Everything I need is on the ground

For me the biggest difference between undergraduate math and PhD life has been something I’ve never seen anyone else talk about: it’s the feeling like I could no longer see the ground.

To explain what this means, imagine that mathematics is this wide tower, where you start with certain axioms as a foundation, and then you build upwards on it. At first learning math felt like slowly climbing up this tower. When I reached a landmark, it felt like I was on the balcony of the 30th or 50th or 100th floor, enjoying the view, with an appreciation of the floors I had ascended to get here.

In theory, proofs in math can be formalized as a long sequence of logical steps from the axioms that could be computer-verified. This turns out to way too cumbersome to actually do in practice given the current state of technology (though this is changing), but I was at least satisfied that all the results I had seen could in principle be compiled to a formal proof.

As soon as I started doing my PhD, this feeling of internal consistency and safety completely vanished. It felt like someone had put me a rocket and boosted me to the 100,000th floor. I had no vision of the ground or any of the floors below me. I could actually be on Neptune for all I know.

It was scary enough having “black boxes” (quoting a theorem without having gone through the proof yourself). At this point, I don’t even know the definitions of half the objects that I’m playing with. It’s like, I’m trying to prove a result about an irreducible tempered cuspidal automorphic representation , except I don’t know what any of the five words before means. So I just take someone’s word for it that the only thing I need for this calculation are the Satake parameters attached to , except I don’t know what a Satake parameter is either, but at least I know it’s a complex number, so yay?

I will tell you a sobering story about my 2016 paper arXiv:1608.04146. I was really nervous when writing it because the arithmetic geometry involved was well above my pay-grade, and even asked a more knowledgeable colleague to sanity-check it before submission. When it reached the journal, one referee said it looked good modulo six minor typos (like “we map pick” to “we may pick”); the second referee never replied.

Then just as the paper was about to be published, the second referee suddenly replied with a document saying the paper “need[s] a deep revision to fix the proof of the main result and to improve the exposition which up to now is not accurate”, followed by a 3-page list of ambiguities and errors. My original paper was only 10 pages! I was extremely grateful to the second referee. (And no slight to the first referee; math is hard, and the point of peer review is sanity-checking, not deep verification.)

I guess I’m scared of heights. I wonder if I’m supposed to just get used to it.