What I Learned When I Wrote About Math

First off, I learned that math teachers have no sense of humor. Seems like every math teacher in the country wrote me to explain, often in excruciating detail, the untoward consequences of removing math from the schools. As far as I know, no one (except me, and I was joking) has ever advocated removing math from the curriculum, yet these folks obviously feel endangered by the dark forces of ignorance.

Second, I confirmed my sense that the foundations of mathematics are shrouded in mystery. As far as I can see, there are four basic approaches: platonism, formalism, empiricism, and dissolution.

Platonism is the view that mathematicians discover truths about an independently existing reality. Frege, for example, held this view. So the reference of "seven" in this case would be an independently existing "abstract" object. But the ontology introduced by platonism is a mite, shall we say, florid, and just what the hell this abstract realm is or how we perceive it is wreathed in profound obscurity. Really, you might as well just worship the Great God Yottle.

Formalism is the view that mathematics does not refer to anything, but merely develops the consequences of a set of stipulations, or consists of a structure of notions that have no meaning except in relation to one another. Mathematics, that is, is a purely formal system. In this case, it's hard to see how or why math connects with our real world. In fact this makes math a mere fantasy; math on this view is something like Tolkien's Middle Earth. So math might be a fun sort of game or hallucination, but nothing more.

Empiricism is the claim that mathematical notions are arrived at through experience and refer to elements of experience. This view has undergone a revival, mostly I think because the alternatives seem...ridiculous. Well let me ask you this: if the world were different, would mathematical truths be falsified? If things had a tendency to fission arbitrarily into multiple entities, would 2 plus 2 no longer be 4? Hardly.

Finally, there is the view, popular among my correspondents, that math needs no foundation. Let us just get on with it, they say. And see how useful are results really are? Yes, I see. No math, little technology etc. I'm there, buddy. Of course you've given up the ship, and there is now no particular reason to take mathematics to be "true" or to privilege it as a model of clarity or precision.

So I assert again that mathematics stands without any agreed-on basis, and treats of entities of which none of us have any clear idea. I don't deny for a second that math is useful or indeed indispensable.

All I'm saying is we got no idea what it's about.