Was a PhD necessary to solve outstanding math problems?(greaterwrong.com)
greaterwrong.com
Was a PhD necessary to solve outstanding math problems?
https://www.greaterwrong.com/posts/p7uv4Ekhf8utFKBxa/was-a-phd-necessary-to-solve-outstanding-math-problems
56 comments
The article seems to treat a PhD merely as a credential - maybe assuming that a person would be equally capable with or without it. Why? The whole point of a PhD in math is to learn to do effective research in math and there are almost no substitutes. This seems like being surprised that so many professional violinists took violin lessons. (I admit my bias here but still.)
The same mindset occurs a lot in the software engineering world, and I think it's somewhat baffling and potentially dangerous. I find software engineering quite difficult, and I am relatively intelligent, studied computer science at top universities, and have decades of experience where I learned a ton. So I'm all for coding bootcamps, but we wouldn't expect someone who went to a violin bootcamp for 3 or 4 months to be able to play in an orchestra, but I don't think that's the standard message coming from these coding bootcamps (understandable, they have a biased motivation) or from higher ups in the tech industry at large (more problematic IMO).
"scratch the surface of programming, which nonetheless equips you for real world jobs" could possibly be fair to say about a bootcamp. It certainly filled a certain demand, although one has to wonder if better software won't just eat 4 month configuration++ bootcamps
I think a better use of coding boot camps would be to go deep in a very specific DSL. Like learn how to do Salesforce or Excel programming deeply. A lot of companies actually need this to improve processes, and restricted environments reduce the cognitive load a novice needs to become productive.
Exactly. Getting a PhD means
1. taking some advanced math classes
2. hanging around some like-minded people for a few years
3. solving a problem or two, big enough to add up to a dissertation
4. publish, with professional help on learning to write
5. getting teaching experience
Except for the last item, all these are things every aspiring mathematician would need / want to do anyway.
Now, not everyone wants to teach, and at some universities TAs are overburdened. But certainly not all, and I'd argue that teaching experience in a well-strucured programs can teach one a lot of useful communication skills.
Working on your own means you can get #3. Not everyone needs #4, and you can get some of #1 online or from books (but harder to find people to talk to). So no, a PhD isn't necessary. But most of the things one would do to get a PhD certainly is useful.
Full disclosure: FWIW I have a PhD and work as an academic mathematician.
Now, not everyone wants to teach, and at some universities TAs are overburdened. But certainly not all, and I'd argue that teaching experience in a well-strucured programs can teach one a lot of useful communication skills.
Working on your own means you can get #3. Not everyone needs #4, and you can get some of #1 online or from books (but harder to find people to talk to). So no, a PhD isn't necessary. But most of the things one would do to get a PhD certainly is useful.
Full disclosure: FWIW I have a PhD and work as an academic mathematician.
I think you’re correct here. The author assumes you have the starting knowledge out of undergrad (or ability to self teach) and everything in grad school is sorting and signaling.
My view of school is more nuanced. Both undergrad and grad have a mix of signaling, sorting and learning. Since the financial barrier to entry to solving math problems is low, the easy problems are solved. To solve the hard ones, you need several more years of post-undergrad coursework to get to the edges of the field, and several more being mentored on how to push the edges. It’s hard to get there without the more advanced guided tour that grad school provides.
My view of school is more nuanced. Both undergrad and grad have a mix of signaling, sorting and learning. Since the financial barrier to entry to solving math problems is low, the easy problems are solved. To solve the hard ones, you need several more years of post-undergrad coursework to get to the edges of the field, and several more being mentored on how to push the edges. It’s hard to get there without the more advanced guided tour that grad school provides.
If u were gifted at math and wanted to do it all the time, would you a) get paid for doing it with peers while getting a PhD or b) work a job as a janitor to support yourself while doing math. With the amount of depth modern math has, i think to achieve results you need to be able to do it full time, so getting a PhD seems to be the most logical route for aspiring mathematicians. Not to mention that most pure math don't have much practical application at the moment, you certainly would not convince your manager in industry to let you solve random problems just for the sake of solving it.
There’s probably a middle ground given how little phds are paid. Consult as a software engineer for 1/3 time, make double janitor comp, 2/3 time on research.
It never ends at 1/3 time.
