Philosophies of Mathematics
Many of the epiphanies in my story that were analyzed raise a different wording of my research question, namely, what is mathematics and consequently, how should it be taught and learned? Even the disparity between dialogues amongst my story and the two w
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Figure 1. Philosophies of mathematics, embodied mathematics, mathematical enculturation
Many of the epiphanies in my story that were analyzed raise a different wording of my research question, namely, what is mathematics and consequently, how should it be taught and learned? Even the disparity between dialogues amongst my story and the two worldviews within the analysis of my story present two very different conceptions of what mathematics is (could be) and how it should (might) be taught and learned. It is this theme emerging from my story that I now explore in more depth, by turning to the literature that speaks to this very question: the philosophies of mathematics. Philosophies of mathematics are concerned with the nature of mathematical knowledge, and thus also consider (albeit sometimes implicitly) the two questions (among others) of “what is mathematics” and “what is knowledge?” The varying answers to these two questions ultimately result in a variety of different philosophies of mathematics. It is also in the answers to these questions that proponents of alternate philosophies frequently find loopholes and paradoxes that, at least for them, render other philosophies discussed here ineffectual, obsolete, or false. In the sections
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that follow, I will be explaining each of the many philosophies of mathematics that have been written about. During these explanations, I will endeavour to avoid the criticisms made by champions of one philosophy or another, but during the analysis of the philosophies I have no doubt that my comments will resonate with my own personal biases based upon my evolving stance with respect to a worldview. It is my belief that these biases will be evident to the reader and that my arguments will not be rendered irrelevant because of my openness in disposition. For interested readers, the specific arguments between and against the various philosophies of mathematics can be found, and quite elegantly in this regard, within the work of many other authors and researchers (e.g., Ernest, 1991; Hersh, 1997; Lakatos, 1978). In his discussion of the philosophies of mathematics, Ernest (1991) categorizes the differing (past and present) philosophies of mathematics into two camps: the absolutists and the fallibilists. The difference between these two camps lies in their response to the notion of mathematical truth. For the absolutists, mathematics is “a body of infallible and objective truth, far removed from the values of humanity” (p. xi); it is seen as “the one and perhaps the only realm of certain, unquestionable and objective knowledge” (p. 3). For the fallibilists however, “mathematical truth is corrigible, and can never be regarded as being above revision and correction” (p. 3). Lakatos (1978) similarly categorizes the philosophies of mathematics, but uses the alternate names of Euclideans and quasi-empiricists for absolutists and fallibilists, respectively. Hersh (1997), alternatively, uses Kitcher and Aspray’s (1988) classification system for categorizin
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