Pluralism and “bad” mathematical theories: Challenging our prejudices

Abstract

Here, pluralism is introduced as a new and independent philosophy of mathematics in its own right. One of the marks of independence from more main-stream philosophical positions is that pluralists take seriously “bad” mathematics. Under “bad mathematics” I include: inconsistent theories, trivial theories, not yet completed theories and intensional theories. Bad mathematical theories are ignored by main-steam philosophers of mathematics because they take it as read that the philosopher’s task is to give a philosophy of successful mathematics. In contrast, the pluralist contends that bad mathematical theories are as much a part of “mathematics” as are the successful parts. Moreover, they are philosophically important. Who is this pluralist? A pluralist in the philosophy of mathematics is someone who places pluralism as the chief virtue in her philosophy of mathematics. She brings the attitude to bear on: conflicting mathematical theories, including different foundations of mathematics, on different philosophies of mathematics and uses (what are usually dismissed as) “bad mathematical theories” to inform her philosophy.