I feel like at 1/3 time you'd spend most or all of it catching up on what happened in the 2/3 of the week you missed. Unless you're not on a team.
I think you would miss out on having a supervisor and peers to learn from. You’d need to replace that. Might be possible with IRC etc. but I don’t know.
Imagine learning from a supervisor, yikes. It is almost always the individual contributors who have to explain things to the supervisor, who might as well be a child at that point.
I’m sorry your PhD supervisor was so terrible. I promise you that’s not universal. I typically learned more in any given ten minutes with mine than in a week with anyone else.
I've thought about this general approach for funding my own research. Unfortunately it seems hard to do in practice.
If the goal is consulting specifically then it's going to take a while to build up the client base needed. I'm just starting consulting myself and I don't expect this to get me more than a couple thousand dollars over the next year.
Generalizing a bit, one could make money running a business, consulting being a possible example. The problem with this is that starting a business is usually more than a full time job. This is only viable in the long-term once all the hard work is done, and only if the business gets off the ground. There are no guarantees.
Another possibility is to work a more conventional job that offers reduced hours and still decent pay/benefits. I don't know of any such jobs and would welcome any recommendations.
If the goal is consulting specifically then it's going to take a while to build up the client base needed. I'm just starting consulting myself and I don't expect this to get me more than a couple thousand dollars over the next year.
Generalizing a bit, one could make money running a business, consulting being a possible example. The problem with this is that starting a business is usually more than a full time job. This is only viable in the long-term once all the hard work is done, and only if the business gets off the ground. There are no guarantees.
Another possibility is to work a more conventional job that offers reduced hours and still decent pay/benefits. I don't know of any such jobs and would welcome any recommendations.
I've known people who (typically) transitioned from being a full-time employee to part-time. Though at that point, they're usually classified as a contractor which tends to have implications for bonuses, RSUs, etc.
I agree with your assessment of consulting. I'd add that, except under special conditions, you pretty much have to keep your hand in or you lose touch with current practices etc. Yes, there are people who are real experts in some specialty. They can semi-retire and "parachute in" for a week to solve some problem that no one else can. But that is, as I say, pretty much the definition of a special case.
I agree with your assessment of consulting. I'd add that, except under special conditions, you pretty much have to keep your hand in or you lose touch with current practices etc. Yes, there are people who are real experts in some specialty. They can semi-retire and "parachute in" for a week to solve some problem that no one else can. But that is, as I say, pretty much the definition of a special case.
Is it really worth it to consult anymore? Considering some FANG engineers are pulling down $300,000 to $400,000 USD?
That translates to between $144/hour to $192/hour, respectively.
Even senior engineers at other non-FANG companies, that makes less than that, still has a high hourly rate.
And what ends up killing you, are taxes, either self employed taxes, S-Corp taxes, or health care costs. The health care insurance costs are perverted, if you don’t have an employer paying for it.
That translates to between $144/hour to $192/hour, respectively.
Even senior engineers at other non-FANG companies, that makes less than that, still has a high hourly rate.
And what ends up killing you, are taxes, either self employed taxes, S-Corp taxes, or health care costs. The health care insurance costs are perverted, if you don’t have an employer paying for it.
> Is it really worth it to consult anymore? Considering some FANG engineers are pulling down $300,000 to $400,000 USD?
Good question. It's tough. Anyone who has 5+ years of experience and works at one of FAANG or Lyft, Uber, Square, Stripe, Airbnb, etc in NYC/SFBA/SEA is almost certainly earning $300k+, yes (probably closer to $400k now).
That being said if you are qualified to work for one of those companies and they're paying you for expertise in a specialty, you definitely have the technical skills needed to earn more as a solo consultant. For example, leading projects in distributed systems as an L6/E6 engineer at Google/Facebook would neatly set you up to start a boutique consultancy like Jepsen/Aphyr.
At that point whether or not you earn more becomes a question of your network, because you have everything else going for you. Very experienced security consultants can also make $500k+ at a 70% utilization rate by charging as much as the larger shops, doing better work and pocketing the spread on the weekly rate.
Speaking as a former consultant who earned more than a FAANG salary at equivalent years of experience: full time work trades off autonomy and uncertainty for lower but relatively guaranteed pay. You're never paid what you're actually worth at these companies, in the sense of value contributed. You're paid according to the cost of labor in your area.
