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A Triune Philosophy of Mathematics A Triune Philosophy of Mathematics Dusty Wilson Highline College June 2015 Abstract What is mathematics and is it discovered or invented? The Humanist, Platonist, and Foundationalist each provide answers. But are the options within the philosophy of mathematics so limited? Rather than viewing and describing mathematics in a mutually exclusive manner, each of these approaches includes components of truth from a greater triune philosophy of mathematics. This paper will briefly outline existing philosophies and then introduce an inclusive triune paradigm through which to explore fundamental questions about mathematics. 1 Introduction My parents were hippies who were leery of traditional education. So out of a desire to both protect and also encourage questioning they put me into alter- native public schools. These weren't edgy enough and so they allowed me to homeschool junior high into high school. This put me on a fast track and I began community college during what would have been my junior year of high school. I jumped right into calculus and worked my way through differential equations. After two years I transferred to The Evergreen State College to continue my alternative education with an interdisciplinary liberal arts degree studying political science, literature, and mathematics. With such an eclectic background, I didn't have a clear direction following my bachelor's degree so I went on to graduate school in mathematics thinking, \If this doesn't work out, I can do something else later." While a graduate student I was given the opportunity to teach and my career path suddenly became clear. I was hired by Highline College right out of graduate school where I became the youngest tenured faculty member in College history. While this makes me sound smart, it really means that I had a lot of growing to do as an educator and colleague. But I was in a supportive environment and by my eighth year I was firmly established as a teacher, in service, and in professional growth, and I generally felt that I knew my professional direction. In 2008 I attended a talk by a colleague [2]. The talk itself was on polling and statistics and not related to this paper. However in the midst of the lecture my coworker said, \I think math was invented by people, not discovered." 1 Is math discovered or invented? In all my non-traditional education as well as traditional community college and graduate studies and then continuing into the first eight years of my professional work, I don't once recall having asked myself the question, \Where does math come from?" But while this was the first time these ideas had ever registered in my mind, I have come to realize over the last seven years of study that I had subconsciously adopted a framework for understanding mathematics. As John Synge said, \[E]ach young mathematician who formulates his own philosophy | and all do | should make his decision in full possession of the facts. He should realize that if he follows the pattern of modern mathematics he is heir to a great tradition, but only part heir." [9, pp. 166] This certainly encapsulates my mathematical journey. As I have come to have \full possession of the facts", I've learned that there are three main ways to explain the origin of mathematics. Within each of these broad categories there is a spectrum of nuance. Others have written compelling descriptions of this, but allow me to outline using broad strokes so that I may synthesize the field. The three broad views are as follows. Foundational philosophies: Mathematics is developed from axioms and def- initions using logic Humanistic philosophies of mathematics: Mathematics is invented by hu- mans who are the source of math Mathematical Platonism Also called 'mathematical realism', this view holds that mathematics exists `out there' to be discovered; perhaps owing its existence is to God, but perhaps not While some readers may recognize or be able to articulate their philosophy of mathematics, others may resonate with my story in that I was years into my career as a professional without realizing that I even had a view. I believed mathematics devoid of presupposition without even having the vocabulary to articulate my own presupposition about the field. So as I clarify the basic views available for later synthesis, I encourage you to ask yourself where these views match your training, intuition, and pedagogy. 2 The Foundational Philosophies The first paradigm is that math is logic | this is the basis of the foundational philosophies: intuitionism, logicism, and formalism. If you do research on the philosophy of mathematics, these three views are described over and over again to the point that they nearly define the field. The intuitionists such as Kronecker and Brouwer held that humans create the axioms of logic/mathematics and that we then manipulate these axioms to construct the theorems of mathematics in a constructivist manner. Because it stems from our work, the intuitionism shows existence by demonstrating a for- mula/algorithm/recipe to explain how each entity may be constructed. Because of this, intuitionists rejected proof by contradiction as well as the existence of 2 Foundational Philosophies Intuitionism Logicism Formalism Figure 1: The spectrum of Foundational philosophies Humanistic Philosophies Biology & Brain Language Social Construction Figure 2: A spectrum of humanistic philosophies an actual infinity. For them the source of mathematics was decidedly human. Or as Kronecker famously wrote, \God made the integers, all else is the work of men." [11, pp. 19] The logicists movement was begun by Frege, reached its height with Russell and Whitehead, and concluded with G¨odel. They felt that the axioms of logic were self-evident truths that were known intuitively to the logician. They accepted the rules of logic apriori. In their effort to make solid their foundation, they held that some axioms were self-evident that are not so evident. Certainly the axiom of choice is on this list. Of the foundational camps, logicism was the most fully developed. For the logicist, the source of mathematics was beyond the human experience, self-evident, and discovered (albeit by a select few). The formalists led by Hilbert were perhaps the largest group. They did not concern themselves with the source of the axioms but worked from these using every clever device they could devise. They had no issue with contradiction or infinity. Hilbert referred to math as a meaningless game. [1, pp. 21] The formalist didn't have a strong opinion about where mathematics comes from; after all, it didn't matter anymore than the source of Chess or Monopoly. 3 Mathematical Humanism The second paradigm is mathematical humanism: all mathematics is somehow human in nature/origin. Unlike the foundational philosophies, the subcategories are not as clearly defined. In part this is because mathematical humanism is more current and thus hasn't had as much time to mature. The spectrum within mathematical humanism that I will discuss ranges from a biology-brain model, to language, and ends with social constructivism. Of these, the idea that math is a language is probably the oldest while social constructivism seems most dominant among educators. According to authors Lakoff and Nunez, our ability to perform abstract reasoning is biological. [8, pp. 347] Mathematics is ultimately grounded in ex- perience. [8, pp. 49] It is effective because mathematics is a product of evolution and culture. [8, pp. 378] Mathematics doesn't have an independent existence. It is culture dependent and only exists through grounding metaphors. [8, pp. 3 356, 368] Consequently the philosophy of mathematics is the realm of cognitive science and not the domain of mathematicians. [8, pp. xiii] Where does math- ematics come from? For these philosophers, the source of mathematics is bio- logical and evolutionary and thus serves only an evolutionary purpose. which is to say it has no intended purpose. Perhaps the most commonly held humanistic philosophy of mathematics is summed up in the phrase, \Mathematics is the language of science." This orig- inates with Galileo who wrote: \[The universe] cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly im- possible to understand a single word of it." [3, pp. 4] Today this phrase is most often used outside of university math departments because it defines mathe- matics through its applications and universities produce pure mathematicians (more akin to the formalists of the foundational movement). The basic premise of the view is that mathematics is invented as a way to describe discoveries in the natural world. Math isn't monolithic and unchanging because language changes. The strength of this view is that it seems to explain the perceived transcendence and beauty in math by tying it back to science. Mathematics is something people do according to Reuben Hersh. [5, pp. 30] The philosophy of mathematics is the study of what mathematicians do. [5, pp. xii] The emphasis of social constructivism is on practice. As a practice there has been an evolution of mathematical knowledge. [5, pp. 224] This extends even to including proof itself. [5, pp. 6] As such, mathematics is a social construc- tion. It draws on conventions of language, rules, and agreement in establishing truths. Mathematical knowledge and concepts change through conjectures and refutations. The focus is on creation rather than the justification knowledge. [5, pp. 228] 4 Mathematical Platonism The third paradigm is mathematical Platonism (or mathematical realism) and is loosely based on the Plato's theory of forms and divided line.
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