Philosophical Nature of Mathematics

A Book Review to What is Mathematics, Really? by Reuben Hersh

            What is mathematics? A common misconception is that mathematics is the study of numbers and figures. Through time mathematics consist of arithmetics and concerned with geometry. Later on, mathematics become the science of patterns in that way it finds a pattern for everything to figure out generalizations. Now, it is renowned that mathemathics is a universal language because it has elements such as symbols and logic to communicate in the universe. But how is mathematics should be understand?

Mathematicians and philosophers of mathematics treat mathematics as isolated, timeless, ahistorical and inhuman. The part one including first five chapters of this book is programmatic, the preceding chapters contained in the part two were historical and the last chapter is the summary. Hersh, in his book comes to repudiate the traditional philosophy of mathematics which is notably the Platonism and Formalism. Platonism is realism; it said that mathematical objects are real and independent of our knowledge. The basic idea is that mathematical entities exist outside space and time, outside thought and matter, in an abstract realm. There are two kinds of Platonism: Platonism of mathematicians which is an empirical scientist, cannot invent only discover because things are already there. And Platonism of philosophers uses logicism. Formalist philosophy of mathematics: “Mathematics is a meaningless game.” He rejected formalism because for him mathematics is not a rule governed game, rules are not arbitrary rather are historically determined by the working societies. On the contrary he showed that from the viewpoint of philosophy, mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. He called this approach as the Humanist viewpoint. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Humanism is a human activity, a product and characteristic of human culture and society. In the humanist philosophy, mathematics is a human activity carried by the society and developing historically. He showed off how the humanist idea of nature of mathematics resembles the mathematics applied to work today. The philosophy of mathematics was said to be a social conceptualism or social-cultural- historic or social-historic-philosophy. Social constructivism views mathematics as a social construction. It is drawn to conventionalism in accepting that human language; rules and agreement play a key role in establishing and justifying the truths of mathematics. The philosophy of mathematics says that mathematics has methods; these are the conjecture and proof. Mathematical discovery rests on proofs and later on becomes theories. A theory is a network of concepts, properties, connections and facts. But Neo-Fregeanism says that theory is the only part of mathematics that deserves philosophical consideration.

Mathematics has a problem and a solution. In this book the title itself imposes a question that should be answered within the context. Reuben Hersh contradicted the two principal views of the nature of mathematics which are the Platonism and Formalism; that was mostly believed by philosophers. He made a philosophical idea to address mathematics which he called Humanism. Humanist philosophy links mathematics with people, society and history. Even if I really got a difficulty of understanding the book having no background of philosophy in mathematics, I think the idea of Humanism is factual. Mathematics has been discovered by the early civilization within their society because they face problems to be answered. Mathematics would always be part of human culture and history for some of the inventions that led to mathematical progress where rooted to the culture of the place it was discovered. An example is the discovery of number zero in India; it was influenced by their religion. And without the history, all those mathematical discoveries would be buried several feet under the ground without being used and developed by the great mathematicians in this century. The main purpose of this book is to answer in what sense do mathematical object exists and to elucidate the mysteries behind the meaning and nature of mathematics.