Abstract But Still Well Read

         

        Believing that mathematics is abstract and concrete at the same time is like words on this text. Symbols of curves and lines are the solid evidence of the words’ existence, seen jumping of the screen and into our thoughts. But the intangible content of their representation, the meanings that they convey, is what gives the words importance in our logical minds. Just as they are subjective to every individual, the philosophy of mathematics had undergone a series of criticisms and contemplations in the minds of many intellectual beings. Even until today the average human may, at one point in time, wonder about the existence or purpose of mathematics. One man, who went by the name of Reuben Hersch, gave out some perspectives and criticisms on what he regarded as mainstream viewpoints of mathematics in a bid to satisfy the question that had reverberated debates throughout time-what is mathematics, really?

        I for one had no interest in mathematical philosophy. It would usually take a big portion out of my time to comprehend such deep matters, let alone the study of philosophy itself. Yet Reuben Hersch’s book, after some painstaking effort of understanding some of its contents, seemed to have irked my inquisitiveness on what he wants to convey. And so, with a sigh of feigned resentment masking my curiosity, I ventured deeper into the pages.

        Hersch was angry. No, more likely frustrated at the philosophies that other thinkers had presented on their ideals of mathematics. He stressed three viewpoints that he had seen quite absurd and unfit for modern math-Platonism, formalism, and intuitionism. Platonism, which was defined as “mathematical entities only exist outside space and time…an abstract realm independent of any consciousness, individual or social,” had illogical agreement with the sciences. He argued that it is unsound for an individual to accept the “entity” without firsthand experience of its presence. It did not associate to “material reality” and failed to explain its interactions with the abstract part of mathematics. I quite agree with his point since, on my opinion, mathematical entities are the brainchildren of intellectual thoughts and contemplations, so therefore it’s hard to visualize the separation of both realms.

        Next in line to Hersch’s rubbish box is Formalism. This viewpoint stated that the rules on making the rules are just random. As what I have seen from The Story of Maths documentaries, each concept in the field of mathematics, ranging from number systems to principles, was derived by systematical thinking rather than through personal whims of the contributors. The author pointed out that the rules were determined based on the agreements in society over time and therefore progressed from one rule to another depending on the logical explanations that prove them valid. Therefore, he rejects the concept of formalism because, to put plainly, it was just so wrong. So invalid and irrational. For the sake of breaking the seriousness over this matter, imagine Hersch sitting in his study room, shaking his head at least once or twice upon every turn of the page on a book that has the [fictitious] title, “Formalism in mathematics.” I might have pitied his grueling dilemma over the matter, but then I would have prevented the development of this book and I would not be writing a review on this matter. Well, back to reality, shall we?

        Lastly is Intuitionism. It stated that the creation of natural numbers is “obtained through a process of finite construction that does not make use of the law of the excluded middle.” I did not get much about what this statement wanted to imply, but I understood that, based on Hersch’s opposition, natural numbers are created through the individual’s perception of abstract construction in numbers during their early lives. With the conclusion of his opposition on these philosophies, he proposed an alternative viewpoint to counter the “mainstream mathematics”- the Humanist viewpoint.

        Hersch wrote: “There’s no need to look for a hidden meaning or definition of mathematics beyond its social-historic-cultural meaning.” I should give him a thumbs-up for this kind of philosophy. It’s evident, after hundreds of years of innovations and discoveries in the field of mathematics, that the science of numbers has become a part of our history and culture. It’s not just secluded to the intellectual scholars who had firmly grasped the ways and philosophies of mathematics, but it was intertwined with the whole biosphere that we and other living organisms resided from the very start. It just so happened that we, the humans, discovered it at one point in history. That’s it.

        As I was typing this review, I got into a dreamlike state of contemplation. What is mathematics, really? I never thought that I’d delve into such an interesting yet nerve-wrecking journey in this field of the academe. Mathematics was, and still is a very difficult subject for me. I had my ups and downs from this subject. Mostly the “downs” part as expected. But just like they say, in the bleak moments of your life you’ll find the answers that you never expected to get from your happy ones. While typing these words, answers and realizations suddenly sprang up from nowhere and embedded themselves into my mind, and I hope that they’ll retain. As I progressed through the book and skimmed over Reuben Hersch’s arguments and dilemmas, I myself had made a conclusion, an unbiased generalization about what the study of numbers and patterns are. What is mathematics, really? Here is my answer: It's the brainchild of humanity’s intellectual prowess and ingenuity as planned by the Almighty Creator.