Sorry to ramble, this is something I like chatting about.
Good question. It's tough. Anyone who has 5+ years of experience and works at one of FAANG or Lyft, Uber, Square, Stripe, Airbnb, etc in NYC/SFBA/SEA is almost certainly earning $300k+, yes (probably closer to $400k now).
That being said if you are qualified to work for one of those companies and they're paying you for expertise in a specialty, you definitely have the technical skills needed to earn more as a solo consultant. For example, leading projects in distributed systems as an L6/E6 engineer at Google/Facebook would neatly set you up to start a boutique consultancy like Jepsen/Aphyr.
At that point whether or not you earn more becomes a question of your network, because you have everything else going for you. Very experienced security consultants can also make $500k+ at a 70% utilization rate by charging as much as the larger shops, doing better work and pocketing the spread on the weekly rate.
Speaking as a former consultant who earned more than a FAANG salary at equivalent years of experience: full time work trades off autonomy and uncertainty for lower but relatively guaranteed pay. You're never paid what you're actually worth at these companies, in the sense of value contributed. You're paid according to the cost of labor in your area.
Sorry to ramble, this is something I like chatting about.
Thanks for your input. Industry insights like these are gold.
The reason I ask is because it’s starting to appear, to not be worthwhile anymore. Yes, some people want the autonomy and independence. But it seems securing the gig itself is tough.
For large companies, with millions of dollars budgeted for a project, they seem to want an off-the-shelf system from a vendor, for a technology qualification selection, with another large company behind it. Because that customer “feels” safer that the larger vendor will not go out of business. It’s the old adage: “No one ever got fired for selecting IBM/Microsoft”.
So, it seems unlikely that a company will hire an expensive consultant, to do something, when they can hire cheaper senior level programmers, and put them on the clock full time.
Granted, some companies do operate at that level, where they try to bring in a specialist to work on something, and get rid of them after a few months. But, they tend to hire these specialists via a contracting agency, where the specialist is deemed an employee of that agency. Hence, the specialist here is not really a consultant, but an employee of a 3rd party. And instead of getting consulting rates, that employee just gets a normal hourly salary.
Also, this allows the customer company, to not have a full time staff, and allows them to easily get rid of people, without violating any lay-off laws. Or having to make some embarrassing lay-off announcement publicly. They just silently kill their staff. Although I heard some states like California, is trying to crack down on this, with new laws, but I’m sure companies will find some ways around it.
Although now, it seems the most expensive hired guns, are the new experts in some deep learning library, or someone pawning off knowledge in some AI or Machine Learning solution.
This appears to be the new valuable gold rush. So if you’ve kept up on this, then it’s time to sell shovels to the desperate prospectors.
The reason I ask is because it’s starting to appear, to not be worthwhile anymore. Yes, some people want the autonomy and independence. But it seems securing the gig itself is tough.
For large companies, with millions of dollars budgeted for a project, they seem to want an off-the-shelf system from a vendor, for a technology qualification selection, with another large company behind it. Because that customer “feels” safer that the larger vendor will not go out of business. It’s the old adage: “No one ever got fired for selecting IBM/Microsoft”.
So, it seems unlikely that a company will hire an expensive consultant, to do something, when they can hire cheaper senior level programmers, and put them on the clock full time.
Granted, some companies do operate at that level, where they try to bring in a specialist to work on something, and get rid of them after a few months. But, they tend to hire these specialists via a contracting agency, where the specialist is deemed an employee of that agency. Hence, the specialist here is not really a consultant, but an employee of a 3rd party. And instead of getting consulting rates, that employee just gets a normal hourly salary.
Also, this allows the customer company, to not have a full time staff, and allows them to easily get rid of people, without violating any lay-off laws. Or having to make some embarrassing lay-off announcement publicly. They just silently kill their staff. Although I heard some states like California, is trying to crack down on this, with new laws, but I’m sure companies will find some ways around it.
Although now, it seems the most expensive hired guns, are the new experts in some deep learning library, or someone pawning off knowledge in some AI or Machine Learning solution.
This appears to be the new valuable gold rush. So if you’ve kept up on this, then it’s time to sell shovels to the desperate prospectors.
