Uncertainties in Mathematical Philosophy

           Mathematics had been part of the social and intellectual evolution of the human race. Its evolution was rapid and historical since it took part in many events of the evolution of the society. Measurements, calculations, forms and patterns are some of the components that make up mathematics. In the past, knowing how to write numbers and solving simple equations are already enough to meet the necessity of the people. In the early days of our history, mathematics took part in the daily lives of the earliest civilizations even in the simplest way possible.

            But as the time evolved, mathematics became complex and broad that is now composed of different theorems, principles and concepts. With its evolution, it can now provide mathematical explanations and models or figures that we needed to answer some of our intellectual doubts that we encounter. For us, mathematics can be simply defined as the study of different numbers and patterns. But as it evolved into a more complex field, definition and its meaning also developed. Some mathematicians define mathematics beyond just a study of numbers or patterns or calculation, they believe that it can still be interpreted deeper and can be given with a better definition.

            In the book, What is Mathematics, Really? by Reuben Hersh, the term mathematics was attempted to be defined by introducing the author’s humanist’s view over the subject. This book was written because of the author’s curiosity as to the real definition of mathematics after he read the book of Richard Courant and Herbert Robbins, What is Mathematics?.

            In Hersh’s introduction, he gave a problem of how to identify the number of the different parts of a 4-dimentional cube. For this to be solved, he used a method where he first considered a 3-cube, 2-cube and a 1-cube where the faces, edges and vertices were counted. These 3 sets of formulas showed an overall pattern that was then applied to solve the parts of a 4-dimensional cube. After identifying the parts of a 4-cube he then lay a question, in which he asked if such form of cube really exist in the real live world. The different types of mathematical philosophy such as Platonism, Formalism and Intuitionism, were then introduced including his own humanism view of mathematics.

            According to Hersh, Platonism has the idea that those mathematical entities exist outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social. Formalism on the other hand thinks that mathematics can be compared to a nonsense game that has clear but random rules. And lastly, intuitionism believes that fundamental unit of higher mathematics is the set of natural numbers that goes through a process of determinate constructions which does not follow the law of excluded middle.

            Among these mathematical philosophies, Hersh believes that none of them is sufficient enough to define the philosophy of mathematics and that his proposed philosophical view of mathematics, which is humanism, is the superior one among the stated philosophies. He pointed out that humanism defines mathematics as a “…human activity, social phenomenon, part of human culture, historically evolved, and intelligible enough in social context.” He believed that using humanism as a viewpoint can include all the beliefs that stating that mathematics is a product of characteristics of human culture and society.

            Personally, before reading this book, I never knew that I actually follow the thinking of formalism. Base on the definition of Hersh, it treats mathematics as a non-significant and useless idea of numbers that does not contribute to the development and evolution of our society. But after I comprehended the points that the author wanted to share, I realized that among the mathematical philosophies, Platonism actually caught my interest. In a viewpoint of Platonism, mathematics is believed to already exist even in the moments that there were still no time and space. It was never invented and developed solely by the mind of the society but rather it was discovered by the great minds that were able to find the connections of hints and clues left by mathematics.

            The topic about the functions of proof was also interesting. He showed that proofs can either be used to prove a certain doubtful idea or it can be used for collecting knowledge about a certain notion. I also find the example of the 4-dimensional cube in the start of chapter one really fascinating. It was able to show that in solving problems in mathematics, going back to the simplest or basic form of knowledge can give way to discovering solutions for the higher and advance mathematical problems. At first, I was not able to comprehend the point of the example but then it turns out that it has a philosophical significance in defining mathematics at the end part of the introduction.

            Although the book was helpful, there were some terms and points of the author that were not that clear and understandable enough due to its too technical terms. As a non-mathematics student, I find it hard to comprehend how this terms are interconnected and what could it possibly mean. At some point, supplementary research was needed for me to understand the topic. Also, for the author to prove his point of having humanism as the superior viewpoint among the other mathematical philosophies, he becomes too forceful for the readers to follow his philosophy.

            Overall, the book is indeed helpful in giving the readers a new knowledge that mathematics cannot be defined as easily as we do today. It involved different philosophers that tried to define the nature of math and yet until now, more ideas are still clashing and are trying to prove that their verdict is the most right among the others.