MATHEMATICS: ITS BIRTH AND HERSH'S PHILOSOPHICAL APPROACH

            When I first learned that there is a book named What is Mathematics, Really? my mind popped some first impressions about the book. Among them would be boring and tedious. I expect that I will face an out-of-the world terminologies and some wacky-doodle stuffs. The book refers to the class book with the same name - What Is Mathematics? which as written by Richard Courant and Herbert Robbins.  I hope my first impressions were wrong.

            The author of the book is Reuben Hersh. He stresses that Courant and Robbins approached the great deal of mathematics treating it similar to a question showing an exquisite type. Meanwhile, Hersh approached mathematics through exploration and its origins. He stresses two points: (a) philosophy established mathematics and (b) mathematicians are vital in foundation in the philosophy of mathematics.

            "Is there such thing a 4D (4-dimensional) cube?" This is the first problem in the book that eventually stuck in my mind. The chapter started with Polya's heuristic problem on a 4-dimensional cube and an inquiry to mathematical existence. It was solved by following the patterns in a 3-cube, 2-cube, and 1-cube by focusing on its dimensions. He used the philosophy of humanism in using possible answer to the said question. This is where he rejected the three mainstream philosophies and considered humanism as the superior.

            Hersh discusses the three types of philosophical approaches - Platonism, formalism and intuitionism. The philosophy of Platonism for Hersh suggests that mathematics lives in the realm of the extremes - outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social. In the field of sciences, we usually do not close ourselves to discovery. We are fond of proving the existence of the non-existent, proving that we are highly dependent of the Platonism. I believe that everyone is inquisitive in nature, making us superior from the other living forms. What makes Platonism beautiful is its mystery of infinity. In Hersh's perspective, he dismisses Platonism because it "violates the empiricism of modern science." He also stresses that there is a unusual "parallel existence on the realities of physical and mathematical" concepts.

            The philosophy of formalism, according to Hersh (1997), described and pointed out that mathematics is a meaningless game played explicit with unpredictable rules. It relies on a system to answer the universe. Mathematics has its own rules, in the form of equations and formula which becomes meaningful if it is applied in solving problems. He stresses that we are nothing without mathematics.

            According to Hersh (1997), the intuitionism philosophy points that we are living in a world of natural numbers. These became the foundational data of mathematics from which all meanings come to life. However, Hersh suggested that the concept of natural numbers for every individual is different. It was strongly supported by a research conducted by Piaget which states that children sets the concepts of natural numbers according to their experiences and style of thinking.

            The main highlight of the book focused on humanism philosophy. Hersh sees humanism as the main tool in understanding mathematics. Mathematics should be seen in the contributions of the society and how people deal and play with mathematics in their daily lives - learning and understanding of it depends on the works of the mathematicians and their concepts. He sees that humanist philosophy is the guiding light of mathematics. He also added that there is a possibility of the union mathematical philosophy with the other philosophies if socio-cultural values are highly regarded. He highly regarded that respected thinkers values humanism. Similar to other fields of knowledge, mathematics is changing - nothing is permanent. Mathematics is everywhere, playing a vital role in our activities and in the society.

            The second part focused on the discussion of Philosophy and Theology. He stressed the points of fifty people - from Aristotle to Wittgenstein. I find this very interesting because we have read the philosophy of some mathematicians and how he interpreted it. He reviewed that Piaget's notion of bases is dependent on maturation rather than cognitive development.

            This book scrutinized the philosophies of Platonism, formalism and intuitionism. He added another a new idea in the philosophy of mathematics - humanism. He showed a perspective of the aforementioned three philosophies but highly showed biasness towards his introduced philosophy.  He elaborated a strong disdain against the three philosophies but highly supported his beliefs. Even if there are differences in the philosophies, it is relevant to say that philosophy became the mother on the foundation of knowledge. Platonism and formalism were comprehended well. The book itself was written well. I suggest the book may be written in a lighter mood so the readers can easily relate to the topics.  Overall, I still find philosophy a tricky field. It came into my realization that we are nothing if philosophy did not came into existence. Interestingly, Hersh had written the book with his passion. He strongly points that the philosophy of humanism makes mathematics work rather Platonism, formalism, or intuitionism. Since new knowledge may pop up once in a while, one may still scrutinize and criticize the works of Reuben Hersh. This book became an eye-opener for me. For a mathematician and a philosopher, the book may seem amusing for them. Even our greatest mathematicians like Albert Einstein, Pythagoras, or Plato have heavily depended on philosophy and found a way of linking it with numbers. Kudos to the Reuben Hersh, the publishing team and his devotion for mathematics.