Sadly, this omits a hard truth that impactful research requires near-total dedication. The payoff is highly non-linear versus time invested; conducting research part-time is a recipe for underperformance, which is (partly) why research funders often contractually cap the hours one may work externally while a staff researcher or PhD student.
Yes phd is just a way of making time in your mife for what you like. The way you use that time is then up to you.
I dropped from grad school with a Master's degree in applied math, and vowed to keep studying for the rest of my life.
It's really hard to study "real maths" on your own. It's harder to understand materials without the aid of someone who has already deep-dove and built some intuitions, metaphors and visual schemas. More critically, it's very hard to know if you're doing proofs correctly without feedback.
It's really hard to study "real maths" on your own. It's harder to understand materials without the aid of someone who has already deep-dove and built some intuitions, metaphors and visual schemas. More critically, it's very hard to know if you're doing proofs correctly without feedback.
The article is missing mathematicians who do not publicize their results. These would mainly be mathematicians working for agencies like GCHQ, NSA, etc.
If you want to work on certain math problems, don't mind working for the government, and don't care about being published you can get good jobs with these kinds of agencies.
If you want to work on certain math problems, don't mind working for the government, and don't care about being published you can get good jobs with these kinds of agencies.
* and don't care about why these results will go unpublished
meaning that everyone who works for the government would agree they are only enabling bad things?
* and don't have anything on your track record making it impossible to land this kind of job due to the security clearance required
Or you are comfortable with why they will go unpublished.
of course, or that
Of course, in science, if it isn't published, it doesn't exist.
Financial markets as well.
I know of a boutique proprietary trading firm wherein a researcher solved an open problem in probability but won't publish the results because it's integral to a trading strategy. That always gives me a chuckle :)
PhD is not strictly necessary, but the process and training that results in a PhD degree is.
Case in point, Freeman Dyson, brilliant mathematician/math-physicist, never got a PhD degree, but did research work all his life. Note, he was deeply embedded in the research community, and pretty much everyone he collaborated with, was a PhD or eventually got a PhD (meaning he went through the motions just like other researchers).
Final note, the process and training required to be able to conduct research is massively undernurtured, i.e., in my opinion, most PhD graduates are barely an iota better-trained than MS graduates these days. This is getting increasingly true as world population grows, and PhD diplomas get handed out willy nilly (a topic I could go on at for hours). In short, Sturgeon's law is in full swing.
Case in point, Freeman Dyson, brilliant mathematician/math-physicist, never got a PhD degree, but did research work all his life. Note, he was deeply embedded in the research community, and pretty much everyone he collaborated with, was a PhD or eventually got a PhD (meaning he went through the motions just like other researchers).
Final note, the process and training required to be able to conduct research is massively undernurtured, i.e., in my opinion, most PhD graduates are barely an iota better-trained than MS graduates these days. This is getting increasingly true as world population grows, and PhD diplomas get handed out willy nilly (a topic I could go on at for hours). In short, Sturgeon's law is in full swing.
> PhD is not strictly necessary, but the process and training that results in a PhD degree is.
I was just about to post the same. I just finished a master's in math (in fact, with additional coursework in physics and statistics I've almost completed two master's) and I don't feel ready to tackle outstanding math problems.
And the PhD students I knew were receiving a lot of help from their mentors as they learned how to tackle outstanding problems.
There's just so much to learn to contribute to research math these days, it's hard to imagine anyone learning it all on their own.
I was just about to post the same. I just finished a master's in math (in fact, with additional coursework in physics and statistics I've almost completed two master's) and I don't feel ready to tackle outstanding math problems.
And the PhD students I knew were receiving a lot of help from their mentors as they learned how to tackle outstanding problems.
There's just so much to learn to contribute to research math these days, it's hard to imagine anyone learning it all on their own.
Freeman Dyson studied in very unusual circumstances and got more support than most PhD students. He was regularly playing billiard and discussing math with famous professors at age 17. From his collection of letters "Maker of Patterns":
>I arrived as a seventeen-year-old undergraduate at Trinity College, Cambridge, in September 1941. It was a great time to get an education, in the middle of World War II. The famous old professors were all there, but there were hardly any students.
