Understanding Mathematics

What is Mathematics, Really by Reuben Hersh is inspired by the classic book ‘What is Mathematics?’ by Richard Courant and Herbert Robbins. Robbins and Courant came up with the answer by exploring the content of math and sharing it through comprehensive description and explanation. Hersh’ approach on the other hand was to go back to the origin and understand the nature of math. As he scrutinizes the three considered mainstream mathematical philosophies: Platonism, intuitionism and formalism, he also debunks these.

In the words of Hersh, Platonism is the idea that “mathematical entities exist outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social.” At times math may seem like something out of this world or often it may seem alien to me but I disagree with this thought much like Hersh does. He explains that this is an inadequate philosophy because: it does not physically interact with mathematicians and it insists that there is a parallel existence between math and physicality but does not elaborate how these are related. In modern science, “experience rather than reason is the source of knowledge”. Platonism contravenes with this idea.

Formalism, as Hersh stated “mathematics is a meaningless game played by explicit but arbitrary rules.” He disagrees with this notion much more than he did with Platonism. To him, the rules are not determined on a whim or by chance, rather, it is determined by the biology and physiology of the world and the product of the interactions of society. He avowed that math is not mindlessly following rules without judgment.  

Intuitionism is the theory that facts are known by intuition and not through their provability. Hersh opposed with this because the intuition of mathematical concepts or numbers is not applicable in all cases. He elaborates that the construction of these in the mind depend on the level of mental capability and on experience. 

Hersh offers a humanistic view as an alternative for the three mainstream philosophies mentioned above. Humanism believes that people have the potential to solve problems through critical thinking. The need for answers leaves no area of thought unexplored, unquestioned or undoubted.  Abstract ideas do not fall from the sky, rather regarded as a human endeavor. Math is much like cash, like belief, like rules and all other things that are existing, yet only regarded as part of the collective human consciousness. Without human beings, there can be no math. 

The book indeed is very knowledgeable but I regret to say that reading it was quite a challenge. I applaud Reuben Hersh and I do not question his great knowledge about math, but I was hoping he would convey his point in a much simpler way. I think the book was too smart for me because it took quite a while and a lot of googling for me to understand the context. Even until now I don’t fully understand what it is he is trying to say. The book is filled with information I wish I could fully grasp. To mathematicians and incredibly smart people, this might be a good read. But to an average college student like me, it is a bit dragging and lacks the ability to capture interest. Though it was a struggle, I have to admit that if it weren’t for this book, I wouldn’t even know about the different philosophies like Platonism, Formalism, Intuitionism and Humanism. So in a way, reading this was not a waste because I was able to gain knowledge I previously did not have.