Was a PhD necessary to solve outstanding math problems?
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.
I don’t think a PhD is necessary for ground breaking math. A more plausible explanation (or so I think) is that academia is a preferable work environment, compared to being by yourself. Even for an introvert, being part of academia will be more convenient. Therefore, everyone who want to do math research will try to find a job in academia, and everyone who is smart/competent enough to do groundbreaking research is also more than smart/competent enough to get a PhD.
I have to say that I also expected some of the work to be done by non-PhDs. But given the result I think that the correlation has at least as much to do with common cause, as with causality from PhD → research.
On the other hand, it could be the other way around? Did you check if they got their PhD before or after that result. If you do a ground breaking research, you can just write it up as a thesis and get a PhD.
I agree that a math PhD is probably mostly for the sake of convenience and companionship and mentorship.
Some of the math discoveries seem to have been the PhD work. Others were produced many years after the discoverer completed their PhD. It’s a mix.
I did this research as much to find out whether I should see a PhD as attractive, as much as whether it’s necessary. I hear lots of people bemoaning their PhD or criticizing the system as a gatekeeping tool. My conclusion is that yes, the PhD system is gatekeeping and yes, it is hard, but that’s because producing new original academic knowledge is hard and the system far from perfect at identifying likely candidates. It’s risky, and many fail, and failure sucks and generates complaints.
The successful ones just continue their work and don’t bother to air their opinions on the system that they’re a part of.
There is also the fact that there are much fewer academic post-doc jobs compared to PhD position. This is probably different in different fields, but my math friend says this is defiantly the case in math. Sure the more successful are more likely to get the next job, but it is more about relative success compared to your competition, than absolute success. I don’t know if the bar to keep going happens to be reasonable in absolute terms.
The way I view a PhD is that it is an entry level research job. If you want to have a research career, you start with an entry level research job, more or less similar to other career path.
I wonder, if you want to do maths research, and don’t do a PhD, what is the alternative? The best thing about a PhD is that you get paid to do research, which is very uncommon every where else, unless you do something very applied.
Do you know of any reasonable alternatives to working in academia for less applied research? Or maybe this is what you mean by gate-keeping, that academia has monopolised funding?
Yeah, I meat “allocating” rather than “gatekeeping.”
I started this project because people complain so much about the PhD system. It makes me think that for the right person, it’s an attractive way to start a research career, but too many PhD students go in treating it like a credential. A way to make more money and impress themselves or other people. A professional degree that guarantees a good job. The only way to contribute to human progress or be a leader.
A PhD is an opportunity to do focused, original research. People should only choose that path if that’s what they really want. It’s underpaid, risky, stressful, and takes a loooong time.
I completely agree. Doing a PhD for credentials is not a good strategy. Doing a PhD for money makes no sense what so ever.
There have been undergrad and grad students who had solved an open math problem before they got their PhD, but for them getting a PhD was not even a question they consider, it’s just something that naturally happens. It’s not about credentialism, it’s about being smart enough, creative enough and hard-working enough to outdo the rest of the very crowded field in a particular area. If you are all that, writing up a PhD thesis is a minor step. And if you are not all that, why pick a field like math to begin with?
Just to TL:DR my comment above: to get a PhD is many times easier than to “accomplish groundbreaking work”, so if the former is an issue, you will never do the latter.
Shminux, I want to disagree with this statement. I guess it all depends on what you mean by “groundbreaking work”. First, it seems to me that most of the Ph.D. dissertations that I know of entailed discovering new theorems. Second, in my math department, it took 7 years on average to get a Ph.D. It often takes a lot less time to discover and write up a good, popular paper (say one with over 20 citations).
If you don’t consider a paper “groundbreaking” unless it has over 100 citations, then maybe you are correct. I just don’t know.
I was using the word “groundbreaking” in the sense of making an unusually significant advance in the field or solving an unusually difficult problem.
I have to admit I know only undergraduate calculus, so I made an assumption that the list I based this off of described such accomplishments.
If not, it still would show that most academically interesting math problems are solved by PhDs, and that people capable of doing the work tend to find the PhD an attractive/helpful way to get there.
The kind of people who are interested in the problems of core math likely do seek a math PhD.
I would expect to the extend that people do groundbreaking work in math without an PhD they are more likely to do it for applied math.
It’s really quite simple. Doing mathematical work at the highest levels requires both extraordinary talent and single-minded focus on mathematics. Mathematics is to some degree a younger man’s game, yet modern mathematics requires knowing vast amounts of previous work to have any shot at solving a serious conjecture.
Doing a PhD is the most straightforward path to acquiring the knowledge to make a serious attack. Most fields will require upwards of a decade to build up enough technical experience to attempt to solve famous open problems.
Spending say 5 years acquiring money to do independent work doesn’t seem like a good plan, when the people who have the talent required to solve one of these problems are almost guaranteed to obtain an academic position.
It’s interesting to consider to what extent mathematics is different from other fields. Perhaps groundbreaking biological research also requires a PhD, but for different reasons.
The “young man’s game” conjecture posits that math is a race to fill your brain with knowledge before it expires. Perhaps the lack of empirical constraints means that many more of the fruits on the tree of mathematical knowledge have been picked. Sheer individual energy, stamina and intellectual ability is all that matters. Getting accepted to a math PhD mainly buys you time to do focused work during your youth.
Other fields have more intellectually-low-hanging fruit.
