The Metaphysics of Platonism

This chapter constitutes an attempt to present a modern version of what Platonism in mathematics really entails. Thus one can view it as an attempt to get a closer approximation to Platonic Platonism and not get stuck in Plato’s Platonism, which admittedly has great historical value, but when it comes to its specific details only such. In particular, the presentation takes exception to the post-Plato development of Plato’s Platonism, the so-called Neo-platonism, which is an exalted version of Plato’s thought but shorn off its irony. In particular, it constitutes a literal reading of Plato’s metaphor positing a separate abode for forms and interpreting the immortality of the “soul” in personal terms. Those religious ideas are but a distraction irrelevant to the issue of mathematical Platonism, but unfortunately providing easy fodder for contemporary attacks and dismissals of it. Physics is of course a human invention, but not a frivolous one, but created in response to a physical reality “out there.” Mathematics as practiced is also a human activity on par with physics, and the content of mathematical Platonism is that it is also created in response to a “reality” which is “out there.” While notions of beauty and virtue are ultimately a matter of social convention, knowledge is not, and just as there exists physical knowledge, we can also talk about mathematical knowledge not only as just being tautological. The latter view becomes seductive if you try to anchor mathematics in Popper’s World One through formalism, when it naturally resides in World Three of Popper, whose discovery incidentally Popper attributes to Plato. While beauty in mathematics is on par with beauty in the arts, it is part of a humanistic aspect of mathematics, not its Platonic. To exclusively place mathematics in the humanistic fold invites the abuses of post-modernism, whose classical antecedents are to be found in the teachings of the sophists, tellingly the mortal enemies of Plato.