Mathematics + Philosophy

“What is mathematics, really?” It’s a simple question. If you asked me, I would likely say “you know that complicated subject that deals with numbers, shapes and etc.” Try to ask other people, you may get a bunch of responses and some may be like mine. Search it in books, you will only find it showing what mathematics is, merely not telling what it really is. In his book, Reuben Hersh provides a different and unconventional answer to the question. He tries to answer by exploring the nature of mathematics. He discusses how its history and what it is, really.

As he explores the philosophy of mathematics, he makes two important points. First is that philosophy is more than an attempt in establishing the foundation of mathematics. Second is that the major players in the development of the philosophy of mathematics is its sources. In his exploration, Hersh contradicts the three mainstreams of philosophical mathematics which are platonism, formalism and intuitionism. He instead states that humans “must understood mathematics as a human activity, a social phenomenon, a part of human culture, historically evolved and intelligible only in social context.” He then describes some standard issues of the philosophy of mathematics, exposing some of its weaknesses and debunking some myths. He then demonstrates how his philosophy deals with this issues better and how it is closely similar to how mathematics works.

In the part one of the book, Hersh introduced list of questions about the existence of a 4-cube. His series of questions makes me interested and also at the same time a bit confused. He then tries to answer these questions. The answers were his way of explaining and helping the readers to understand the philosophies of math. He continues this by giving his reasons on why he rejects the three mainstream philosophies. He first explains 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. He then points out that he didn’t mean Platonism is adequate as a philosophy. He rejects this because it does not relate to reality, it violates the empiricism of modern science and it creates a strange parallel existence of two realities. For the second mainstream which is formalism, he states that formalism says mathematics is a meaningless game played by explicit but arbitrary rules.  He contradicts this, saying that it is not arbitrary but rather they are determined by the workings of society that evolve because of the interactions of the social groups and the environment.  Intuitionism, according to Hersh, “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.” He rejects this idea with his view that intuition of natural numbers is not universal. This is supported by the research of Piaget.

                With his contradictions and arguments on the three philosophies of mathematics, he offers the humanist philosophy. He says that we do not need to look for other meanings of mathematics beyond it social-historic-cultural meaning. This means that as we humans try to understand the significance of mathematics in our society and in our daily lives, we can answer the big questions. With this, Hersh added social as the third in physical and mental as the standard kind of existence. To prove his humanist philosophy, Hersh he used it as a basis for investigating and rejecting commonly ideas of mathematics.  According to him, mathematics has a front part and back part. The front part is the results that are shown to the world while the back is what is obtain in the results. The three mainstream philosophies relate only to the front while the humanist focuses on the back. Hersh also discusses infinity shortly. He points out that infinity is different from physical reality and it only comes out of our head. He then said that the brain is a finite object which means it cannot contain anything infinite.

                In the second part, Hersh gives fascinating account of the philosophy and ideas of many important individuals and philosophers of mathematics. It ranges from Aristotle and Pythagoras to Locke and Wittgenstein. But Hersh focuses more in the account and work of Piaget. He acknowledges his great work and its impact in the world of mathematics especially in cognitive psychology. He agrees with his works especially on his notions of stages.

                For a first timer in the world of mathematics philosophy like me, it was a bit confusing to read. I had to read every paragraph twice to understand it. With all the formulas and philosophy, it takes me a while to process it. When I all processed it, I began to appreciate his work and passion in answering this one question. I salute him for his passion in mathematics and for his awesome brain. I just wish that he tried to explain more about infinity and why it is not possible. Also his discussion on how his book will be applied to education is somewhat lacking and not convincing. He only gives few details about its application. It may have been nice that he paid more attention to it because it is really important for educators and teachers to know what math really is. In conclusion, I think the book is really great and has great significance in the world of mathematics and also, for me. The book may have been boring and difficult to understand at first but it helps me to reflect my views on mathematics and what is it in my life. And more importantly is it helps me to know the answer in the question what is mathematics, really?