This is a useful nuts-and-bolts discussion of the academic routine, but I'd disagree with this conclusion:
> "It’s your company. Mathematicians each run a small business. We work in
malls called math departments. Your company sells theorems. Your advisor is a free
consultant for your company, not your boss, not your employee. Ultimately YOU are
responsible for the success of your company..."
The adoption of corporate entrepreneur model of academic research is one of the worst, if not the worst, things about the current American academic system. Why?
First of all, who is funding this 'company'? Not you, in all likelihood - it's the tax-paying public who finances the majority of the funding train that keeps academic institutions afloat. What does the public expect in return, ideally?
Well, how about producing highly skilled teachers, that seems important in a technological society, where mathematical skills are in general critically important. Hence, a critical part of the PhD student's job (and the professor's) is the instruction of the undergraduate population. This article doesn't even mention the importance of that role.
Secondly, corporate research is highly secretive and non-collaborative in nature, as the goal is creation of intellectual property that can be sold on for a profit. This attitude is entirely at odds with the most successful acadamic model for making scientific progress, i.e. collaboration and rapid sharing of results. If you spend your time in academia worrying about people stealing your work and not trusting anyone, it's going to be a fairly miserable experience.
My advice is to toss the corporate enterpreneur model in the garbage bin.
The highlighted sentence is a metaphor and it’s the right one.
You are selling in your academic career. You are selling results, papers, conference presentations, whatever counts among your peers in your field.
You will not have a career as an academic in your field unless you can sell your work to your peers in the field.
No one cares about your teaching, certainly not of undergrads (exceptions exist at some liberal arts colleges). Your department and your dean care about your research. That’s it.
You have to sell it both to get it published and to get it cited. Do that and you’ll be fine. Don’t and you won’t have a career in a university.
What you say is true... but that's also why American academics is widely viewed as corrupt and unproductive, ridden with fraudulent con artists, and falling into the same kind of failure mode seen in, for example, Soviet science under the regime of Lysenko.
Now if you wanted to rise to a position of academic power under the likes of Lysenko, you'd want to follow this careerist path, you'd want to observe Lysenko, sing the song that Lysenko wanted to hear, and you'd be rewarded appropriately.
That's the way of authoritarian bureaucratic institutionalism, not the way of high-quality science. It also doesn't really matter whether the structure is 'capitalist' or 'socialist', it's all about who sits at the top of the hierarchy in such systems. Crushing independent thought is what they have in common.
> What you say is true... but that's also why American academics is widely viewed as corrupt and unproductive, ridden with fraudulent con artists, and falling into the same kind of failure mode seen in, for example, Soviet science under the regime of Lysenko.
Citation needed. I'm a former academic, and seldom a defender, and I entirely dispute this statement. American academia is widely considered the international leader by any ranking I'm aware of.
The reluctance of the American academic bureaucracy to admit to the prevalence of fraud in academic research is understandable, but there are many examples. Some of the recent notables are the serotonin hypothesis of depression, and aspects of the amyloid theory of Alzheimer's. Data falsification is not uncommon, and it's often done for obvious reasons - a graduate student may want that positive result that finishes off their PhD work and gets them a needed publication, a professor may want to support their past work with new data and results, an academic startup may want to prop up their latest drug development results to draw in investors and big corporations, etc.
The reason academic institutions don't want to talk about it is the same reason that religious institutions don't want to talk about child abuse - it makes a mockery of their whole enterprise. With the former, it's the search for truth via the scientific process, and with the later it's upholding moral standards and guiding society. Both institutions cover up such things in order to preserve their public image - which, of course, allows such behaviors to persist.
Note also that the corporatization of research, i.e. the focus on generating lucrative patents and the resulting emphasis on keeping raw data secret, tends to amplify opportunities for fraud.
Citing a few highly damaging cases of fraud isn't enough to make your point. You have to show that it's more prevalent and damaging in the US system than in other systems.
That seems an extraordinarily unlikely bar to meet based on my cursory knowledge of where large scale fraud is happening. But perhaps you know more and would like to enlighten me.
I wasn't aware that the serotonin hypothesis was "fraudulent," nor can I find credible evidence of this on Google. Instead, it seems that the scientific process worked as intended. A reasonable hypothesis was initially proposed, then opposing evidence accumulated over time.
