PHILOSOPHY and MATHEMATICS: A Book Review on Reuben Hersh’s “What Is Mathematics Really”

In today’s world, the perspectives of people have been drowned to mainstream media and entertainment. Even I, myself am certain that the world of fantasies had somehow altered the way I live and how I think. With this condition, I have really struggled into reading a book that could not even relieve me from stress, could not even sweeten my love life and could not even alter the way I live my social life. However, the book, “What Is Mathematics, Really” by Reuben Hersh has amazingly changed my way of thinking. One impact it has given me was dealing with things in a philosophical and intellectual way so as to deeply understand concepts that exist universally.

            The book consists two major parts with a summary consisting of one chapter. Part one consists five chapters and part two consists seven chapters. The book has mainly 13 chapters. As a quick overview, let’s take a look at what’s inside the book. The book, as I read focused mainly on four philosophies: Platonism, formalism, humanism and intuitionism. Chapter one entitled “Survey and Proposals” discussed the nature of math. It scrutinized the existence of a 4-cube and how to deal with proving its existence. The author showed series of interconnecting questions to answer the problem and ended up going back to the query if it really exists. Moreover, it discussed the two major philosophies in mathematics – formalism, wherein it describes Mathematics as a ‘meaningless game’ and Platonism, which projects math in realism and universally existing. Here, it has come into my mind that math displays a complicated mass of arguments on its existence.

The attack of the next chapter is interesting as it could boggle up a reader’s mind. It evaluated the 13 major criteria for a philosophy of Mathematics. It was said that to assess the philosophy of math is to ask test questions. I find these questions as those that could guide us to the idea of how is math distinct to other bodies of knowledge. Interestingly, the third chapter dealt generally to mistakes and misunderstandings in the mathematics world as the chapter’s title suggests. The chapter provided the basic theory of Erving Goffman of “front and back”. I was really interested upon reading this part of the book. One example given was that at a restaurant, the front is the dining room for costumers and the back is the kitchen. It was done not only to separate the costumers from the kitchen but to keep them from knowing much about the cooking. Similarly, the chapter presented that math has also the “front and back” concept. The front are the finished textbooks and journals while the back are the working mathematicians.  With this basic knowledge, the chapter was followed with discussing four more myths about mathematics then the possible mistakes in the concepts and ideas of math and ended up with concluding that there is no absolute certainty in math but rather virtual. While the last two chapters of part one was comparative. Chapter 4 compared the view of formalists and Platonists to the view of humanists in dealing with the different meanings of proof, intuitions and certainty. In chapter 5, the book discussed the 5 classical puzzles as pondered by Platonists and formalists. One main thought that I have learned from the chapter was that “Platonists don’t see math as created by people while formalists believe that it is.”

The next part of the book generally dealt with the points and ideas of different mathematicians towards mathematics. The author decided to divide the ideas of philosophers into two streams: mainstream and “humanists and mavericks”. He explained that mainstream deals with the eternal view of math while humanistic view sees math as a human activity. The chapter tackled the ideas of 7 philosophers and mathematicians. Plato presented the Socratic method wherein Socrates was asking a slave boy with questions that deal with the analogy of math. I caught myself reading diligently on Euclid’s idea. The chapter discussed the myth of Euclid about Elements and related this to the battle between rationalism and empiricism. It explained that mathematics is indeed an interconnecting system since one field is a foundation to the other. Later, the book argued about the new philosophers that came about. It was said that mainstream math bloomed later and the concepts of the previous philosophers were somehow covered with the new ones. It also tackled structuralism that defines math as a “science of patterns”. Thus, math’s definition lies on how things are studied.  Interestingly, factionalism contradicts the existence of mathematical ideas and facts. Chapter 10 and 11 discussed many philosophers who saw math in a humanist’s point of view in which math is a human activity. Also, it was several times reiterated in this chapter that math is a “social-cultural-historic” body of ideas. Chapter 12 was the most difficult for me to understand since it assessed contemporary humanist and mavericks. It discussed new fundamental learning that came about. And the last chapter briefly summarizes the book.

            Having read the book amused me in different ways. I have learned many ideas about the nature of math. At first, I did not like the choice of words of the author since I could hardly understand deep terms, but when I moved on from chapter to chapter, I have already adapted to his words. Although it’s quite technical, I believe that a typical college student could understand it. I also liked the chronology of the topics. I have easily seen the interrelations of the discussed topics to one another.

            Now, how did it affect my way of thinking? We know that change is constant. I have learned that no matter how established a mathematical theory is, through finding proofs, it could still be proven wrong. However, the concepts we have today are very established enough. I have also found out that the way one looks and understands mathematics fully depends on his point of view. As pointed out several times, that a Platonist sees math as a universal entity but a humanist sees math as a human activity. I have discovered many ideas of different philosophers and mathematicians and I was very amused with what they have imparted. One example was the concept of “front and back”, since we could see this concept in our daily living.

            To sum it all up, I believe that the author was very passionate upon sharing his thoughts in order for individuals to understand the philosophy of math. It has affected me in such a way that indeed, upon dealing and understanding things and knowledge, philosophy should first be learned.