Mathematics: Redefining its Real Essence (Book Review)

Mathematics: Redefining its Real Essence

            If a student hears the word “Mathematics”, what comes directly to his/her mind? Simply, it would convey a generalized answer, a subject full of numbers and theorems making our lives miserable. But in a different perspective, how can we really define Math? Or in a sense, what is Mathematics, really? A book written by Reuben Hersh elaborated mathematics through explorations of its humble beginnings and the philosophies creating its solid foundation that is valuable up to the present. Hersh imposed two points in connecting philosophy to the universal language, mathematics. One, that philosophy should not just support the underlying concepts of Math but would strengthen the subject thus establishing a firm principle. And two, for each philosophy to reach its goal, information provider should play the primary role in its development. The book “What is Mathematics, really?” showcased and explained the roles of the three primary philosophies, Platonism, formalism and intuitionism or constructivism in mathematics and compared it with his philosophy called humanism manifesting that mathematics should be viewed as a human exertion, a product of culture and society. (Hersh, 1997). In this book, concepts in the philosophy of mathematics like proving, intuition, actuality of finite and infinite mathematical entities, certainty, object versus process, and invention versus discovery were retraced and compared it with the use of the primary principles against his humanist philosophy.

            The book started with a conversation of Laura and Reuben about numbers and infinity as well as connecting a certain color with how numbers are arranged. They were interested of the thought, “Up to what extent could a gazillion be?” and “Who could ever count a number that high?” Retorting to their confusions in mind, they exchanged ideas concerning the existence of the biggest number a human mind knows and a human device could count of. They began thinking of constructive images that would satisfy their concerns and came up with a philosophy of mathematics.

            Formalist philosophy of mathematics defined mathematics as a meaningless game pointing out that math doesn’t have a strict property. In this kind of philosophy, rules were determined by the workings and challenges met by the society developed under pressure by social group interactions as well as works of nature. This definition was objected by Hersh because he believed that is was not the real definition of mathematics. He termed this philosophy as deceitful when applied to real life. Platonist philosophy of mathematics, as how Hersh describes it is that mathematical entities do exist beyond the bounds of space and time, as well as thought and matter. In this philosophy, mathematical objects are said to be existing and real in our knowledge. Curves, infinity as well as infinite sets all constitute a definite object with certain properties whether recognized or not. According to Platonism, a scientist can’t invent because everything was already given, what they can do is to only discover what was laid off on them. Platonsism acknowledged that the facts of math can stand by his or her own wishes making it extraordinary. Hersh had a different view in this philosophy; the definition itself does not satisfy the requirements being a philosophy of math. Hersh rejected this with the counter-statements that the ideas gathered do not really relate the reality, that it desecrated the logics of modern science. Intuitionism, on the other hand, accepted the fact that the fundamental datum of Math came from natural numbers where all concepts of math were formulated in a process of numerous constructions. Four more general myths were also considered to be a part of this philosophy: unity, universality, certainty and objectivity which played a vital role in improving and elaborating the concepts under this philosophy.

After his rejection with the different kinds of mainstream philosophies, Hersh introduced humanist or sociohistorical point of view. In his viewpoint, he persuaded people to go into what is and has been done by mathematics to mathematicians and those people who have been on a mathematical problem in their lives. Thus, the existence of mathematics doesn’t just talk about having the mental and the physical capacity to do and apply math but also aims to have the social capacity in the use of mathematics. Humanism just doesn’t show one side of mathematics or elaborate one philosophy for instance, humanism sees the Platonism, formalism and intuitionism form in every aspect of mathematics making it better and understandable by more people.

            In this book, Hersh also suggested that philosophies in mathematics have parts: the front and the back. The front signifies the results publicized to the world which was applied by mathematical knowledge through concepts and ideas and the back signifies how the results were gathered and tested. Based on Hersh, the three discussed mainstream philosophies only focus on the front part of mathematics while his philosophy focuses on the back, on how the results were achieved. Hersh also believed that mathematics does not contain truth about world and universe and that they are clear and unquestionable since people have their own perspective in each phenomenon.

            The book also uncovered the relationship of philosophy to theology, more likely on the concept of how Platonism was formed. Since most of the oppositions of the concepts behind this philosophy came up with the thought that the basic principles of math came from God, and that math became existent through the presence of a deity or supernatural, it can be said that the concepts underlying mathematics provides perpetual truths about the universe. But as years passed, many mathematicians and even scientists did not believed that God existed and that math was formed through divine intervention so the most powerful principle holding Platonism was removed. Formalism, for others, and intuitionism, for others, became the philosophy followed by the mathematicians.

            What does this book imply to me? This book gave me the idea of the concepts of variation in mathematics. Each mathematician had shown what concept would actually fit mathematics and what philosophy does math follows with. This book enlightened me of the various philosophies in math with its differences as well as how it was related to it and how significant it is to the natural world. Reading this book gave us the hint to further study and analyze each philosophy to derive to a general idea that would really define mathematics and establish a foundation that would connect it to the world.