This seems moreso an issue with the pharmaceutical industry. For example: "In 2005, two academics published a piece of research in which they compared the information on pharmaceutical websites with the pronouncements of certain researchers, and found a ‘disconnect’ between the marketing and the experts’ views." - https://joannamoncrieff.com/2022/07/24/how-to-take-the-news-...
> The reason academic institutions don't want to talk about it is the same reason that religious institutions don't want to talk about child abuse - it makes a mockery of their whole enterprise.
This is why in science, replication of past work is essential. Every new Ph.D.
student should replicate a paper's results in their area of interest as a warm-up exercise.
In Information Retrieval (the part of Computer Science concerned with search for information - search engines etc.), conferences like ECIR have regular reproduceability tracks, where such works of scrutiny get published, and are highly regarded.
There needs to be a cultural change to the effect that genuine errors and fraud can happen, but we can weed it out to an extent by independently repeating past work.
You may be accurate (or not) in your assertions but I would be wary of citing "usnews.com", when using it as proof that US universities are the "best".
I think Americans are generally very biased when looking objectively and rationally at their own country. E.g. Looking at Iran as a barbaric backwards country, when the US just banned abortion and has spent the last 50 years killing more people in international wars than any other country.
If what you claim is true true, public funding of universities should end. Universities should serve the public interest, and teaching is an essential part of that.
The public spends untold billions funding universities and we should at least get a well educated workforce out of it.
The thousand year old college system’s time is over if what you say is true. There’s no point to it.
Due to the bomb, US academics got a big dose of money aimed at:
Making sure the US is never behind in anything in math, science, or even medicine in any way that could hurt US national security or even the US economy or US health care. Or "Never again will US academics be permitted to operate independent of the US military" -- or some such, and I do recall the thought but just now don't recall the source.
So, the departments of math, physical science, and engineering get grants from the US NSF, DARPA, DoE, NIH, etc.
The university takes 60% or so for overhead and, thus, pays for the nice campus landscaping, the limo for the President, the glass walled coffee shop, the theater group and the string quartet concerts, the social science departments, the humanities departments, the art departments, the artist in residence program, the library, etc.
But the US stays up in number theory, mathematical and theoretical physics, computer science, ....
"Education"? There, sure, the first requirement is that the prof knows what the leading edge results are and understands them well enough to teach them and, more important, direct research in them.
How to make sure the profs are not just wasting time? Use competition, have all the profs in competition with all the other profs in the world and insist that the US profs getting funded look like the best profs in the world.
How to evaluate a prof? Count published papers. Weight the counts by the prestige of the journals. Count citations. Count prizes. See what job offers the profs are getting.
If the US starts to fall behind, then Congress can vote more money. Congress can vote lots of money, plenty to be sufficient to keep the US ahead.
Uh, Congress has lots of money because, net, the US is rich. E.g., the last time I added the retail cost of the ingredients for one of my pizzas for one, the cost was $0.39, that is, 39 cents. When a lot of US people are getting paid $30+ an hour, the 39 cents represents some astoundingly high economic productivity -- for the flour, cheese, sausage, trucks, railroads, farm machinery, wheat, corn, soy bean seeds, etc. involved.
Education for good jobs in the economy for the students? Naw, .... Instead, the main issue is the bomb, the hydrogen bomb!!!! Did I mention the bomb?
As it stands, more and more college level teaching is being done by non tenure track employees. Also, the non-tenure-track teaching jobs are a miserable grind. The thousand year old college system has evolved continuously from decade to decade, and is not the same thing that it was, even 50 years ago.
You misinterpret the spirit of the quote. What it means is the following.
> Having very good research is a necessary condition to succeed in academia, but it is not sufficient.
The reason why this is emphasized to starting academics is that many naively believe the system is purely meritocratic and that their work will speak for itself. This is borne out of an ignorance of how fundamentally important it is to communicate the importance of your work to others.
The reality is that there is a total glut of research. Even the best do not have time to sift through all recent results in their broader field. Even if they scan abstracts, understanding the true value of results is a hard process that takes time. The most effective way to understand value of research is to interrogate the researcher directly (e.g., in a seminar talk). Thus, self-advertisement through seminar talks and other modes is crucial to getting your message out. Otherwise you will wilt into obscurity.
I can assure you, the space for some sort of corporate style huckster selling poor research with flashy branding in pure math is very small. This stuff is not easy to fake.
