Antirealism in the philosophy of mathematics

Realism in the philosophy of mathematics is the position that takes mathematics at face value. According to realists, mathematics is the science of mathematical objects (numbers, sets, lines and so on); mathematicians, to use the old metaphor, are discoverers, not inventors. Moreover, just as there may be truths about physical reality which we can never know, so too, realists say, there may be truths about mathematical reality which we can never know.

It is this claim in particular which antirealists find unacceptable. Equating what can be known in mathematics with what can be proved, they insist that only what can be proved is true. (Only what can be proved: different accounts of what this ‘can’ means, facing different difficulties, generate different positions.) This leads antirealists to recoil not only from realism but also from the practice of mathematicians themselves. For the orthodox assumption that every mathematical statement is either true or false would be invalidated, on the antirealist view, by a statement that was neither provable nor disprovable. Not that antirealists themselves can see it in these terms. For if a statement were neither provable nor disprovable, that would itself be an unprovable truth about mathematical reality. Antirealists must learn how to be circumspect even in defence of their own circumspection.