Perspectives

A Review on What is Mathematics, really? by Reuben Hersch

                    Define mathematics.

                I thought that that will be the main goal the author wants to achieve in this book, to define Mathematics. I thought that he will give me list of definitions, mathematical concepts, formulas, and equations. I thought it will tell me stories of the author’s encounter with math and how it changed his life. After reading it, I realized that I was wrong. It is not a math book. It is not a romantic love story on how the author, Reuben Hersch, fell in love with Mathematics. It was more than that. It dealt with the nature of Mathematics.

                The book “What is Mathematics, really?” is written in a very different and radical point of view of the author, Reuben Hersch. The tone of the author is serious and formal. This makes the book very informative and worthy of being read. This suits his target audience which are, mainly, mathematicians and educators. One of his aims was for his book to be used as a tool by teachers for educational reform.

                What makes Hersch’s work different is that he rejected the mainstream philosophies and used his own to explain the nature of mathematics. These mainstream philosophies he rejects are Platonism, Formalism, and Intuitionism or constructivism. On the other hand, his alternative for these is humanism. He believes 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 used this alternative philosophy to explain issues such as existence of finite and infinite mathematical entities, intuition, proof, and truth. However, some of his points and opinion were not strong enough.

                The author opened the book with a conversation between him and Laura dealing with the existence of mathematics. This idea was good because it makes readers dig into the book for more. In the first chapter, the existence of the 4-dimensional cube was also questioned. He also explained using humanistic point of view. He explained that using the method of solving the vertices of a 3-D cube, a 2-D cube and a 1-D cube, a pattern can be observed. This pattern allows the vertices of the 4-D cube to be determined. This gives proof to its existence.

                This is followed by the part where Hersch justifies his “alternative” philosophy against the mainstream philosophies. The three mainstream philosophies he pointed out were Platonism, Formalism, and Intuitionism. His argument with Platonism is that it states that “mathematical entities exist outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual, or social". He believes that this idea is not enough to prove the nature of mathematics. Platonism explains the idea of the physical and the mathematical. However, it does not explain the bond connecting them.

                On the other hand, he views Formalism as having a philosophy that “mathematics is being a meaningless game played by explicit but arbitrary rules”. He believes that this kind of philosophy is pretentious and has no strong basis. He also argued that following rules without any judgment is fiction. I agree with him in this. For example, a person wants to prove a certain theorem. Using the established proofs is not enough. A person needs to evaluate, first, if this proof will lead to the answer. He has to fin a reason. So, the philosophy of Formalism does not really apply to real life in all occasions.

                In the case of Intuitionism, he sees it as a philosophy that “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 intuition of natural numbers is not universal”. He used the studies of Piaget, a famous mathematician, to prove his point. Piaget explained that children, for example, view numbers not as natural numbers given by God. Instead, they view them as constructions made by their mind through their experiences. There is a varying conception of numbers. Therefore, this philosophy is not applicable to all.

                For his “alternative” philosophy, humanism, he pointed out that “there's no need to look for hidden meaning or definition of mathematics beyond its social-historical-cultural meaning". What has been laid down by mathematicians is what you use to answer questions. The social aspect adds on to the mental and physical aspects which need to be considered. However, I am not agreeing to this. I believe that the emergence of mathematical concept is not because of social interaction. They are mentally constructed by the minds of the people and expressed by the physical aspect.

              More of his thoughts in his Humanist philosophy are told in chapters 3 and 4.  He believes that mathematics is like humans with a front and a back. The front represents what is being shared to the world while the back represents where the concepts are obtained. He pointed that the front is the focus of the mainstream philosophies while the back is the focus of the humanist point of view. In my opinion, mathematics does not have a front and back. Mathematics is one-faced. Where and how the concepts are obtained is what is being told to the world.

              Also, the book has provided details on the “alternative” philosophy of Hersch explaining the nature of Mathematics. He also used the ideas of mathematics philosophers which is very good. It is very informative especially about the philosophies of mathematics which only few books are written about. However, in my opinion, Hersch did not successfully reject the mainstream philosophies using the ideas of the humanistic philosophy. His opinions and justifications were useful yet were not strong enough. Moreover, his goal of making mathematicians and educators understand the philosophies of math to aid in educational reform was not met. There were only few details to justify this claim.

             In totality, the book is worthy of being read. It made me look at the different perspectives of different Mathematicians. They have constructed their own perspective of Mathematics and relayed it to the world for better understanding of Math. For me, it is not about whose philosophy is more convincing and efficient. Each of this philosophy laid foundation to the existence of Mathematics. Each of their perspective changed the world.