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? . . . Plato is the standard-bearer for the believers in discovery. The Platonic notion is that mathematics is the imperturbable structure that underlies the very architecture of the universe. By following the internal logic of mathematics, a mathematician discovers timeless truths independent of human observation and free of the transient nature of physical reality. . . . But if the mathematical ideas are out there, waiting to be found, then somehow a purely abstract notion has to have existence even when no human being has ever conceived of it. Because of this, Mazur describes the Platonic view as “a full-fledged theistic position.” It doesn’t require a God in any traditional sense, but it does require “structures of pure idea and pure being,” he says. Defending such a position requires “abandoning the arsenal of rationality and relying on the resources of the prophets.” . . .
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