Book Review: What is Mathematics, Really? by Reuben Hersh

             Reuben Hersh is a mathematician who devotes himself in studying the nature, practice and social impact of mathematics. His writings mainly challenge the typical philosophies behind this discipline. After reading the book entitled, What is Mathematics? By Richard Courant and Herbert Robbins, he was not satisfied with their teachings and simply did not answer what is mathematics in the first place is all about. This gave him the urge to write a book having the same title but with a greater emphasis on the additional word, “really”.

         The book was divided into two sections. The first one mainly talked about the different philosophies where mathematics revolved. These are Platonism, formalism and intuitionism. Upon investigating his own philosophic insight, he opened his mind in the concept of front and back views of Mathematics and finally arrived in a social-historic philosophy or the humanistic approach where he rejected the aforementioned philosophies and insisted that humanism is much better to be accepted.

            Platonism is the view that there exist such things as abstract. Hersh said that “mathematical entities exist outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social.” He rejected Platonism because it does not relate to any material entity and does not make physical contacts. It violates the theory of knowledge which states that knowledge comes only or primarily from sensory experience.
            Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of Euclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of interference" to generate new strings from given ones), one can prove that the Pythagorean theorem holds (that is, one can generate the string corresponding to the Pythagorean theorem). According to formalism, mathematical truths are not about numbers and sets and triangles and the like -- in fact, they aren't "about" anything at all. Hersh also rejected this kind of view because he thought that the rules must not be arbitrary.

          Lastly is intuitionism which is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds. This view on mathematics has far reaching implications for the daily practice of mathematics, one of its consequences being that the principle of the excluded middle is no longer valid. He rejected this because of his anthropological point of views.

           In his humanistic approach, Hersh considered it as the greatest philosophical view. He denies that mathematics has any kind of reality independent of human minds. For him, it is a social phenomenon and co-exists with man. We cannot argue that math is part of human culture because it truly is. I somewhat agree with him because without our mathematical thinkers, then math would not be fully explored and developed at all. It all started as a way of life because it is based on man’s needs or necessity.  

         The second section of the book was all about the different views of our philosophical thinkers whom Hersh have taken account with and have also widened my mind. It is truly helpful if one wants to study the in depth contexts of mathematics because it was taken from the brilliant minds of our mathematicians.

            I would be lying if I say that the book really interests me because I find it quite hard to understand. Though the author presented it very well, and has his own writing style that could catch the attention of the readers, especially the first part where he had a conversation with a girl about the existence of a gazillion and one. Thereupon I had second thoughts and I pondered on the question, “Is there really such a number?” I think that it all began in the human mind and we could fully imagine things out of the abundance of it. So we could think and move beyond our knowledge and beyond what our mind is capable of and I could say that abstracts and certain arbitrary forms exist. And this is somehow in agreement with what Behrens said which is “Humanists have extended some links between mathematics and the physical world, but most mathematicians still believe they operate in an immaterial realm of the mind, with unquestionable logic and abstract thought.” On the other hand, the book is really very informative and provides an understanding on the different views of mathematics that could change one’s perspective towards the said discipline.