Personas of Mathematics

          Growing up, we were thought that mathematics is a science. It is and only is a science. From preschool up until we step into university, we are taught that mathematics is nothing more but a science discipline. We were taught all throughout our life that mathematics is a mere science that deals with numbers, formulas and even x’s and y’s. To us, mathematics is a branch of science, nothing more. But this idea is challenged by Hersh in his book, “What is Mathematics, Really?”

           “What is Mathematics, Really?” is a thought provoking book about the philosophy of mathematics. (Singh, S., n.d.) Written by Reuben Hersh, an American academic and mathematician, this book challenges and complements the mainstream philosophy of mathematics. (Wikipedia, 2013)
Long before Reuben Hersh became a mathematician, he encountered a classic text entitled “What is Mathematics?” by Richard Courant and Herbert Robbins. And his dissatisfaction from how he felt cheated after reading the classic text inspired him to write “What is Mathematics, Really?” (Singh, S., n.d.)

          In this philosophical book, Hersh explores what he considers to be the three mainstreams of mathematical philosophy – Platonism, formalism, and intuitionism or commonly known as constructivism. And as an alternative he proposes what he calls “humanism”, the notion that “mathematics must be understood as a human activity, a social phenomenon, part of human culture historically evolved and intelligible only in a social context.” Hersh also talks about some of the standard issues on the philosophy of mathematics such as the existence of finite and infinite mathematical entities, intuition, proof and truth. Then he compares them with the humanism approach to the philosophy of mathematics, implying that it deals with these issues better than the three mainstreams of philosophy. (Dubinsky, E., 2000) Aside from these, Hersh also records the historical background on the philosophy of mathematics, as well as the philosophers attached to it.

          Throughout the book, Hersh considers a number of generally accepted properties, principles and philosophy of mathematics. He also tries to debunk them in the hope to explain why he rejects these three mainstream philosophies: Platonism, formalism and intuitionism, as inadequate to explain the philosophy of mathematics. By contrasting these philosophies with his proposed alternative, the humanism, he implicitly elucidate why his humanism approach is superior. (Dubinsky, E., 2000)
The book is generally divided into two parts. Part one talks about the three mainstream philosophies of mathematics and discusses his ideas on humanism. The Second part tackles about the history of the philosophy of mathematics, dividing the mainstream philosophies of mathematics from the “humanist and mavericks.” (Vila, J., 2007)

          Part one of the book deals with the mainstream philosophies. The three major points of view is in debate about the nature of mathematics. The Platonist saw mathematics as an “almost experimental science, studying objects that really exist though clearly they don’t exist in a physical or material sense.” (Gouvea, F. 1999) Platonism, or realism as it has been called, according to the book, is “the most pervasive philosophy of mathematics.” This believes that mathematical entities “exists outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social.” The book also says that these objects aren’t physical or material. That objects are outside space and time, and that they are immutable and uncreated. One cannot invent anything because they are already there, as mathematicians, we must only try to discover it. (Hersh, R. 1997) The second mainstream philosophy, formalism, argued that “mathematics was really simply the formal manipulation of symbols based on arbitrarily-chosen axioms.” (Gouvea, F. 1999) According to Hersh, formalist philosophy of mathematics simply views mathematics as axioms, definitions and theorems – for short, formula. Some formalist would say that there are rules to derive one formula from another but these formulas are not about anything. They are mere strings of meaningless symbols. The formalist philosophy of mathematics can be condensed to this: “Mathematics is a meaningless game.” (Hersh, R. 1997) And the intuitionistic philosophy of mathematics says that mathematics is a human creation, therefore like all human creation, is essentially finite.  (Gouvea, F. 1999)

          However, Hersh attacks these three dogmas by presenting his alternative, humanism. He want to involve the readers as to the debate on the nature of mathematics. He does this by doing a number of things. First, he argues at most of the writings on the foundation of mathematics is ignorant of actual mathematical practice. Then he tries to challenge the dogmas by proposing a new concept as to what mathematics really is. (Gouvea, F. 1999) According to Hersh, “mathematics is a social-historical-cultural phenomenon, without the need of anything abstract-Platonic-nonhuman, without the need of formalist or intuitionist reduction.” His view implies that mathematical objects are just like other objects; they are also processes. “They change, whether in plain sight or too slowly to be noticed.” (Hersh, R. 1997)

         In the part two of the book, Hersh strengthens his claims by running through the history of the philosophy of mathematics, arguing that “(a) his position is not really new, but has a distinguished pedigree, and (b) that all the other positions are clearly wrong.” And in this part, he “connects philosophical positions on the nature of mathematics to broader philosophical and political issues.” (Gouvea, F. 1999) In order to do this, Hersh had included the “historical account of the mainstream philosophy – ranging from Pythagoras, Descartes, and Spinoza, to Bertrand Russell, David Hilbert, and Rudolph Carnap--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, and Lakatos.” (goodreads.com, 2013)

          In this book, Hersh challenges the idea that mathematics is only science, that it is limited to being only a part of science. And as readers, we would meet a new persona of mathematics when we read this though provoking book. Here, mathematics lends itself into two different personas, all of which, I have not met before. Personas that, when understood, would bring us closer into understanding what mathematics really is. Two personas that would unravel the mystery behind the identity it holds. Two personas that would widen our view on what mathematics is. Two personas that will not limit mathematics to being a science rather it will view mathematics to have meaning through its philosophy and to have social context and experience change through its being a human activity.