> The adoption of corporate entrepreneur model of academic research is one of the worst, if not the worst, things about the current American academic system. Why?
That's not how I read the paragraph you quoted.
I think the point is that you're basically working for yourself when you do a PhD, not for your boss. You're given carte blanche for a few years, and it's your responsibility to produce value out of this time. The value here isn't measured in money but in publications.
> Well, how about producing highly skilled teachers, that seems important in a technological society, where mathematical skills are in general critically important. Hence, a critical part of the PhD student's job (and the professor's) is the instruction of the undergraduate population. This article doesn't even mention the importance of that role.
That makes zero sense. If you want to make teachers and lecturers, why not create the path of a pedagogy bootcamp or a MSc if you like fancy titles and wasting time? Why would you push people through PhDs? PhDs spend 3-5 years mastering a single narrow domain area, it has absolutely zero utility wrt teaching unless we are talking about graduate-level courses that PhD students take…
> Well, how about producing highly skilled teachers, that seems important in a technological society, where mathematical skills are in general critically important. Hence, a critical part of the PhD student's job (and the professor's) is the instruction of the undergraduate population. This article doesn't even mention the importance of that role.
They don't mention it because it's unimportant. Your thesis committee cares about your research and that's it. If your advisor is well funded, you can easily finish a stem PhD and never teach a single class. If you do have to teach, getting perfect feedback or terrible feedback from student evaluations has zero bearing on whether you earn a PhD.
It can, at the same time, be correct as an (individual-level) descriptive statement but counterproductive as a (system-level) normative statement.
As in, it's probably realistically good advice to a PhD student, because it reflects the state of affairs, but it's possibly unfortunate that that is the case.
Brit here. It seems a little different than US. I didn't take any exams (except viva at the end). I was given some reports by PhD students/researchers working in a similar field in the same department as background reading. I thought that I'd never understand the stuff, but after awhile, it seemed fairly straightforward.
My supervisor chose me rather than the other way around. It seemed an interesting subject, so I went for it. He set the subject and had a general idea of the direction it should be going.
An alternative approach is to look at the PhD opportunities at various universities, and pick one that looks interesting. I don't think it's rocket science.
The overwhelming number of theses will be a bunch of results that are "quite interesting" and extend the field incrementally. Don't expect to redefine the way people look at physics, mathematics, or whatever.
My number one tip for getting a PhD in a timely fashion: work hard and don't arse around on the internet all day. You don't need the intellect of Gauss to get a PhD; although the brainier the better, obviously.
Comprehensive entry exams are being done away with a lot of schools. At the university I went to (before I dropped from the program for financial reasons) I simply had to have gotten A's in my last 4 semesters of CS classes. The last 4 semesters are graduate courses, and undergrads simply do marginally less work.
Comprehensive exams in my opinion are extremely dumb. You take some classes and to take more classes you need to take sometimes up to a year to cram 4 years of various CS trivia, most likely unrelated to your core interest, to sit a 4-5 hour long exam to prove you can regurgitate things. It really is trivia too. My advisor issued these exams and showed me one of the old ones. I am not sure how regurgitating the finer details of computer architecture will ever help me produce research in computational mathematics but apparently it was required.
It's just yet another shibboleth of academia. It provides the illusion of exclusivity for a group that is so smart, they're too dumb to see comprehensives prove nothing.
> « You will like any subject once you get to know enough about it. » this is a really important observation
And unfortunately false, as I've witnessed on multiple occasions. Don't underestimate mental health problems in academia, telling people that everything will fall into place if they just keep pushing can lead to bad outcomes.
> Don't underestimate mental health problems in academia
Having dropped out of my PhD due to - mostly - mental health reasons, I don't fully understand the preoccupation with the mental health situation in academia. Not that it is not bad, but is it worse than in most other occupations? Or is it just that academia is mostly populated by people of at minimum middle class upbringing who are not supposed to experience hardship of any kind?
I suspect it is in part the combination of academia being a somewhat "normatively standard" path for people to take but, unlike most other such paths, inherently requires a lot of self-discipline and self-motivation. The result is that mental health issues may cause worse extreme outcomes, for the sort of person who can go into a PhD. The "standard" alternative path for somebody like that is some office job where you can just sort of muddle through: you may be deeply unhappy, but the money still comes in and it's easier to find satisfaction in other aspects of your life.
There are other paths where self-discipline and self-motivation are important, like being an independent consultant, but those are generally not considered "normatively standard" paths for people to take.
