Questioning the Foundation of Mathematics

What is Mathematics, Really?. This book was written by the author for the sole purpose of exploring the philosophies which are essential to the foundation of mathematics. Reuben Hersh, the author, made important points regarding the philosophies that served as the foundation of mathematics for decades and introduced his philosophy of Humanism. He also runs through the history of the philosophy of mathematics to question and make his points and then connecting these philosophies to broader political and philosophical issues today.

In the first part of his book, he wanted to inquire mathematical existence. He wanted to ask about the existence of a 4-cube by first analyzing the other cubes that could be seen and formed in the physical world. For the 3-cube, 2-cube and 1-cube, he counted the vertices, edges and faces. He showed three sets of formulas which show patterns that can be generalized to four dimensions which lead to a number of questions regarding the existence of a 4-cube. He questioned: if it exists, where is it? How does it exist? If it does not exist, how could we obtain such detailed information about it? And a lot more questions similar to that followed. After imposing these questions, he used possible answers and also helped explain various philosophies of mathematics: Platonism, Formalism, and Intuitionism, and also offered his own Humanism. That we have to focus our attentions on things we can actually observe.  And after this example, he proceeded to the main point of his book. To explain why he rejects the three mainstream philosophies as inadequate for a philosophy of mathematics and also why he believes humanism should be the most appropriate and adequate philosophy of mathematics. Along the course of his discussion, he considered a number of properties of mathematics and tried to debunk each one of them.

The idea that “mathematical entities exist outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social” is Platonism, as what the author explains it. For me as a biology student, it is impossible to avoid the feeling of urgency to discover something that is out there, something that has an existence independent to many things. In order to deal with this, a Platonist view should be exhibited. We engage in Platonism without even thinking or realizing that we are. What more to those mathematicians? They wanted to solve and understand complex things which exist and which could possibly exist in this world. But then he rejects this philosophy as such because it violates the empiricism of modern science, it insists on the acceptance of physical and mathematical realities but does not explain how these to realities interact with each other.

“Mathematics is an otherwise meaningless game played by explicit but arbitrary rules”. This is Formalism. He argues that the rules are not arbitrary, but are “historically determined by the workings of society that evolve under pressure of the inner workings and interactions of social groups, and the physiological and biological environment of earth.” He said that following rules without judgment is a fiction and that it is misleading to apply it in our real lives.

The acceptance of the set of natural numbers as the fundamental datum of mathematics where all meaningful mathematics is obtained by means of finite construction is Intuitionism. And then again he argues that the intuition of the natural numbers is simply not universal. This view was supported by the research made by Piaget. For Piaget, the natural numbers are not given by God but are constructed in an individual’s mind by coordinating the concepts of set inclusion and ordering.

He debunked and rejected all these three mainstream philosophies of mathematics. He then proposed his philosophy of Humanism. He said that “there’s no need to look for a hidden meaning or definition of mathematics beyond its socio-historic-cultural meaning.”  And by this statement, we can deduce that we can easily find the answers without thinking too much of it by just looking all the things that were done in the society by mathematicians in our everyday life. He added the social to the standard kinds of existence. Mental, physical and then there is social.  In the problem regarding on the existence of the 4-cube, in humanist’s view, the 4-cube exists in the shared consciousness of people as a kind of shared thought or idea.

The second part of his book revolves around the different point of views of many mathematicians ranging from Aristotle to Wittgenstein. This part of the book is really helpful to those who wanted to appreciate and engage deeper into the philosophies which revolved around mathematics.

As for me, I think this book is very helpful since there are a number of philosophical ideas on mathematics mentioned and discussed in here, and the author’s proposition of the philosophy of humanism is quite interesting. This book may have drawn much attention to the mathematics-based communities regarding on how he argued and debunked the greatly accepted philosophies of mathematics. How he thought and put Platonism, Formalism and Intuitionism in his own point of view. Even I thought it was really brave and intelligent of him to do that.  His proposal is getting interesting as he progresses with his discussions. It makes right claims for mathematics which corresponds to mathematician’s human experiences and that mathematics is learned and taught. The humanist view allows us to focus on things we can actually observe and wholly experience. This is a type of book that should be read and discussed and argued with. A book that can generate many emotions, that is in relation to the academic field, by just going through with different point of views of great mathematicians.