Imagine the following dialog between a mathematician and a philosopher. Both are actually me since I’m making the whole thing up, but play along because there are a few good points to be had, and I end with a dirty little secret. The mathematician goes first.

Mathematics is about the construction of truth. You start with a set of axioms, which are claims that are so basic that you can’t imagine them being false. Then you combine the axioms with each other in such a way so as to build further truths out of them, until you have a system of thought that is incontrovertible.

The philosopher smells blood and replies like so.

Yes, but everyone thinks they know the truth. The concept of “truth” is problematic and must be unpacked, because

atruth is alwayssomeone’struth. These combinations of axioms are only incontrovertible because a bunch of mathematicians agree that they are, and those mathematicians’ personal agendas, conscious or not, are an inextricable part of the whole business.

There are many ways the mathematician can respond. Some responses are good and some are not so good. Here are examples of each.

Good | Not So Good |
---|---|

Maybe you’re right. Mathematics is a human endeavor, and mathematical truth is ultimately reckoned by consensus. But the procedure for combing the axioms is designed to be as clear as possible, so that mathematicians’ biases have fewer places to hide. Mathematics unpacks itself. |
What you’re failing to see is that what one does in mathematics is proceed from axioms step by step so as to produce theorems that are incontrovertibly true. For example, you can prove the binomial theorem by assuming that its truth for any integer exponent n implies truth for n+1, then proceed by induction from the ground case of n=0. Many really smart people have seen how to do this since days of Euclid in ancient Greece. If you want to know more you should take some classes so that you can understand how it all works. |

I’m exaggerating for the sake of exposition, but nevertheless these two responses serve as illuminating examples and are not so far removed from conversations I’ve actually had.

The good response addresses the objection. In it the mathematician says to the philosopher, “Yes, and…” The not so good response merely restates the original claim as if the philosopher was hard of hearing. The good response actively seeks a point of contact (“Maybe you’re right…”) while the not so good response hunkers down inside a protective shell of assertion. The good response makes an effort to adopt foreign technical jargon (“unpacks”) on the spot to illustrate concepts within its domain. The not so good response whips out its own native jargon (“binomial theorem”, “exponent”, “n+1”) for use as a bludgeon. The not so good response makes a few Appeals to Authority in passing (“Euclid”, “ancient Greece”, “really smart”) before implying that the philosopher is unqualified (“take some classes”) to make criticisms. This last part is the most egregious, not because it’s condescending–we’re all big boys, we can take it–but because it forecloses on the possibility of dialog. If I have to become you before I am allowed to question you then I’ll just never question you because honestly who has the time?

If you have enough like-minded compatriots in your field, you can happily spend your entire life in the right-hand column. The only thing you sacrifice is the range of people you can have conversations with. If you don’t care, you don’t care. If you do care, however, be prepared to put some serious effort into meeting others halfway, because that is hard work in itself. And that hard work often has no immediate payoff. You forego the heady finishing-each-other’s-sentences excitement of being totally on the same page with someone else. Your employer/tenure committee/klatch of fellow artists may not understand why you insist on dumbing things down, and may come to suspect that it’s because you yourself are dumb. It also won’t win you any arguments. I guarantee that regardless of the response, the philosopher and the mathematician in this example will both walk away convinced the other guy is wrong. So why bother? Because all this high-flown big-brain talk is just another aspect of life, and life works better when you open yourself up to other people. Specifically in this arena, the more people you are able to have reasonable conversations with, the greater chance that one of them will plant a seed of doubt that one day blooms in a way you could have never imagined. Just because you can’t win doesn’t mean you don’t want to play.

The dirty little secret: in my experience, the roles of mathematician and philosopher are usually reversed.