James Franklin has an interesting piece today at Aeon, asking what exactly mathematics is. He looks at Nominalism and Platonism, but discounts both in favor of Aristotelian Realism, which is something I’d not heard of before but seems equivalent to the idea that mathematics is empirical.
What is mathematics about? We know what biology is about; it’s about living things. Or more exactly, the living aspects of living things – the motion of a cat thrown out of a window is a matter for physics, but its physiology is a topic for biology. Oceanography is about oceans; sociology is about human behaviour in the mass, long-term; and so on. When all the sciences and their subject matters are laid out, is there any aspect of reality left over for mathematics to be about? That is the basic question in the philosophy of mathematics.
I’m in the final chapters of Max Tegmark’s ‘Our Mathematical Universe’ and will have a lot of thoughts about this soon. Right now, I’ll just note that I do happen to think we learn mathematics initially through perception (some of which is instinctual) but that many derived mathematical structures built on top of those empirical foundations are non-physical.
The question is are those non-physical concepts “real”, meaning Platonism, or are they simply structures built on an incomplete or simplified understanding of our reality, that is built on quantitative relationships and patterns we have observed, but ignoring or in ignorance of constraints that prevent the physical embodiment of those derived structures?