REAL DEFINITION

What is Mathematics, Really?

This book was created by Reuben Hersh. This contains his reaction about the book by Richard Courant and Herbert Robbins entitled “What is Mathematics?” in this book, Hersh dealt with the question (What is Mathematics?) by exploring the nature of mathematics. Hersh made two important points in appealing to the philosophy of mathematics for his exploration of the nature of mathematics. First is that philosophy, as one way of answering the question, must be more than an attempt to establish a foundation for mathematics. Second is that the purveyors of mathematics must be the major players in the development of its philosophy.

In this book, a serious and accomplished mathematician explores deeply, and rejects, what he considers to be the three main streams of mathematical philosophy which are the Platonism, formalism, and the intuitionism or constructivism. As an alternative, Hersh offers the “humanism”. Humanism is the notion that states that mathematics must be understood as a human activity, a social phenomenon, a part of human culture, is historically evolved, and is intelligible only in a social context. He described some of the standard issues of philosophy of mathematics, and tried to show that his philosophy deals with these issues better than those philosophies that he rejected.

Part One

The opening of the book is about a conversation between the author, Reuben, and a twelve-year old girl, Laura about the philosophy of mathematics. The beginning gives the problem about a 4-dimensional cube: on how to count its various parts and reflect on what kind of sense the calculations could make. Hersh solved it by switching to the 3-dimensional cube and counted its vertices, edges, and faces. He did the same in solving for the 2-dimensional and 1-dimensional cube. In solving the three cubes, it showed a pattern that is easily generalized to solve the 4-dimensional cube. The solution of the parts of the 4-dimensional cube led to a list of questions about its existence: if it exists, where is it?; if it doesn’t, how could we get such information about it? Later, Hersh used those possible answers to help explain various philosophies of mathematics, including the one he called “humanism”.

After the 4-dimensional cube example, Hersh turned to the main point of the book, which is to explain why he rejected the three mainstream philosophies as inadequate for a philosophy of mathematics, and the reason why he believed that his “humanism” is superior. While he was studying, along the way, he considered a number of generally accepted properties and tried to debunk them.

Part Two

Hersh gave us a capsule account, ranging from a few sentences to a few pages of the philosophical thinking or nearly 50 individuals, starting from Aristotle to Wittgenstein. This book is particularly helpful because this is written from a point of view of a mathematician. Hersh omitted the work of Orestes and the Scholastics maybe because he is focusing on the analysis strand of mathematics and so, is less interested in the arithmetic, number theory, or algebra line of development. Hersh gave greater attention to the work of Piaget than others who wrote about the philosophy of mathematics. He acknowledged the tremendous impact of Piaget’s writings on cognitive psychology. He seemed to feel that Piaget’s notions of stages are based on maturation rather than cognitive development and so, misunderstood its role in education. Hersh focused on Piaget’s epistemology but rejected the book that Piaget wrote with Beth on the same topic because of his disagreement with the philosophical position taken in the first half of the book. What he missed is the first half about the foundations of mathematics which was written by Beth. The second half which was written by Piaget is a wonderful account of his epistemology. In that book, Piaget considered, in some depth, all of the questions Hersh is interested in and more.

Conclusion

When I first heard the title of this book, something came into my mind. Something I thought as the answer of that question. I thought that mathematics is all about numbers and just focuses on the numbers and the fields where numbers are related. But I had learned that I was wrong. I learned that the real answer is that, mathematics a language of the universe, and it is the king and queen of all sciences. Studying mathematics is great whenever you are already used to it.

When I started reading this book, I thought this will be boring because this is about the history of mathematics, but I found this book an interesting one. The first problem is that "how many sides and points a 4-dimensional cube has?" and "what will be the shape of this 4-dimensional cube, if it really exists?". Supplied with the information of the 1-, 2-, and 3-dimensional cubes, i found out how many sides and points a 4-d has,but i was not able to find its shape.

Well, I can say that this book just proves that mathematics has really a lot of discoveries and is not just created by some people who are considered mathematicians, but is discovered and proven by them.

This changed my perspective about the real meaning of mathematics. At first I thought that mathematics is just all about numbers, but when I have read this, I have known that it is not just about numbers, but also about the rules and philosophies that mathematics is following.

Reference

·         Ed Dubinsky. Book Review (5 pages). < http://www.ams.org/notices/199909/rev-dubinsky.pdf>.12/10/13