What is mathematics, really? Book Review

The book, “What is Mathematics, Really?”, is a philosophical work that revealed the content of mathematics by discovering its nature. It is said that there are three elements of mathematical philosophy: Platonism, formalism and intuitionism & constructivism. However, Hersh proposed an alternative termed as “humanism” which said “that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved and intelligible only in social content”.

  There are two points that Hersh made: ‘(1) Philosophy must be more than endeavor to set up a foundation for mathematics and; (2) The purveyors of mathematics must be major players in the development of its philosophy’.

The book commenced with an interesting exercise to count the various parts of a 4-dimensional cube and to contemplate on the sagacity the calculation made. He said it is ‘worked exercise in Plya’s heuristic’ and “an inquiry into mathematical existence’. Hersh used 3, 2 and 1- dimensional cubes and tallied their faces, edges and vertices to find a pattern that generalized a 4-dimensional cube. This made it more confusing if such unimaginable cube really exists.

In the whole book, Hersh attempts to reject the three mainstream philosophies of mathematics—Platonism, formalism and intuitionalism. For him, these are insufficient for a philosophy of mathematics insisting that humanism is superior.

Platonism for Hersh are the mathematical entities that exist outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social’. This somewhat made me agree because we can’t really feel mathematics. It is just a knowledge floating out there that helps us live on our physical world. However, it does not connect the interaction between physical and mathematical. There may be a floating knowledge there but the people in the physical world can’t connect with it. This makes it a hanging entity.

According to the author, Formalism is an “otherwise meaningless game played by explicit but arbitrary rules.” Here, Hersh insisted that the rules are not arbitrary but rather they are historically determined. Mathematical solutions that are discovered by our ancestors to be able to cope up with the environment are passed on and applied. For Hersh, “the notion of strictly following the rules without any need for judgment is fiction and is misleading to apply in real life”. Although we think there are rules to follow in order to get a solution for a problem, there aren’t. There are a lot of ways on obtaining an answer explaining why there are different versions in each ancient civilization. This element of mathematical philosophy I do not fully agree on because for me mathematics is also an art where the one solving it has the freedom which way to use.

The last element, intuitionism, is where natural numbers are accepted as the fundamental datum of mathematics. It would only become meaningful if is obtained through a process of finite construction. This concept I really can’t fathom. What I understood is maybe the process of obtaining the knowledge of mathematics should be done systematically, in other words from simple to complex.

The humanist philosophy or sociohistorical view of mathematics is what Reuben Hersh believes on. He rejects the other mainstream philosophies. For him, “there’s no need to look for a hidden meaning or definition of mathematics beyond its socio-historic-cultural meaning. This means that answers are obtained by considering what has been done by mathematicians or the people who have been dealing with mathematics in their everyday life. So, aside from mental and physical, Hersh added a third standard kind of existence which is the social. It is the interaction considering the ones interacting.

Reuben Hersh, compared to other philosophers of mathematics, has understood what the real stand of mathematics in our society is. It is for us a shared consciousness or a shared idea that is standard for the common good and not just a pigment of the imagination. According for Hersh, the mainstream philosophies only concerns the front, ,whereas humanism insist to focus on the back.

            Using the humanism view, mathematics is not unique because as I’ve said earlier, contradicting the formalism view, there are different approaches on how to study the same phenomena. Hersh claims here that formalism does not describe where a mathematical result comes from because he already has knowledge of the concept of what should be obtained before writing a formal proof.

“What is Mathematics, Really?” is a very long book. I am not much of a reader which made it very boring for me. To be honest, I’ve only understood one-fourth of the book but I greatly appreciate the effort of the writer. I personally think he is mad at Plato and his ideas. It made me wonder on some things and made me realize that there are several ways of acquiring solutions. I have a different view of mathematics and this book made me understand it more. It showed that there are a lot of ways on viewing mathematics. But personally, I agree on Hersh that humanism is the most superior element of the philosophy of mathematics. Other philosophers just made it more complicated by not looking into the deeper foundation of it. This book is not only an eye opener but also run-through on what has mathematics have become. Now if they would ask me what is mathematics, really? I would give them the book and say find out for yourself.