A Mathematical Dilemma

As a biology student, the book What is Mathematics, Really? by Reuben Hersh in my opinion is difficult to understand, especially that Mathematics is not my forte. But as I tried to read it, I found it quite fascinating as the author pointed out the discrepancies of mathematical philosophies, and as well as emphasizing his stand on what mathematics’ true philosophy should be.

But, that’s impossible!

Does a 4-cube exist?  This is the first question the author presented on the first part. As he explained that by calculations, following the patterns done in a 1-cube, 2-cube and 3-cube properties, we can produce a 4-cube, but it is impossible to create such representation. Even a 3-cube cannot be made as a physical object, so how can it be? It was then presented the three philosophies that the mathematical society has known through time, namely: Platonism, formalism, intuitionism, and adding his own philosophy: humanism. Hersh’s point of writing this book, as I see it is to explain why he does not accept the abovementioned ideas and insists of his.

Putting oneself in deeper thoughts

“None of the three can account for the existence of its rivals.”

According to him, mathematical objects in the view of Platonism are “…objects [that] are real and independent of our knowledge.” He emphasized that a mathematician is similar to a botanist in a sense that “He can't invent, because everything is already there. He can only discover.” I somewhat find this explanation agreeable since mathematics have existed long before as we knew it. Numbers and values are there, just waiting to be named. Whether the most genius person and the slow learner may vary in time or method solving a certain problem, they will arrive at a same, universal, correct answer. As I quote him, answers to math problems are “Right because they are right.”

“Mathematics is a meaningless game.”  This is how formalism describes its philosophy according to Hersh. As he said, games have rules, but these do not apply in mathematics because they are arbitrary and explicit. That basically, mathematics is all about established formulae, which only gains sense if interpreted. In geometry for example, we are tasked to identify the different types of congruent angles, and it always needs a strong proof on why your answer could be side-angle-side, side-side-side, etc. Why is this so? Wouldn’t it make any difference if we just leave it there because it is what it is? As he quoted Lazakus on his work Proofs and Refutations, “Formalism denies the status of mathematics to most of what has been commonly understood to be mathematics, and can say nothing about its growth.”

In intuitionism or constructivism, natural numbers are the foundational data of mathematics. In the viewpoint of an ordinary person, Hersh describes that intuition for numbers is not similar to everyone. Numbers are not made up of a child’s mind, but of a proper mind of an individual by bringing set inclusion and sequencing in harmony.

Humanism: The Light of Mathematics

“To the humanist, mathematics is ours—our tool, our plaything.”

Going to Hersh’s own philosophic view is humanism. Unlike the abovementioned three, it is unique since “Humanism sees that constructivism, formalism, and Platonism each fetishizes one aspect of mathematics, insists that one limited aspect is mathematics.” Its basic idea is that the real meaning of mathematics lies on the works of mathematicians, the people who deal with daily mathematical concepts and their society. Proofs do not need long explanations, but must be brief and concise. He emphasized that mathematical philosophy will be bonded to other philosophies if socio-cultural interactions are valued. For him, humanism values “respected thinkers”. I think this idea cannot be easily applied to the study of mathematics today, as the mathematical society continually evolves with these three mainstream philosophies, but considering it may not be bad at all. Mathematics in humanism, as he said, is a part of human activity and its society, which I find acceptable since it is how I see it, as an ordinary human being.

Conclusion

I considered that this book is written to criticize the three mainstream mathematical philosophies: Platonism, formalism, and constructivism and set a new idea, in which Hersh had created which is humanism. He has presented the gist of the mainstream philosophies, but he was clearly biased on his newly-established philosophy. The example set in Part One really caught my attention, but as I go on reading it, his explanations were a bit dull, especially if the reader is not quite interested in philosophy and mathematics. The emotions he poured out on how he really dislikes these philosophies as well as his determination to present humanism as a new idea to define mathematics is really expressed on his proofs and criticisms. His evidences are also valid since he cited other mathematicians that believe on these ideas as well as the ones who may have accepted his. I find the knowledge on Platonism and formalism explained thoroughly, but constructivism lacks some details in terms of its real message. But in the end it was quite good since he left the decision to the reader on what philosophy he or she is comfortable in believing with.

Overall, this book is a very intellectual piece of writing that will make you think and help you in deciding, especially if you are a mathematically-oriented person, if the ideas passed on from generation to generation is still accepted or a new mathematical horizon is set forth.

*Total word count: 900 (excluding title and subtitles)*