One possible reason is that the data takes so long to gather that sheer intellectual ability matters less than opportunity (access to training, lab space, collaborators and funding). That’s not to say these researchers are less intelligent, but that intelligence brings diminishing marginal returns in their line of work and is not the bottleneck for faster progress. Getting accepted to a PhD in other non-math scientific fields mainly buys you resources and contacts to develop into a long-term research career.
Another possible reason is that scientists lack the resources or incentive to invest in efficiency. They use the same old tools rather than trying to invent better ones. They do conservative research that’s easy to turn into a paper, rather than what’s hard but truly useful.
If this model is true, then it suggests three avenues for speeding scientific progress.
To speed mathematical progress, assuming that IQ is real and fixed, we should scour the world for child mathematical prodigies in countries that don’t have the capacity to identify them and give them access to opportunity. Mathematicians Without Borders? Is this already a thing?
To speed data-gathering, we should automate, encourage specialization in data collection vs. analysis vs. engineering, expand the number of PhD positions, increase funding, create tools that diminish ethical issues (e.g. organoids, which could replace some animal testing, and iPSCs, which avoid some of the ethical issues with embryonic tissue), and remove red tape.
To encourage efficiency, more funding should be awarded in the form of bounties for certain specific tools, techniques, or applications that are yet to be invented. Scientists who have perfected a certain rare and useful technique should create startups and commercialize their work, rather than try to further their career by being a sort of glorified technician on future projects. Outsiders should create companies that hire scientists to train others in the techniques they’ve perfected, increasing the division of labor between education/training/technique and creative research.
I’ve read this after I wrote my own reply. This seems like a reasonable hypothesis too. One thing a PhD supervisor is great for, is telling you what has already been done, and what papers you should read to learn more about some particular thing.
One way that mathematics is different from the other sciences is that, since the last time it had to repair its foundations around 1900, progress within it doesn’t get obviated by new technology.
Biologists who’ve spent a career using one tool can be surpassed quickly by anyone who’s mastered the new, better tool. Not the case for mathematicians. (Maybe computer-verified and computer-generated proofs will change that, but they really haven’t yet in almost any domains.)
That means that someone who’s put years of work into a mathematical field has a strong advantage over someone who hasn’t; and if you’re going to put years of full-time effort into mathematics anyway, why not get a PhD for it?
This claim as stated stands in tension with the idea that mathematics is a young person’s game. If true, we’d expect either a positive or no correlation between age and mathematical output.
By contrast, in the biological sciences we’d expect to see a phenomenon of accelerating individual output (as knowledge and resources accumulate) followed by a sharp decline (as the techniques an individual biologist is an expert in become obsolete).
My guess is that it’s a mix. Some older biologists will indeed get outmoded, while others will continue to invest in new techniques. Mathematics doesn’t have this burden. Another reason to expect it to have picked more of the high-hanging fruit on the tree of mathematical knowledge.
I think that math is a “young person’s game” because it is intellectually demanding requiring a fair amount of energy to process what is going on. It takes years to learn the background necessary to do the research, so I think that most discoveries are made by mathematicians in the age range of 30 to 45.
I’m 55 and I have mostly given up on doing research. I just don’t have the energy and I’ve never been a professor, so I just don’t have the time to do it for fun anymore. My former advisor continues to publish in his 70s.
Brain plasticity. I wondered whether I should put “given that the two are the same intelligence and age” into the last paragraph.
Some what related:
This trailer for the documentary “Death (& Rebirth) Of A PhD” claims that getting a PhD used to be great, but is now crap.
And here’s an almost finished blogpost I’m working on: “Should you do a PhD?” Where I try to sort out some misconceptions I’ve seen, and give some very general advise.
However, neither of these exactly address the question of the post.
However again, I think it is probably more useful to ask the question: If I want to solve outstanding maths problems, is a PhD my best choice.
I started my Ph.D. at age 27. I would have made a lot more money during my career if I had instead just joined a large company like Microsoft because I was a good programmer, but I really have enjoyed mathematics.
No clue if they might fit well with your thinking or if it is even common, but where would you put the case where the person has a Ph.D. but in a different field from their main study?
Would that fit better with the non credentialed case? I would like to think so but perhaps just having the paper gets you into the club.
No idea, but I didn’t see that for any of the people who I sampled for this study. I don’t think it’s common.
There was a mathematician called Ramanujan, https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
He did some ground breaking work in Number theory and he didn’t even have basic formal education.
Same applies to Issac Newton, He invented calculus at least 200 yrs before the term PhD was Invented.
The title “Doctor of Philosophy” is older than calculus, though neither of its inventors, Newton and Leibniz, held it. (They had BAs though.)
You are right https://en.wikipedia.org/wiki/Doctor_of_Philosophy
The term appeared after 1652 in Germany, although Doctorates existed before that but these were used interchangeably for Masters Degree.
I’m not sure what you mean by “a basic formal education”. Ramanujan attended Government Arts College, Kumbakonam and Pachaiyappa’s College and then he became a mathematical “researcher at the University of Madras”. About 3 years later, Ramanujan wrote his first letter to Hardy. It is certainly true that he did not have the normal formal background of a Ph.D. student, but I think he did take a number of undergraduate math classes.
University education in India for pure sciences is 3 yrs for an undergraduate course, so even if he attended any college he would taken up courses only for the first year. Once he failed his exams in the first yr he cannot continue the education and study second and third yr courses in mathematics, which is why i think he was probably an autodidact in mathematics.