(My personal experience with academia was great. I was genuinely interested in the topics, had great colleagues and a great advisor. It probably helped that it was in maths, where there is money for decent pay but no lab work with which PhD students are routinely burdened.)
The poor do not have a monopoly on suffering. The middle class constitutes a large/majority portion of a lot of western countries, it's odd to believe they are living perfect lives free of pain and hardship.
Academia is the worst environment I've ever worked in, worse than any industry job I've ever had whether blue collar or white collar.
Doing a PhD is a bit like doing a startup, in the way that you can work on something for years and produce nothing of value. This is way more stressful than regular job where your effort is validated every month with a paycheck. Many people are not built for this level of stress and uncertainty, but they're too young when they join the PhD program to know that about themselves yet.
I wasn't actually saying anything about outside academia.
Makes me wonder, is there a name for that sort of reasoning where one person asserts 'A is associated with B', and the response tries to turn it into an argument about !A? I guess it's not a strawman, but I'm sure the philosophers have come up with a name for it.
Academia has a uniquely personal importance: your work is a reflection of your personal effort and interests for years. It's also very isolated: the only people who did what I did were hundreds of miles away at other institutions.
It's not a statement about your general motivation for a PhD -- I don't think you can easily find a subject for which you will find a non-stop motivation 24h/7days for 3-4 years (or so, eqv. period required to get a PhD). If you encounter a dip in interest, which most likely will happen, you need to understand that changing a PhD is not always a best option -- just because it sounds good now, it doesn't mean it will in 2 years time. You supervisor should be there to guide you into interesting areas of research and may play a substantial role in rekindling your motivation. Naturally, mental health issues are another problem -- but I think they have often different origins than "just boredom" with subject. I may be completely wrong though.
That bit about researchers being the CEOs of their own little companies really resonated with me. I'm myself not a PhD, but I worked at IBM Research for a while in a business development role, and I amazed at how incredibly good the researchers in the lab were at sales (when they took the time off from their research to find a customer for their idea). I came to the conclusion that it was their training as PhDs and their need to constantly be pitching for grants. The process forces you to sell your ideas, even when there is no concrete 'product' yet. And that is not so different from how a startup CEO works. Not surprisingly, many of them have gone on to start their own companies very successfully, desipite never having managed anything bigger their lab printer before.
As a mathematician who works with many engineers and computer scientists, I wanted to expand on one of the points under the “Getting a Job” section. While it is certainly true that a mathematics education provides a great background for understanding other STEM fields, I would caution math Ph.D students who expect these jobs to be open to them because of their STEM connection: the onus is completely on you to bridge the gap between what you do and the field you want to work in. While it may be true that someone will hire you for your critical thinking skills (though I will personally say that I have never seen this happen), it is more likely that your deep specialization in a tangentially connected field (coupled with not being involved in the culture/conferences of the community you wish to enter) will be an impediment to entering a new field: you expect to be paid like a Ph.D., but will potentially require years of training to get up to speed.
As an example, I remember the advice of “just go into data science” being handed out like candy to students interested in industry around the time I was in grad school (10 years ago). To be sure, there was a period where a STEM background + interest could get you in the door, but that time is over. These days you are competing with many equally brilliant students who have taken multiple courses and done research in this area, and it is highly unlikely that an employer will take a chance retraining an e.g. algebraic geometer with no precious data science experience to suit their needs.
All this to say, if you have an interest in another area, you must know the players and their work in that area while simultaneously knowing your area in math. It is not easy by any means, you are essentially signing up for twice as much work learning your field and theirs, but the rewards are great - as a connector between two fields, you have precious expertise that is very employable across a broad range of industries (my first job out of grad school was essentially providing advice on research programs helping connect different STEM communities to government funding agencies, but I was able to use my connections from that job to get back into research).
Yes, the onus is on you to learn the skills you need to show up and produce on day 1. That's 100% true and it is a lot of work. Having a math phd alone and interest is not enough.
However, there are still plenty of paths from math phd into data science and adjacent fields. Those courses you mention are things that someone with a math phd can 100% self teach, almost certainly to a level of understanding that is stronger than someone coming out of a DS masters. Learning how to interview on those concepts is important but ultimately they are very easy for a math phd once they know what the rules of the game are.
