When is math like a microwave?
When I worked as a research mathematician, I was always flabbergasted by the speed at which other people would seem to absorb mathematical theory. I had then, and pretty much have now, this inability to believe anything that I can’t prove from first principles, or at least from stuff I already feel completely comfortable with. For me, it’s essentially mathematically unethical to use a result I can’t prove or at least understand locally.
I only recently realized that not everyone feels this way. Duh. People often just assemble accepted facts about a field quickly just to explore the landscape and get the feel for something – it makes complete sense to me now that one can do this and it doesn’t seem at all weird. And it explains what I saw happening in grad school really well too.
Most people just use stuff they “know to be true,” without having themselves gone through the proof. After all, things like Deligne’s work on Weil Conjectures or Gabber’s recent work on finiteness of etale cohomology for pseudo-excellent schemes are really fucking hard, and it’s much more efficient to take their results and use them than it is to go through all the details personally.
After all, I use a microwave every day without knowing how it works, right?
I’m not sure I know where I got the feeling that this was an ethical issue. Probably it happened without intentional thought, when I was learning what a proof is in math camp, and I’d perhaps state a result and someone would say, how do you know that? and I’d feel like an asshole unless I could prove it on the spot.
Anyway, enough about me and my confused definition of mathematical ethics – what I now realize is that, as mathematics is developed more and more, it will become increasingly difficult for a graduate student to learn enough and then prove an original result without taking things on faith more and more. The amount of mathematical development in the past 50 years is just frighteningly enormous, especially in certain fields, and it’s just crazy to imagine someone learning all this stuff in 2 or 3 years before working on a thesis problem.
What I’m saying, in other words, is that my ethical standards are almost provably unworkable in modern mathematical research. Which is not to say that, over time, a person in a given field shouldn’t eventually work out all the details to all the things they’re relying on, but it can’t be linear like I forced myself to work.
And there’s a risk, too: namely, that as people start getting used to assuming hard things work, fewer mistakes will be discovered. It’s a slippery slope.