Have you ever sat in a mathematics lecture and thought to yourself, “Wow, that is some real cowboy mathematics?”
I have. Not just once or twice either: this curious pastiche has several times jumped from the deep recesses of my mind to surprise me. So what in math counts as cowboy? Certainly not subjects that have been a historical favorite among the more intellectually inclined cattlemen. Even though I’m not an expert in these matters, I’m suspicious that there is any sort of tradition of mathematics there. I mean, maybe the more eclectic personalities like Doc Holiday dabbled with some maths from time to time, but it couldn’t have been a common thing.
No, for me cowboy refers to a particular style of mathematics. It’s the sort of style you’d imagine Clint Eastwood bringing to a lecture: “Notherian hypotheses? Of course, if it gets the job done more easily. Is the definition of the Gauss-Manin connection less technical in the complex analytic setting than the algebro-geometric? Then use the complex setting. But you want to work with varieties over other fields? Well, work the details out later — I’m too busy proving theorems at the moment.”
Cowboy mathematics is characterized by an openness or, more precisely, an eagerness to avoid excessive generality or technicality. Eagerness is important here. Many lecturers wisely sweep some details under the rug and inform the audience of technical omissions only in passing. This is sage lecturing, not cowboy mathematics. No, a cowboy is happy to run headlong through a lecture without pausing to point out the finer details, or more general constructions, or what happens in characteristic two. In such a lecture technical omissions tend only to come out via audience questioning. The theorems snared along the way justify this approach and, for a true cowboy, the results should be hearty nuggets that one would gladly submit to top journals.
This style isn’t confined to the lecture room, of course. In my mind Deligne’s proof of the Weil Conjectures is a famous example of cowboy mathematics in the literature. Rather than wait for Grothendieck’s school to complete their exposition of etale cohomology, for the sea to rise so to speak, once the peak was visible Deligne forged ahead and completed the long ascent on his own. If a real cowboy saw a wild stallion in a field up ahead, do you think he’d wait for his buddies to come help him lasso it? No, he’d go ahead and do it himself. And so did Deligne.
Are you unsure if a particular mathematician is a cowboy? Here’s a test that might help you decide: sit in on one of their lectures and bring up the axiom of choice. If this precipitates a weighty discussion, the speaker is definitely not a cowboy. If the speaker brushes the question aside, it’s inconclusive. But if they laugh in your face about it, they’re definitely a cowboy.
Cowboys just don’t care about the axiom of choice.