Personally, I spent the last year or so of my postdoc obsessively leetcoding and doing side projects in DS and landed a non-entry level data science position as my first job out of academia at a FANG. This is with a pure math research background totally unrelated to the position.
So it is still very possible. I think painting it like you do is a bit pessimistic and will discourage the wrong people. From personal experience, I saw comments like yours over the past year while I was job hunting and found them very discouraging.
The most important things are:
- have a network of similar people who have also made the transition (recently!) to get good advice and maybe also some referrals. These are the people you met in grad school a few years ahead of you.
- know exactly what type of position you want (or converge quickly) and focus on it relentlessly.
- understand the value you bring and the value others perceive someone like you to bring. Talking to people in hiring positions for different roles is the fastest way to learn what you have that is valuable. Do that as much as possible. Then you can line up how you value yourself with how a hiring manager values you, which will be the happiest result.
- take as many interview opportunities as possible to get that interview experience.
- Work relentlessly to interview better than those people from DS masters or whatever other sources they might come from.
I think this is good advice but also not the only possible path towards completing a PhD in math. I get a lot of fruitful research done while binge-watching television, for instance. The key is that you have to find something that works for you. For some that may be logging hours but I know a lot of people for whom logging hours is a hindrance. There are mathematicians who have to actively hold back from just doing research all the time, and for these types there's no point in logging hours. My point is that if you don't think the work ethic described in this article is agreeable, don't let that dissuade you from pursuing a PhD in math.
> Computer programming work only counts at half the normal rate because debugging isn’t quite the same as deep thought.
This mentality is the reason why most software originating from academia is awful. Productivity is measured by research output (i.e. conference/journal papers) rather than the artifacts (i.e. code/software) built to support said research.
There really is one question that people should grapple with early in their academic path, and think about hard (because it is rarely deliberately posed as a question to students):
Unless you are extremely talented and have some gift for a very specific topic or subfield, I would argue that finding a good environment (e.g. lots of adjacent collaborative research opportunities, and other productive students) and a good advisor is far more important than any particular thesis topic.
What I have observed is that few people at the end of their academic careers (esp. those who decide to do something else later) would say that the topic of their thesis was really all that important. They might have equally chosen some other related topic out of sheer luck or accident.
Far more important were the experiences of:
-- developing critical thinking
-- how to organize your life and work methods while trying to get a research project done
-- how to collaborate with others and use self-initiative to seek out opportunities
-- learning how to pick yourself up after ideas fail, or how to power through adversity and difficulty phases in your work
Almost all academic programs (or just tradition) make it seem like the thesis question you answer is the entire goal of your career. I doubt it.
As I said maybe for the rare few with a calling to be expert in <xyz tensor manifold number theory> etc. etc. it matters. But given that so few people reach the pinnacle of academia, for many, many others, these things are far more important outcomes that you keep with you in life.
Maybe it's a similar logic to "it doesn't actually matter that much what particular app you're building, it's that you launch something and have the experience of doing that, in prep for your next thing".
No but seriously, it was one of the best periods of my life. A lot of creativity, freedom and long nights working on things that are far away from the corporate world.
I feel like writing this in LaTeX is a way of saying, "someone should study this."
My takeaways are the 20 hours a week. That's a hard wall to break through, beyond which you can't really consolidate, I'm running into that, and I don't have any BS to put up with.
On finding an advisor -- why would a graduate even apply to a school without having selected a research interest / target? Undergrads are all research-exposed, so I don't know why you'd subject yourself to more if that wasn't your purpose! Yikes! What happened to the 3-4 LetRecs to even get in the door?!
Mathematician here. For me, writing in latex is the lowest possible barrier to making a document, because I write all my documents in latex. Professors get free and easy static webspace, fancier web services are a headache. It's not a statement. It's the familiar and accessible technology.
Not every Ph.d requires the same amount of deep thought. Experimenting and hypothesizing and writing code is just not the same as logically deriving a new proof, but not any less worthy of a Ph.d.
Of course if you do theory you barely have to work more than 4 hours a day can be counterproductive unless you are embarking on a Lurie or Grothendieck style project and work 12 hour a day.
This article seems great. I can't help but think that given the job market, the "best choice" for somebody in the program can often be to get a masters and go into industry, which honestly seems quite a bit easier than finishing a math PhD. The academia jobs are hard to get and they pay relatively terribly.
