What is Mathematics, Really?

What is Mathematics is a work made by Richard Courant and Herbert Robbins. Courant and Robbins show some exquisite exposition about the nature of Mathematics. Hersh explore the nature of Mathematics- where it comes and what it is. In exploring Mathematics he makes two important points. The first one is answering the question of Courant and Robbins. The second one is the sources of Mathematics must be a major player in developing philosophy.

 In this philosophical work, a mathematician accomplished the main stream of mathematical philosophy: Platonism, formalism, and intuitionism or constructivism. He calls the “ humanism”  as a notion that “ mathematics must be understood as a human activity, social phenomenon. A part of human culture that historically evolved. He also designate the issues of philosophy as an existence of finite and infinite mathematical articles, intuition, proof, and truth.

The book is presented as an “aninquiry into mathematical existence”. The problem of this is the counting of the parts of 4-dimensional cube and what the calculation could be make in this. Hersh switches the 3-cube and counts its vertices, edges, and faces. He also applied it to the 2-cube. These three sets of formula show the pattern of a 4-dimension cube. The main point of this book is to explain why he rejects the three philosophies which is the Platonism, formalism. And intuitionism.

Platonism as what Hersh describe as an idea that “ mathematical entities exist outside space and time, outside thought and matter. Hersh reject it because of several reasons: it is not related to material reality. It violates the impiricism of modern science and insists on an acceptance of the realities- physical and mathematical.

Formalism according to Hersh ia a meaningless game played by subjective rules. The formalism is more serious than Platonism for him. He argues that rules are not subjective but a factually determined works of the society that evolve through the workings of social groups.

Intuitionism accept the set of natural numbers as fundamental datum. Mathematics obtained through the a process of finite construction. Hersh only adopt thepoint of view of intuitionism of natural numbers but simply not its universe. Hersh offers a humanist   point of view as an alternative to the main stream he rejected. He said that there’s no need to look the hidden meaning of mathematics beyond its social-historic-culture meaning.

Humanism argues that Mathematics is not unique because in mathematical situations, mathematicians do not understand each other and develop different approaches to study the same phenomena. But proofs are incomplete, people cannot fathom its axioms, and these are only alternative.

An important issue that every philosophy of mathematics had to consider the mathematical intuition and Hersh provides some interesting insights. For Platonist, Intuition is the mechanism for accessing the postulated world. For intuitionist, the source of natural numbes is intuition. For Formalist, intuition is the source of correct theorems for which formal proofs are formulated. For the humanist, mathematics is the study of mental objects. Intuition is the affect in mind or brain. Thus according to Hersh, in Humanist viewpoint, intuition consists of mental representation and ideas.

Mathematics becomes interesting and important when it explains the essential aspect of mathematical experience. Hersh treatment of this important topic is superficial. Infinite object cannot contain anything infinite.

Mathematician is seriously interested in philosophy of mathematics Hersh performs a great service in this book. He account a few sentences about Aristotle and Wittgeinstein. Hersh pay attention on works of Piagets than those who write philosophy of mathematics. He acknowledges the tremendous impact of Piaget’s writings. Piagets notion of stages are based on the maturation rather than intellectual development and misunderstood the role in education.

For almost in this period, mathematics was confidentially connected, as well as in scientific thinking, with theology. This book is appreciated because it sketch the number of philosophical ideas relating to mathematics and Hersh introduction of Humanist philosophy of mathematics.