Is mathematics natural?

While perusing the Uncommon Descent blog, I noticed a post

• Mathematics challenges naturalism, says math prof

This seemed a strange thing to say.  It perhaps even involves what Gilbert Ryle would have called “a category mistake”.  Browsing through that post, I saw that it referred to an article in Aeon magazine

• Some philosophers think mathematics exists in a mysterious other realm. They’re wrong. Look around: you can see it (James Franklin)

Clearly, professor Franklin and I have very different ways of looking at mathematics.  And that’s what I will be discussing in this post.

Mathematics and naturalism

Let’s start with that reference to “naturalism”.  Franklin adopts a philosophy that he describes as Aristotelian realism.  And it is in relation to that philosophy, that he makes his comment about naturalism.

Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience.

I’ve never fully understood what people mean by “naturalism”, nor have I understood the arguments about it that seem to be part of the culture wars.  The idea that the world and human knowledge can be explained in terms of physics, biology and neuroscience seems to me a non-starter — again, I see it as involving something like a category mistake.