>I have fixed up all my lectures now, they are: Hardy on Fourier series, Besicovitch on integration, Dirac on quantum mechanics, Pars on dynamics. The lectures are very select; Hardy has an audience of four, Besicovitch three, Pars four, and Dirac about twenty
Here's a video of him speaking about the PhD system, part of a great interview series: https://youtu.be/DzC1IRYN_Ps?t=98 (“It is an evil system and it has ruined many lives”)
>I arrived as a seventeen-year-old undergraduate at Trinity College, Cambridge, in September 1941. It was a great time to get an education, in the middle of World War II. The famous old professors were all there, but there were hardly any students.
>I have fixed up all my lectures now, they are: Hardy on Fourier series, Besicovitch on integration, Dirac on quantum mechanics, Pars on dynamics. The lectures are very select; Hardy has an audience of four, Besicovitch three, Pars four, and Dirac about twenty
Here's a video of him speaking about the PhD system, part of a great interview series: https://youtu.be/DzC1IRYN_Ps?t=98 (“It is an evil system and it has ruined many lives”)
This doesn’t prove that PhD is needed. This just proves that people prefer to get paid.
The author clearly articulated that it is needed in practical terms. That is, empirically (according to their small sample) people who don't do a PhD don't feature in the list of notable maths achievements.
Why then did he only mention money once in passing? Sounds like a very practical concern to me.
There's a difference between a degree and an education.
The problem is how we consider the destination (the degree), to be a proof of competence and knowledge, while it's really the journey (the education) which actually matters. That's why degrees should be abolished, and a certificate of education be given to the student as long as he/she is present.
School should not act as a social filter. As long as you go to school and go to classes, and shows motivation to learn, it's enough. There are no proper ways to evaluate how a student really learned and absorbed the knowledge that was given. There are many students who love the knowledge, but cannot accept scholasticism, the competition, the selection and the filtering. It often ends up being about "belonging to a group", and honestly it was never the goal of education.
It's up to companies to really check if someone if competent and has the knowledge, it's not the job of universities. Higher education is an enormous source of inequality, and an immense social barrier.
It's really easy for people with degrees to disagree with this, I can only answer with survival bias.
The problem is how we consider the destination (the degree), to be a proof of competence and knowledge, while it's really the journey (the education) which actually matters. That's why degrees should be abolished, and a certificate of education be given to the student as long as he/she is present.
School should not act as a social filter. As long as you go to school and go to classes, and shows motivation to learn, it's enough. There are no proper ways to evaluate how a student really learned and absorbed the knowledge that was given. There are many students who love the knowledge, but cannot accept scholasticism, the competition, the selection and the filtering. It often ends up being about "belonging to a group", and honestly it was never the goal of education.
It's up to companies to really check if someone if competent and has the knowledge, it's not the job of universities. Higher education is an enormous source of inequality, and an immense social barrier.
It's really easy for people with degrees to disagree with this, I can only answer with survival bias.
I have a degree, and I agree with you. At least for software engineering.
What there could be is a standardised test, that is a national or international standard, not linked to a uni. Learn to program and do the test and that is your credential. Maybe it costs $200 or something.
That way you can skip uni, self teach but be able to show you can code without employers needing to find novel ways to evaluate.
Just spitballing!
What there could be is a standardised test, that is a national or international standard, not linked to a uni. Learn to program and do the test and that is your credential. Maybe it costs $200 or something.
That way you can skip uni, self teach but be able to show you can code without employers needing to find novel ways to evaluate.
Just spitballing!
Ultimately if you standardize something, people will gamify it. The point of employers finding novel ways to evaluate candidates is to win that game and beat the average. Good for them! It can't/shouldn't be otherwise (imo).
They aren't mutually exclusive approaches, e.g. the standardized test could be used just to get past the initial resume filter.
Similar to the bar exam for lawyers. It used to be you could be a lawyer by passing the bar even if you had no degree.
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> It's up to companies to really check if someone if competent and has the knowledge
Except that the way these companies do this is to throw stupid LEET code questions at you. And expect you to solve it in 10 minutes, otherwise you’re not smart enough to handle their boring CRUD cruft.
Except that the way these companies do this is to throw stupid LEET code questions at you. And expect you to solve it in 10 minutes, otherwise you’re not smart enough to handle their boring CRUD cruft.
If you're interested enough to do math research, you're likely to get a PhD in mathematics because it's, well, full time math research?
>> 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)?
