Where do mathematical objects live?
Think too hard about it, and mathematics starts to seem like
a mighty queer business. For example, are new mathematical truths discovered or
invented? Seems like a simple enough question, but for millennia, it has
provided fodder for arguments among mathematicians and philosophers.
Those who espouse discovery note that mathematical statements are true or false regardless of personal beliefs, suggesting that they have some external reality. But this leads to some odd notions. Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it?
On the other hand, if math is invented, then why can’t a mathematician legitimately invent that 2 + 2 = 5?
Many mathematicians simply set nettlesome questions like these aside and get back to the more pleasant business of proving theorems. But still, the questions niggle and nag, and every so often, they rise to attention. Several mathematicians w