Was a PhD necessary to solve outstanding math problems?

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.