Was a PhD necessary to solve outstanding math problems?

by AllAmericanBreakfast3 min read10th Jul 202024 comments

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Logic & Mathematics EducationRationalityWorld Modeling
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My composition teacher in college told me that in some pottery schools, the teacher holds up your pot, examines it, comments on it, and then smashes it on the floor. They do this for your first 100 pots.

In that spirit, this post's epistemic status is SMASH THIS POT.

Previous post: Was a terminal degree ~necessary for inventing Boyle's desiderata?

This is my second post investigating whether a terminal degree is practically ~necessary for groundbreaking scientific work of the 20th century.

Mathematics seems like a great field for outsiders to accomplish groundbreaking work. In contrast to other fields, many of its open problems can be precisely articulated well in advance. It requires no expensive equipment beyond computing power, and a proof is a proof is a proof.

Unlike awards like the Nobel Prize or Fields Medal, and unlike grants, a simple list of open problems established in advance seems immune to credentialism. It's a form of pre-registration of what problems are considered important. Wikipedia has a list of 81 open problems solved since 1995. ~146 mathematicians were involved in solving them (note: I didn't check for different people with the same last name). I'm going to randomly choose 30 mathematicians, and determine whether they got a PhD on or prior to the year of their discovery.

The categories will be No PhD, Partial PhD, PhD, evaluated in the year they solved the problem. In my Boyle's desiderata post, 2/15 (13%) of the inventors had no PhD. I'd expect mathematics to exceed that percentage.

Results:

Robert Connelly: PhD Anand Natarajan: PhD Mattman: PhD Croot: PhD Mineyev: PhD Taylor: PhD Antoine Song: Partial PhD Vladimir Voevodsky: PhD Ngô Bảo Châu: PhD Haas: PhD Andreas Rosenschon: PhD Paul Seymour: PhD (D. Phil) Oliver Kullmann: PhD Shestakov: PhD Merel: PhD Lu: PhD Knight: PhD Grigori Perelman: PhD Haiman: PhD Ken Ono: PhD Ben J. Green: PhD Demaine: PhD Jacob Lurie: PhD Harada: PhD McIntosh: PhD Naber: PhD Adam Parusinski: PhD Atiyah: PhD Benny Sudakov: PhD John F. R. Duncan: PhD

Contrary to my expectation, all of these mathematicians had a PhD except Antoine Song, the only partial PhD. He finished his PhD the year after his work on Yau's conjecture.

So either:

a) This list is not in fact an unedited list of important mathematical conjectures and who solved them, but instead a list retroactively edited by Wikipedia editors to select for the the credentials of the discovers, or 

b) A PhD is an almost universal precursor to groundbreaking mathematical work.

Discussion:

First, the bad news. It's a problem that I have no way to verify that the list I used was not cherry-picked for problems solved by PhDs. The suspicious may want to look for a list of open mathematical problems published in a definitive form prior to 1995 and repeat this analysis.

My model for why a PhD would be necessary to achieve groundbreaking work is:

These degrees come with credibility; access to expensive equipment, funding, and data; access to mentors and collaborators. A smart person who sets out to do groundbreaking STEM work will have a much lower chance of success if they don't acquire an MD/PhD. Massive, sustained social coordination is ~necessary to do groundbreaking research, and the MD/PhD pipeline is a core feature of how we do that. Without that degree, grant writers won't make grants. Collaborators won't want to invest in the relationship. It will be extremely difficult to convince anybody to let someone without a terminal degree run a research program.

Authoritativeness of the proof, access to expensive equipment, and access to data don't seem to be very much at play in mathematical discoveries.

Perhaps the reason these mathematicians enrolled in their PhD is that the academic environment is both conventional and attractive for genius mathematicians, even though it's not actually necessary for them to do their work. My guess is that funding, the sense of security that comes with earning credentials allows risk-taking, and access to long-term collaborators and mentors also play an important role.

37 of the discoveries (46%) are credited to a single mathematician, giving some perspective on the extent to which access to collaborators is important.

How did the two inventors of Boyle's desiderata who didn't hold a terminal degree manage to do their work without a PhD? The fact that they both worked in the field of robotics seems relevant.

Maybe the story is something like this:

Earning a PhD is both attractive and helpful for people doing basic research in established fields.

A PhD is less important for doing groundbreaking applied engineering and entrepreneurial work, especially in tech.

It's hard to overstate the extent to which business contributes to academic work. How many mathematical, biological, and physical discoveries would never have been made, if it weren't for robotics (invented by someone with no higher education) and cheap compute (provided by the business sector)? How much has economic growth expanded our society's capacity to fund academic research?

Conclusion:

Let's think about the situation of a STEM student with lots of potential, but no money and few accomplishments.

If they do a PhD, they'll get enough money to live on, and some time and mentorship to try and prove their intellectual leadership abilities. Coming out of it, they'll have a terminal degree, which will give them the option of continuing in academia if they like it, or leaving for industry if they don't.

If they go straight into industry without a PhD, they might earn more money early on. But they'll also have to work their way up from near the bottom, unless they can join a small startup early on. They might get caught in the immoral maze of some gigantic corporation. They won't have the same leeway a PhD student has to choose their own project. And they likely won't have the same long-term earning potential.

From that point of view, the PhD concept itself doesn't seem like empty credentialism. Instead, it's a mechanism for sifting through the many bright young people our society produces, giving a certain percentage of them a boost toward intellectual leadership and a chance to take a crack at a basic research problem. It's also a form of diversification, a societal hedge against an overly short-term, profit-oriented, commons-neglecting capitalist approach to R&D.

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