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.

do you think that the people who are up there know what theyre doing?

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It’s hard for me to tell. They certainly know way more than I do though, that much is clear…

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This is certainly some real phenomenon because my chemistry teacher told me his similar experiences. He said that after he was done with the advanced chemistry during his postgraduate years (sorry not an expert about these things e.g., studies, syllabus and all), maybe a niece or someone asked him “What is Matter?” and he could not reply. By that time he had lost connect with the ground.

He is also unable to point out his “favorite portion” of chemistry but I am not sure if you too have trouble in pointing out your favorite portion of mathematics!

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I’m always wondering how much of the ground people see – it always seems like everyone understands everything which is of course always very intimidating…

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Do you feel that areas like graph theory are closer to the ground?

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Probably, but I don’t know if that’s saying much.

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