What is Mathematics, Really?: A Book Review

A Communication Arts student should know the elements in writing a substantial book review. It should be descriptive and evaluative as possible. It should contain points on the author’s style of writing and parts of what you loved or hated in the book.

However, as much as I want to write a substantial book review, I just can’t. I have to admit that I have not finished reading the book. And honestly, I do not even fully understand and grasp the parts that I have read. Well, perhaps, it’s because I do not have a Math or Math-related subject for two years. All thanks to Math1 for watering my ‘Math-barren’ heart and mind, and for giving me such a challenging read.

            To start off, one could not get a simple reply from asking a short yet thought-provoking question, “What is Mathematics, Really?” The same question was the title of the book Reuben Hersh wrote. In his book, Mathematics was not just defined, described and explained; it was presented with the help of Philosophy. Or, it was more of the Philosophies of Math. The book was divided into two parts. The part one was about the three main streams of Mathematical philosophy: Platonism, formalism and intuitionism and constructivism, and how he rejected those three and offered a new one, which he called, humanism. I do not have any idea what the part two has to offer since, unfortunately, I have not finished the entire book.

            Hersh began by introducing and presenting a problem on counting a 4-dimensional cube and even doubting its existence. But then, this problem was faced with possible answers which Hersh also presented. Later on, he explains the mainstream philosophies of maths which he still rejects anyway.

            Formalism, the first principal view of the nature of mathematics, is more known and understood when related to the slogan, “Mathematics is a meaningless game”, in which both the words meaningless and game were undefined. It was further mentioned that Mathematics is partly a rule-governed game. However, in reality, there are no such things as rule-governed activities. According to Hersh, “It is misleading to apply it in real life.” The belief of following rules without judgment is pure nonsense.

            Platonism, on the other hand, asserts that mathematical entities exist in an abstract world and are independent of our rational activities. But as Hersh would say, this platonic view does not suffice the philosophy of mathematics. He rejects this idea because he thinks it goes against the empiricism of modern science and does not even relate to mathematical reality.

            Lastly, Intuitionism acknowledges that natural numbers are the fundamental datum of mathematics. Still, Hersh rejects this idea since, according to him, intuitionism is not universal. Because of that, he proposed humanism. In this, Hersh adds, aside from the mental and physical, social to the standard kinds of existence as discussed by philosophers.

            Hersh also lists criteria for the philosophy of mathematics. These are the first three essential elements: Breadth, or recognizing the scope and variety of math; Links to epistemology and philosophy of science, and should be valid and compatible into mathematical practice. The next five elements in the criteria are not necessarily essential but are desirable: elegance, economy, comprehensibility, precision and simplicity. Another element is Consistency. It is essential but it is not as difficult to attain as the first three elements. However, he rejects novelty and originality since it is inessential and unattainable, and certainty and indubitability because it is misleading.

            Overall, I’d like to think that Hersh has written such an appealing book but I just couldn’t take that much information on philosophy and mathematics all at once. It didn’t interest me that much. Well, if, at least, he has written in such a way that even high school students would understand, I bet I could appreciate it. But, that’s not the case. He wrote it for those who are in the field of Mathematics and Philosophy, apparently. For if, as he writes the book, expects the ‘greater public’ to understand, he should’ve written it simply and slowly, without shoving the philosophies and mathematics all at once in our throats.

            It’s such a shame that I have written such an unfair review on Hersh’s book. I honestly think that I am not the appropriate ‘target audience or reader’ for the book. I know that book reviews should be as substantial as possible, but you could not blame me if I’d rather read articles on Intercultural Communication than a book on the Philosophy of Mathematics.