Was a PhD necessary to solve outstanding math problems?
greaterwrong.com
greaterwrong.com
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.
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 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 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.
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.
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.
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.
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.
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.
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.
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 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”)
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.
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!
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.
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?
____________
[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...
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.
(2) having the knowledge to attack it.
would be enough.
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/