Who does the article mean, by "robotics (invented by someone with no higher education)"? Wikipedia tells me that [in] 1948, Norbert Wiener formulated the principles of cybernetics, the basis of practical robotics [1], but Wiener had a PhD from Harvard [2] and certainly much education, at all levels.
The wikipedia article on robotics has a number of other names of people who contributed in various ways to robotics from ancient to modern times, but I'm not sure who fits the article's description. Did Heron of Alexandria have a "higher education", sensu stricto?
____________
[1] https://en.wikipedia.org/wiki/Robotics
[2] https://en.wikipedia.org/wiki/Norbert_Wiener
Who does the article mean, by "robotics (invented by someone with no higher education)"? Wikipedia tells me that [in] 1948, Norbert Wiener formulated the principles of cybernetics, the basis of practical robotics [1], but Wiener had a PhD from Harvard [2] and certainly much education, at all levels.
The wikipedia article on robotics has a number of other names of people who contributed in various ways to robotics from ancient to modern times, but I'm not sure who fits the article's description. Did Heron of Alexandria have a "higher education", sensu stricto?
____________
[1] https://en.wikipedia.org/wiki/Robotics
[2] https://en.wikipedia.org/wiki/Norbert_Wiener
It's really unclear who they could mean. A few of the people in the History of Robotics Wikipedia article[0] didn't have PhDs[1], but seemingly all of them had some higher education. Maybe they are referring to someone who wrote fiction about the idea of robots before robotics really existed: Karel Čapek (who came up with the word "robot") had a PhD though[2]. Perhaps it refers to L. Frank Baum, as he appears to have had no higher education[3].
[0] https://en.wikipedia.org/wiki/History_of_robots
[1] See William Grey Walter for example: https://en.wikipedia.org/wiki/William_Grey_Walter
[2] https://en.wikipedia.org/wiki/Karel_%C4%8Capek
[3] https://en.wikipedia.org/wiki/L._Frank_Baum#Childhood_and_ea...
[0] https://en.wikipedia.org/wiki/History_of_robots
[1] See William Grey Walter for example: https://en.wikipedia.org/wiki/William_Grey_Walter
[2] https://en.wikipedia.org/wiki/Karel_%C4%8Capek
[3] https://en.wikipedia.org/wiki/L._Frank_Baum#Childhood_and_ea...
[deleted]
Hint, quiet, don't let this out, a Ph.D. is not really a knowledge degree but a research degree. From that degree on, in blunt terms, everyone knows that no one can carry the whole library around between their ears and so no longer much cares what you know but cares what you can create!
There can be and are some exceptions, but overwhelmingly successful research in math requires the background of a Ph.D. for (1) finding a suitable problem and (2) having the knowledge to attack it. And it helps to be in a relatively good school so that will get relatively good versions of (1) and (2).
But with everything in good shape, apparently there is one more challenge -- being successful in the actual research. For a hint at this challenge, buried in D. Knuth's The TeXBook is:
> The traditional way is to put off all creative aspects until the last part of graduate school. For seventeen or more years, a student is taught examsmanship, then suddenly after passing enough exams in graduate school he's told to do something original.
That is, the research is work that is suddenly different, maybe for some people quite different and challenging, than all the academic work before. E.g., there are cases where a student made A's and was the darling of all the teachers from kindergarten through college but in all that time never encountered anything like having new ideas. Bad such cases can lead to stress, loss of self-esteem, crippled ability to work, more stress, burn out, clinical depression, and ... suicide. No joke.
For me, part of what helps in research is some qualified respect for some of the existing material. So, I look at what is there as needing improvement and try to do that. If look at the existing material as some nearly perfect construction, then maybe won't feel confident should or could improve on it!
One thing rarely taught in math is the importance of intuition: It is needed to do well at guessing, guess a suitable problem, broad outlines of a solution, attack, tools, etc. Good guessing is important since that's most of what there is to do, and good intuition helps with good guessing. Sure, when the results are obtained and in clean form with polished proofs, there can be little or no view of the sources, the intuition.
There can be some question about how good some Ph.D. research is: The professors don't want to grant Ph.D. degrees for poor research but don't really know how to ensure good work, indeed, for either the students or sometimes themselves. So one standard that can remove some possibly painful ambiguity is that the Ph.D. research should be "an original contribution to knowledge worthy of publication" with the usual standards for publication being "new, correct, and significant". If a student does some research and the professors question if it is publishable, then the student can settle the issue in an objective way -- try to publish the work.
