The Best Writing on Mathematics 2012 by Pitici Mircea; Mumford David;

Author:Pitici, Mircea; Mumford, David;
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2012-08-15T00:00:00+00:00

Acknowledgment

The author was supported in part by NSF Grant DMS-0905087.

How Your Philosophy of Mathematics Impacts Your Teaching

BONNIE GOLD

“My philosophy of mathematics? I don’t have one! I’m a mathematician, not a philosopher. I leave philosophical questions to the philosophers.” Maybe. Or perhaps you are among those mathematicians who are interested in the philosophy of mathematics. Whatever your attitude toward the philosophy of mathematics, when you teach mathematics, you do in fact take, and teach your students, positions on philosophical issues concerning mathematics. If you do not think about them, then you probably acquired your positions from your teachers when they imposed them on you without discussion. Furthermore, you may find, if you do examine the positions you are taking, that they contradict each other or disagree with positions you would say are obviously true.

Many mathematicians’ distaste for philosophy of mathematics comes from one of two sources. One is our preference for questions that, if worked on seriously, eventually receive definitive answers, as opposed to questions (as in philosophy) that, at best, clarify what the issues and alternatives are. The other is the detour the philosophy of mathematics took into foundational issues starting in the second half of the nineteenth century. This detour lasted until about 1975, when work on questions beyond foundations resumed. The detour led to a substantial growth in the area of mathematical logic, and some consensus on what was not going to be solved. (For example, we are not going to be able to show in a finitistic manner that mathematics is consistent.) However, most mainstream mathematicians lost interest after Gödel’s work.

I use a rather broad definition of philosophy in this article: It includes anything involving our attitudes toward mathematics. I am not trying here to convince you to start working in the philosophy of mathematics. But I do hope to make you aware of a range of issues that we end up taking a stand on when we teach a relatively standard calculus or introduction to proof course. Many of these issues are closely related to issues that beginning students are confused by. Perhaps a lack of clarity on philosophical matters adds to students’ confusion.

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