A Review on the book: What Is Mathematics, Really?

 The book, What Is Mathematics, Really? tried to counter the book read by its author, Reuben Hersh, when he was still a machinist’s helper. For him, the authors of the said book were not able to answer what Mathematics really is. The two authors answered the question by showing Mathematics and not by telling what it is really. Reuben Hersh wished to answer it in a radically different way. According to the author, there are three main streams of mathematical philosophy and those are Platonism, Formalism and Intuitionism or Constructivism. He had rejected these three and he made his own philosophical view on mathematics and this is what he called as “Humanism”. His philosophy is to make the people understand that Mathematics is a human activity. In that case, since it involves human activity, it is a social phenomenon and these social phenomenons as part of the people’s culture.

As what any philosophers would do, he claimed that his kind of philosophy is better than the philosophies he rejected. Since he wanted not just to show but to tell what Mathematics really is, he presented answers to problems like on counting the parts of a 4-dimensional cube and he describe some of the standard issues like the issue on finite and infinite numbers. To answer these, he made comparisons on the three main streams of Mathematical philosophy together with his very own philosophy. Speaking of the main streams, I think it is better to talk about it first so that we’ll know why for him his “Humanism” is superior. According to him, Platonism is a philosophical idea where “mathematical entities exist outside space and time outside thought and matter, in an abstract realm independent of any consciousness, individual or social.” Well for him, this Mathematical philosophy is inadequate for it is something that is not real or something that can be grasped concretely.

The second main stream for him, Formalism, is still not acceptable as Mathematical philosophy because for him formalism says Mathematics is a meaningless game played by explicit but arbitrary rules. This time, he is more serious compared to other main stream philosophies. This is because, still in relation to his idea of “Humanism”, the rules are not arbitrary (contrary to what Formalism believes) but rather determined by the society as generations pass and evolve. The last mathematical philosophy that he considered as inferior to his “Humanism” is Intuitionism or constructivism. Here his view is supported by Piaget’s research. According to Piaget, the concept of natural numbers was only constructed based on experiences and modes of thinking. Now, let us take a look at Reuben Hersh’s “Humanism”. He states that in order to understand the meaning and definition of mathematics, one doesn’t have to unravel a hidden meaning. Instead, just by looking at the society of mathematicians and the people dealing with mathematical situations in everyday life can answer the big questions.

Let us compare and contrast the ideas of the four philosophies on a 4-dimensional cube. The idea of Platonist as pointed out by Hersh is that 4-cube exists as “transcendental, immaterial, inhuman abstraction.” For intuitionist and formalist, “there is no real 4-cube but only a representation without being represented.” As for the humanist, 4-cube exists “at the social-cultural-historic level as a kind of shared thought.” Another issue mentioned by the author is about finite and infinite. For him, infinite is different from physical reality and it just comes out of our minds. Our brains cannot tell anything infinite because it is a finite object.

From what I have read, this book mainly reflects the author’s own view of Mathematics and his desperateness to at least relay what Mathematics really is in his own point of view. Well that is expected since we are talking about philosophy and in philosophy it is normal to critique other philosophies as long as you have your own argument. The author’s inspiration to create this book is because he wanted to tell the readers what is mathematics really which, for him, was not really answered when he read a certain book. I think he is not successful in further explaining what mathematics really is. When he read What is Mathematics?, he said that the authors were just mainly showing what is mathematics but they were not able to tell what is mathematics. Well, this time around, for me he went the other way around. Although I know that he tried his best to relay his point of views about mathematics, I think this time he was more on telling and not showing. I cannot help but think that he was just being rhetoric. For me, it would have been better if the explanation were of showing and telling. 

I think his mathematical philosophy on “Humanism” focuses more on mathematics as part of human culture and not mathematics as science. His views on mathematics were also those that can be of grasp. He is not so much on things that are not physically reachable. Generally, I think his book focused more on differentiating the three main streams of mathematical philosophy from his own “Humanism”. At the end, it looks like he just wanted to make it look like his own philosophy is superior or better in understanding Mathematics. As a consequence, he was not able to stick to his objective which is to provide us a better and more radical answer to what Mathematics really is.