What is mathematics really? -- A book review

          When you see the cover page of this book, I think readers would readily become interested. Mathematics has gained a lot of misconceptions usually from students who are having a hard time doing calculations and solving problems.

          The book “What is mathematics, really?” by Reuben Hersh answers the question using philosophies. Platonism, Formalism and Constructivism are the three mathematical philosophies he considered mainstream.  He explained these three and rejects the ideas of each and offered another one—Humanism.

           Humanism says that the concept of math should be seen and understood as a human activity and a part of our culture. Hersh pointed out that “There’s no need to look for a hidden meaning or definition of mathematics beyond its social-historic-cultural meaning.” From what I understood in this line, mathematics is defined by looking at the society and the people using and applying mathematics. One thing that makes math hard is the thought that there is something concealed. We tend to think of something very complex when the answers of mathematical problems were just right in front of our eyes.

         From the kinds of existence of math which are physical and mental, he added a third one—social. I agree with his point. Humans made math, MATH. Math came from our society and is being applied to improve our society.

          Hersh explained 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.”. Because Hersh looks at mathematics as a way of life of humans and it exists in the society he rejected this idea. One of his reasons was Platonism does not speak about to reality.

          In formalism, he described it as a “meaningless game.” Mathematics was just a gate with a set of rules to be followed. Hersh rejected this philosophy as well since he believed that rules are not subjective but 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.” For him, this does not apply to real life since people do not just follow rules without questioning and looking for proofs.

      The third mainstream mathematical philosophy was Intuitionism. Intuitionism recognizes the fact that the set of natural numbers are the basic datum of mathematics. It is in natural numbers where all mathematical processes are obtained through a process of finite construction which does not use the law of the excluded middle. Hersh also rejected this one since he believes that the intuition of natural numbers is not universal.

           I don’t think a can choose one out of the four mathematical philosophies discussed. From the explanations I got from the book. Mathematics is a little bit of every philosophy. Not everything about each philosophy is right but it’s not all wrong either. In formalism, for example, mathematics was governed by a set of rules. This is true for mathematics. In solving problems, rules are followed. Intuitionism on the other hand, can also be used in understanding mathematics since you also need to view individual perspectives in dealing with mathematics.

          For me the book cannot be understood easily by everyone. You need to learn other bodies of knowledge in order to cope up with what the author has been saying. The first part of the book already makes your brain think deeply about the question “Is there a four cube?” If was not required to read this book, I would definitely close it after reading the first line.

        All in all, Hersh’s book was very informative since he was able to define and provide thorough explanations on why other philosophies should not be used in studying mathematics. I think the thing that was lacking was his humanistic approach. It was also very interesting since collision of other disciplines was done to come up with a new idea of understanding math.