What is Mathematics, Really?: A Book Review

               I am neither a wizard in math nor a person who loves math. I am just a learner who tries to understand mathematics. For a better grasp of “What is Mathematics, Really,” I used Ed Dubinsky book review of Hersh’s book to aid my understanding of this book.

According to Hersh, mathematics is a social-historic phenomenon- it is not mental neither a physical object. It is like education or war that exists from human minds and bodies. Through these human minds and bodies, came culture, society or history. Let’s say education, it is not a physical object. We go to school, which is a physical structure, but the school per se is just a part of what we understood of education. Another part of education is “learning” which is in our minds. But just because of that, we cannot categorize education under as mental object. It is a social phenomenon that people deal with in their lives.

Overall, this book is explaining what mathematics is, really. Hersh explains the three main streams of mathematics which he rejected: Formalism, Platonism and Intuitionism or constructivism; and introduced his humanist viewpoint. The book is composed of two parts.

The opening of the first part was the question whether the 4-cube exists. Hersh tries to answer the existence of this 4-cube to better understand the different philosophies in mathematics as well as his humanism.

Hersh explained that Platonism is 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”. According to Ed Dubinsky, who also reviewed Hersh’s book, that the reasons for Hersh to reject the Platonism philosophy are: “it does not relate to material reality or make contact with  flesh-and-blood mathematicians; it violates the empiricism of modern science; and it insists on acceptance of a ‘strange parallel existence of two realities- physical and mathematical’, but does not explain  how the two interact."

Formalism, as Hersh says that mathematics is an otherwise meaningless game played by explicit but arbitrary rules. Reasons for objection of Hersh according to Dubinsky: “rules are not arbitrary but rather ‘historically determined by the workings of society that evoke under pressure of the inner workings and interactions of social groups and the physiological and biological environment of the earth’; he asserts that this is not how mathematics is actually done, that the ‘notion of strictly following rules without any need for judgment is a fiction’ and that it is misleading to apply it to real life’”

According to Hersh, “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.” The reason why Hersh objects the Intuitionism is: the natural numbers are not universal, according to the anthropological viewpoint which Hersh adopts. In Piaget’s work, children construct in their minds a concept of the natural numbers based on their experience and modes of thinking.

On the other hand, the Humanism of Hersh states that “There’s no need to look for a hidden meaning or definition of mathematics beyond its social-historic-cultural meaning”. This means that the answer can be found in the society. He stresses the importance of the social aspect-not just the mental and physical aspect.

The second part of the book, there is part of “Philosophy and Theology”. In this section, it says that mathematics is connected with theology. Dubinsky said that, “mathematics exists as the thought of God, and therefore any knowledge of it provides eternal truths about the universe. “ Until the eighteenth century, mathematicians did not believe in the existence of God.  People started thinking, “if it does not come from God, where does it come from?”; and they said from the logical foundation of the subject. For me, I believe that everything comes from God. He is the source of everything- the source of the wisdom that man has. Without Him, we cannot think because He created our brain- our minds and everything in it. If we say that it came from a great philosopher or a mathematician, it couldn’t be possible because no human being has greater mind than God. I still believe that even mathematics and mathematicians exist, we need not to compromise the existence of God also who is the source of everything.