Hersh: Coz I Hate Mainstream

                Mathematics is a subject which involves the study of numbers, quantities, shapes, etc; we could settle to that definition and end our talk here. But of course, we would rather discuss more because that lexical meaning simply isn’t sufficient for our ever curious brains. So, “What is Mathematics, Really?” - that’s exactly the title of the book written by Reuben Hersh which talks about the philosophy of mathematics. It is actually a reaction to “What is Mathematics?” by Richard Courant and Herbert Robbins. Quite a long reaction, I would say, as it composes of 360-pages filled with information to contemplate what Hersh thinks math should be.

                The book is divided into two parts and the first part talks about why he rejects the mainstreams philosophies of mathematics. Hersh identified the three mainstreams as Platonism, Formalism, and Intuitionism. Platonism says that mathematical objects exist outside physical space and time. The author argues that this way of thinking math is inadequate because it does not make contact with reality, it violates the practicality of modern science and it does not explain how physical and mathematical realities interact. Formalism on the other hand, as Hersh would put it, is condensed to the short slogan: "Mathematics is a meaningless game." He is rather enraged and expresses this by saying: “this is a gross error”. The author thinks it is way more than just a game with arbitrary rules. Mathematics is the product of inner workings and interactions of social groups since time immemorial, determined by the physical and biological properties of this planet and therefore is definitely not arbitrary. Intuitionism accepts that the set of natural numbers are the foundations of mathematics without including the law of the excluded middle. Hersh argues with anthropological background supported by Piaget’s research. That research concluded the children make numbers in their minds based on certain way of thinking and experiences. Thus, natural numbers are made in an individual’s mind and not given by God.

                Hersh suggests an alternative way of taking in mathematics: the humanistic kind of way. He says that traditional philosophy only sees the front of math and not the back. He emphasizes that a humanist sees mathematics as a way of life, “a social-cultural-historic activity”.  He also discussed on the standard issues of philosophy of math. One of these is the existence of finite and infinite. Hersh believes that infinite and finite dimensional space differs for only one or two but totally different properties. He mentioned that infinity comes from our heads but also emphasizes that it’s not really the infinite that comes from our mind but notions of the infinite. Another issue he mentioned was the discrepancy in the meaning proofs. Hersh stresses that proof has two meanings. One, “it is the transformation of certain symbol sequences according to certain rules of logic. The other one states that proof is “what we do to make each other believe our theorem”. The author expresses his disappointment of this unaddressed issue and gives possible assertions. And of course, intuition is also part of the story. Platonism thinks of intuition as the one connecting the human awareness and mathematical reality. Hersh criticizes this because it fails to answer many questions. For the intuitionist on the other hand, they take a stand that natural numbers come from intuition. This stand fails to answer the same question Hersh asks the Platonists. The formalists say intuition is the basis of formal proofs derived from correct theorems which intuition provides. This is again unsatisfactory for Hersh. He creates a stand in the humanist point of view where intuition is the mental representation of mathematical objects acquired from repeated experiences.   

          So we’ve barely covered all of Hersh’s thoughts but now it is time for mine. I am in no place to criticize an author of a detailed philosophy-of-mathematics book so I would have to give comments as a mere book reader. Hersh often excludes non-mathematicians like yours truly in his words. This is probably because the book is addressed to mathematics teachers as he once said: “we mathematics teachers”. And so I find the book unrelatable most of the time. It didn’t help that there was so much calculations involved. It gives me the thought of how dumb I get when it comes to arithmetic because it seemed so easy for the author. So much for the bad parts, I would also have to give it to Hersh that he’s written this book with so much passion. He eagerly tries to convey his message as a humanist and his view on how mathematics is supposed to be seen. As shallow as it is, I would have to commend that he was rather interactive with his reader compared to most philosophical writers that I’ve read. This book is very much informative and it gives its readers a good argument. Nonetheless if you are after an informative and good argument on the philosophy of maths, then this book is what you’re looking for.