          Mathematics has meaning through its philosophy. Philosophy, according to Merriam Webster dictionary means “the study of ideas about knowledge, truth, the nature and meaning of life; a particular set of ideas about knowledge, truth, nature and meaning of life; a set of ideas about how to do something or how to live.” (Merriam-Webster.com, 2013) The philosophy of mathematics basically means that not only is mathematics limited to numbers, shapes or addition, but it has a theoretical basis, it has fundamental nature and knowledge that makes mathematics, mathematics. That it has deeper thoughts, principles and nature rather than the number that lies on its surface. It is not enough anymore to just say that mathematics is just a study of number because it in fact has a philosophy attached to it. There is much more to mathematics that just number, for once, through this book, we can encounter meaning through mathematics because of the fundamental nature attached to it. Meaning. It is something that I have never affiliated with mathematics in any way. But it does. Mathematics has meaning because there is philosophy of mathematics. Mathematics matter not only because it is a science discipline or it contains number, but it matters too because it has meaning through the fundamental philosophy attached to it, through the interwoven nature of knowledge, reality, and existence.

          Mathematics has social context and experience change through its being a human activity. Can mathematics be part of the humanities? This question has been daunting me since I took my language and culture class. Mathematics is a science, but can it be part of humanities? And this was answered by the book. Hersh says that mathematics is a as a “human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context.” (goodreads.com, 2013) Therefore, since it is part of our daily life, since it is part of our culture, it has then become part of us, by saying that, then we could in fact view it in the light of the humanities. Therefore since we can view it as a human activity, as part of our culture, it only means that it too has a social context. A social context that varies within the different culture and different society. And this social context implies that mathematics too involves a process. But aside from this, the most important trait of a social context is that it is dynamic, therefore mathematics is dynamic, and it is susceptible to change. What change do I mean? Well, it is evident in the history of both mathematics and the philosophy of mathematics that there are changes that happened and that is happening to mathematics. And this change will continue as mathematics continue to strive in this world. As long as mathematics live, change upon mathematics is inevitable.

          Growing up, we were thought to look at mathematics only in a scientific perspective, but now through books, films and researches, we can now slowly get acquainted to the many personas and faces that mathematics contains. Through this book, Hersh allows us to see mathematics in a different light. Through this book, we see that mathematics has meaning through its philosophy. We also see that mathematics is dynamic because it has social context because it is a human activity. We may now know new personas of mathematics but there are yet so many more to discover. It is our duty to unravel those personas for it is our obligation to reveal the mystery that revolves around mathematics.



References:

Singh, S. (n.d.). What is Mathematics Really?. Simon Singh. Retrieved December 11, 2013 from http://simonsingh.net/books/recommended-books/mathematics-books/what-is-mathematics-really/

What Is Mathematics, Really?. (2013). In goodreads. Retrieved December 11, 2013 from http://www.goodreads.com/book/show/750042.What_Is_Mathematics_Really_

Philosphy. (2013). In Merriam-Webster. Retrieved December 11, 2013 http://www.merriam-webster.com/dictionary/philosophy

Hersh, R. (1997, August 21). What is Mathematics, Really? [PDF File]. Oxford University Press. Retrieved December 11, 2013 from https://www.dropbox.com/sh/qwobqaw4bcy58v7/epqUsLqjUv/2013-2014-2/book%20review/Reuben%20Hersh%20What%20Is%20Mathematics%2C%20Really%20%201997.pdf

Gouvea, F. (1999, February 18). Reviewed by Fernando Gouvea, on 02/18/1999 [Review of the book What is Mathematics, Really?]. Mathematical Association of America. Retrieved December 11, 2013 from http://www.maa.org/publications/maa-reviews/what-is-mathematics-really

Dubinsky, E. (2000, November 18). What is Mathematics, Really, by Reuben Hersh [PDF File]. Retrieved December 11, 2013 from http://www.math.kent.edu/~edd/HershReview.pdf

Vila, J. (2007, January 14). An alternative philosophy of Mathematics [Review of the book What is Mathematics, Really?]. Amazon. Retrieved December 11, 2013 from http://www.amazon.com/What-Mathematics-Really-Reuben-Hersh/product-reviews/0195130871

Reuben Hersh. (2013, March 31). In Wikipedia. Retrieved December 11, 2013 from http://en.wikipedia.org/wiki/Reuben_Hersh