Book Review: What is Mathematics, Really?

Mathematics as a form of life, it is basically the thought that the book is trying to imply. According to Hersh, Mathematics is never about the mental or the physical aspect that most of us consider. Instead, it is about the socio-cultural-historic terms. The book was indeed a philosophical work; Hersh tried to reject three mainstream mathematical philosophies, namely; Platonism, formalism, and constructivism. Instead, he suggested a different philosophy, “humanism”. He wants Mathematics to be accepted as a normal human activity or a natural phenomenon, in other words, a form of life. Hersh tackled the issues in mathematical philosophy and showed the reader that his philosophy (humanism) deals better than those philosophies he rejected.

First, he talked about something that we know has values, has dimensions, has measurements, but we also know does not exist. How is it possible for us to know very detailed information about something that does not exist? Or, if it exists, why can’t we see it, where is it then?

Hersh discussed about Platonism which states that “mathematical entities exist outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social.” I think it is a bit impossible for an individual to avoid himself from having thoughts about searching for something that we consider “out there” that is not just a fruit of mere imagination. Then, if that is the case, I think Platonism is indeed right. But according to Hersh, Platonism does not support the fact that there are “flesh-and-blood mathematicians”, real Mathematicians. It also does not support the practicality of modern science. Another thing is that it keeps on insisting about two parallel realities, namely; physical and mathematical but never explained how they interact with each other.

If Hersh did not agree with the Platonism, he objected more with the formalism that says “mathematics is a meaningless game with arbitrary rules”. He said that Mathematics is not arbitrary; instead they evolved to become like what it is now the same time as man did.

Constructivism, otherwise known as Intuitionism is the third philosophy Hersh tried to reject. He opposes this philosophy and his opposition was even made stronger by Piaget’s experiment. Piaget used children from the western culture from one to seven years old and asked them to do things for affirmation of his theory that knowledge is acquired not readily there in one’s mind. By this experiment, he proved that numbers are not ‘naturally given by God’, instead, numbers are acquired, learned and is being placed in one’s mind as it grows.

Hersh rejected all of the three philosophies above. He believes that Mathematics has no hidden meaning or definition; instead we must only consider its social-historic-cultural meaning. Two standard kinds of philosophy that the Philosophers discussed are obviously the mental and physical standards. But for Hersh, there is still one more standard, the most important of them all, the social standard.

He used a 4-dimensional cube as an example to differentiate the three philosophies from humanist. For the Platonism, 4-dimensional cube exists only as a “transcendental, immaterial, inhuman abstraction”. The same as for both formalism and intuitionism, 3-dimensional cube does not represent anything real; it even said that it is a “representation without a represented”. For humanist on the other hand, a 3-dimensional cube exists at the social-cultural-historic level, shared in peoples mind and thoughts.

            Frankly, the book was not that interesting for me. Maybe because it is Philosophy and Philosophy for me is very boring. I also don’t like the way he states bluffs and promises things that he cannot fulfil, including those he mentioned in the preface. He analysed and criticized the other philosophies effectively, yes, I’ll give you that, but he was not able to defend or give more information why he considered humanist as the superior philosophy. I think he focused or criticizing the other philosophies and forgot that offered ‘humanist’ as an alternative and he still need to defend it more. On the bright side, he was still able to tell why ‘humanist’ is different from all of the other philosophies.

This book’s title was from the book What is Mathematics? by Richard Courant and Herbert Robbins. Hersh mentioned that the book did not really answer the question they asked; it is because Courant and Robbins’ approach of answering the question is by showing us, not by telling us, that is why he decided to try to answer it himself. Hersh dealt with the question by discovering Mathematics’ nature, finding out its origin and knowing what is it, really. He did all of these in a philosophical manner.

So, apart from the negative comments I mentioned, the author at least was able to answer the question, What is Mathematics really? Also, the fact that he is just an assistant machinist and still, he was able to do such work amazes me. It just means that one does not need to be extraordinary to do something great.

All in all, I was given an idea on what Mathematics really is. Only an idea since Philosophy is just a way people think. For me, Mathematics being ‘form of life’ is a very good definition and I would claim that he had convinced me with his definition of Math. But, I will not limit myself to that particular idea because for me Mathematics is too broad to have only one meaning.