THE MYSTERY OF MATHEMATICS

"Still more astonishing is that world of rigorous fantasy we call mathematics." - Gregory Bateson

“What Is Mathematics, Really?” is a book written by Reuben Hersh that addresses the question of the true nature of mathematics. This book can be considered as a reaction to Richard Courant and Herbert Robbins’ book called “What Is Mathematics?”. Unlike its predecessor, this book tries to explain the nature of mathematics instead of just showing how it works. Hersh believes that the philosophy of mathematics answers this question. Hersh explains how Platonism and Formalism are inadequate to be used in the philosophy of mathematics. He believes that viewing mathematics as an everyday human activity that we do or viewing it through humanism will give us the answer to our question.

In part one, he talked about how mathematics comes from inherited concepts and is a result of historic evolution, cooperative and competitive work of generations.  He talked about the difference of applied mathematics and pure mathematics. He also believes that examples, problems and solutions come first before the axioms on which the theory is based and that the vice versa of this is wrong. He said that the mystery of mathematics lies in its objectivity, seeming certainty and near independence. He then talks about the criteria for a philosophy of mathematics. He also wanted to get his point through that the philosophy of mathematics is part of philosophy so it should be evaluated with its standards. He then reports on how mathematics looks to the present day mathematician. Mathematics has what we call a front stage and a back stage area. Hersh also talks about the myths of different people and tries debunking them. He also expresses that how one thinks of mathematics is affects how that person portrays and presents it. Both Platonism and Formalism expresses mathematics as truth with absolute certainty, yet based on our experiences with mathematics, there is always that doubt and uncertainty. Mathematics can’t be certain because it changes and with it its point of view changes as well. Then he talks about mathematical proof in principle and shows what it is in practice.  

In part two, Hersh gives the accounts of the philosophical thinking of almost fifty people. The mainstream philosophies of mathematics presented here are very informative. This part also talks about the “foundationism” of mathematics; some mathematicians truly believed the delusion that mathematics has a firm foundation. Its roots are ancient and are found tangled with religion. He then talks about the period of the peak of mainstream philosophy and discusses three topics on Kant. Hersh talks about what makes Kant different from Descartes and Leibniz is that Kant didn’t use the certainty of mathematics to serve as evidence of the certainty of a religious deity. Mathematics and science should be studied independently from religion. Hersh then proceeds in explaining rationalism and empiricism, though he is convinced that mathematics is somewhere in between the two.  Then he talks about the humanist trend in the philosophy of mathematics. He reviews the works of other mathematicians and other humanists and mavericks. 

          I find reading this book very challenging. It required knowledge of Platonism, Formalism, other different views of philosophy and even knowing what philosophy is beforehand for the reader to fully comprehend and appreciate the argument or point that the author was making. Also there were parts in this book whose topics seem redundant and unclear. This reader had to use dictionaries, books and other sources online to understand the terms and the people Hersh was talking about. Though it was very difficult to read, there are topics in this book that I was able to relate to. For example, when Hersh talked about how mathematics had a front and a back stage. This part enlightened me because I realized that we judged mathematics by its cover just like how we judge books by their cover. Just because the cover or the front of mathematics seems boring and unfamiliar to us, we condemn it badly and negatively. Little do we know that what we are seeing is just the front or cover of the entire thing; that interesting things lay within it and that many different people worked hard behind the scenes to give us their beautiful handiwork. Students misjudge mathematics because they don’t really know what it is, what it really is. Reading this book made me realize that just merely reading someone else’s explanations on what the true nature of mathematics is will not help me understand or appreciate what mathematics is. For that to happen, one must first experience mathematics on hand himself and see mathematics through his own eyes. That is the only way one can truly appreciate and know the “real” mathematics or the mathematics within. That is the only way one can know what mathematics really is.