E.g., computer science is concerned with computational time complexity, i.e., good algorithms where good means running time that grows no faster than some polynomial in the size of input data for the problem (rough statement -- more details in the famous
Michael R. Garey and David S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, ISBN 0-7167-1045-5, W. H. Freeman, San Francisco, 1979.
and more recent sources).
IIRC that polynomial criterion came from J. Edmonds. More IIRC, he left his Ph.D. program early and did and published some of his work on networks. Eventually a committee of his former professors came to him and said that should he stack his publications and put a staple in one corner, that stack would be accepted as his Ph.D. dissertation and he would get his Ph.D.
There can be and are some exceptions, but overwhelmingly successful research in math requires the background of a Ph.D. for (1) finding a suitable problem and (2) having the knowledge to attack it. And it helps to be in a relatively good school so that will get relatively good versions of (1) and (2).
But with everything in good shape, apparently there is one more challenge -- being successful in the actual research. For a hint at this challenge, buried in D. Knuth's The TeXBook is:
> The traditional way is to put off all creative aspects until the last part of graduate school. For seventeen or more years, a student is taught examsmanship, then suddenly after passing enough exams in graduate school he's told to do something original.
That is, the research is work that is suddenly different, maybe for some people quite different and challenging, than all the academic work before. E.g., there are cases where a student made A's and was the darling of all the teachers from kindergarten through college but in all that time never encountered anything like having new ideas. Bad such cases can lead to stress, loss of self-esteem, crippled ability to work, more stress, burn out, clinical depression, and ... suicide. No joke.
For me, part of what helps in research is some qualified respect for some of the existing material. So, I look at what is there as needing improvement and try to do that. If look at the existing material as some nearly perfect construction, then maybe won't feel confident should or could improve on it!
One thing rarely taught in math is the importance of intuition: It is needed to do well at guessing, guess a suitable problem, broad outlines of a solution, attack, tools, etc. Good guessing is important since that's most of what there is to do, and good intuition helps with good guessing. Sure, when the results are obtained and in clean form with polished proofs, there can be little or no view of the sources, the intuition.
There can be some question about how good some Ph.D. research is: The professors don't want to grant Ph.D. degrees for poor research but don't really know how to ensure good work, indeed, for either the students or sometimes themselves. So one standard that can remove some possibly painful ambiguity is that the Ph.D. research should be "an original contribution to knowledge worthy of publication" with the usual standards for publication being "new, correct, and significant". If a student does some research and the professors question if it is publishable, then the student can settle the issue in an objective way -- try to publish the work.
E.g., computer science is concerned with computational time complexity, i.e., good algorithms where good means running time that grows no faster than some polynomial in the size of input data for the problem (rough statement -- more details in the famous
Michael R. Garey and David S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, ISBN 0-7167-1045-5, W. H. Freeman, San Francisco, 1979.
and more recent sources).
IIRC that polynomial criterion came from J. Edmonds. More IIRC, he left his Ph.D. program early and did and published some of his work on networks. Eventually a committee of his former professors came to him and said that should he stack his publications and put a staple in one corner, that stack would be accepted as his Ph.D. dissertation and he would get his Ph.D.
From what I believe, many of the old days Mathematicians were given PhD by various institutes after they got famous.
Not all had PhD, but most of good one got it after being recognized. Also don't forget many people of that time did not publicize their work.
I'd agree with the hypothesis that a PhD let's you hang out and get paid to do a lot of math. My brain is pretty spent from writing code -- I don't have time to spend doing more math.
It also depends on the person, no?
"A PhD is less important for doing groundbreaking applied engineering and entrepreneurial work, especially in tech."
I'm just going to sit here and ponder "groundbreaking entrepreneurial work" for a while. Is it like "financial innovation?"
Anyway, I'll also point to Matt Might's illustrated guide: http://matt.might.net/articles/phd-school-in-pictures/
I'm just going to sit here and ponder "groundbreaking entrepreneurial work" for a while. Is it like "financial innovation?"
Anyway, I'll also point to Matt Might's illustrated guide: http://matt.might.net/articles/phd-school-in-pictures/