-historical meaning. This means that one answers the big questions by looking at what is done on the society of mathematics and the people dealing with math situations in everyday life.

                Going back to the example of the 4-cube, it can be used to illustrate some differences between the different philosophies.  For Platonist, the 4-cube exists as a transcendental immaterial, inhuman abstraction. For intuitionist and the formalist, 4-cube is not real but only a representation. For humanist, the 4-cube exists at the social-cultural-historic level as a kind of shared thought or idea.

                The real test for the humanist philosophy is its ability to serve as a tool for investigating important questions. In chapter 3 and 4, Hersh tries to use this for looking at and rejecting ideas about mathematics. He suggests that the mathematics has a front which has good and polished results that we show to the world and a back which consists of what we do to obtain those results. According to him, the mainstream philosophies belong to the front while humanism focuses on the back. Mathematics is capable of making mistakes because mathematicians make mistakes.

                Mathematical intuition is an important issue that every philosophy of mathematics has to consider. For Platonist, it needs intuition to connect human awareness and mathematical reality. For formalism, it eliminates intuition by concentrating on refinement of proof and dreaming of an irrefutable final presentation. For intuition, it is not a direct perception of something external. We have intuition because we have mental representations of mathematical objects. For the humanist, mathematics is the study of mental objects with reproducible properties.

                Mathematics is full of infinite. As Hersh points out, infinity is different from physical reality and comes out of our heads but our brain is something finite and could not contain anything infinite. The huge mystery of infinity is an artifact of Platonism.

                In the second part of this book, Hersh discusses Philosophy and Theology. He gave some points and ideas of the philosophical thinking of almost 50 individuals, from Aristotle to Wittgenstein. It is somewhat helpful because it is written from the point of view of a mathematician. He acknowledged the impact of Piaget’s writing on cognitive Psychology and pays greater attention to it.

                This book mainly reflects Hersh’s own views of the philosophy of mathematics and he wants to tell the readers what mathematics really is in his own perspectives. He wants to relay ideas of what is mathematics which was not really answered by the authors of What is mathematics?. I can see that he writes the book from a deep love of mathematics and deep concerns to make it understandable to others. Hersh’s view deserves careful consideration and you can learn from him whether you agree to it or not. But the readers are left wondering what new results in the philosophy of mathematics can be obtained from his humanist approach.