What would have really helped me during my PhD, would have been to learn some project management skills. Just basic stuff like tracking where you’ve spent your time and using documents to track various projects. I wasn’t exposed to any of these things until I entered industry and by that time it was too late.
(1) Pick a practical problem in physical science, engineering, medicine, some other field, e.g., ecology, or business, a problem not yet solved.
(2) Get a good solution to this practical problem, mostly using math, hopefully at least a little new math.
For publication, the math is supposed to be "new, correct, and significant" -- get the "significant" part from how much the math does for the practical problem. In medicine, your work might save thousands of lives. In business, your work might make/save millions of dollars. In ecology, you might help clean up a river.
(3) Surprising blunt academic fact of life: Academics is very short on money and understands clearly that very little of their new theoretical work has any near term practical value. So, academics understands that they are short on money mostly because their work is short on valuable, current applications. Indeed, the words and applications are commonly in the titles of journals. So, publish in such a journal.
Some high end universities say that the requirement for a Ph.D. dissertation is "an original contribution to knowledge worthy of publication" -- so, remove all doubt and publish your work.
Advisor? Show them the letter that your paper was accepted.
Done, fully, de facto, essentially, or nearly so.
Where to get the practical problem? Typically outside of a math department and maybe off campus.
For a job?
Long the main source of good jobs for a Ph.D. in math was US national security, heavily near DC. That may still be the case.
If you have any hope of a career that lets you be financially responsible, e.g., support a family and provide for retirement, stay away from academic careers.
Academics just quietly assumes that any recent Ph.D. is just obsessed with research, even apparently useless research, and will pursue it neglecting nearly everything else in life including being financially responsible.
Instead, go into business, e.g., start a business, own 100% of it, and make it successful. The business is likely a sole proprietorship and LLC (limited liability company).
In that business, maybe some math, new, or new to the field of the business, will be an advantage.
In particular, currently take advantage of the not fully exploited amazing technology of current computing and the Internet.
> Academics just quietly assumes that any recent Ph.D. is just obsessed with research, even apparently useless research, and will pursue it neglecting nearly everything else in life including being financially responsible.
This isn't even an assumption. It's borderline enforced. I dropped from the program after paying my own way because my lack of interest in being financially irresponsible put me at the bottom of the bin for good research opportunities.
Academics force you to take 9-10/hr. and work 60 hours a week (of course on a contract for only 30 hours), because if they don't have the constant meat-mill of H1Bs and young financially irresponsible people the entire industry would collapse into itself. It's not unlike tech itself which relies on exploiting the revolving door of new grads to get cheap labor. The only difference is the new grad working for BigCo for sub-par dollars can usually afford to eat. Where I went to school the MAJORITY (not an exaggeration) were on some kind of assistance. Either food or housing. Post-docs were barely able to afford rent.
In that case do not do a Ph.d in math at all. Do it within an engineering department instead, much more funding and you get paid accordingly. And then it is much easier to network your way into an industry job as well.
Frankly, there is nothing in math research that is going to be directly applied to industry. Even applied math is entirely theoretical. To get to the level of your work actually being used, you need to wait for it to siphon through to the engineering department.
> In that case do not do a Ph.d in math at all. Do it within an engineering department instead, much more funding and you get paid accordingly.
The math I listed can be one heck of an advantage in doing research in engineering and other fields. First, get to apply math known poorly or not at all in the other fields. Second, get to do some research in math that in a math department does not look significant but still is regarded as significant for what it does for the problem selected from the other field.
> "It’s your company. Mathematicians each run a small business. We work in malls called math departments. Your company sells theorems. Your advisor is a free consultant for your company, not your boss, not your employee. Ultimately YOU are responsible for the success of your company..."
The adoption of corporate entrepreneur model of academic research is one of the worst, if not the worst, things about the current American academic system. Why?
First of all, who is funding this 'company'? Not you, in all likelihood - it's the tax-paying public who finances the majority of the funding train that keeps academic institutions afloat. What does the public expect in return, ideally?
Well, how about producing highly skilled teachers, that seems important in a technological society, where mathematical skills are in general critically important. Hence, a critical part of the PhD student's job (and the professor's) is the instruction of the undergraduate population. This article doesn't even mention the importance of that role.
Secondly, corporate research is highly secretive and non-collaborative in nature, as the goal is creation of intellectual property that can be sold on for a profit. This attitude is entirely at odds with the most successful acadamic model for making scientific progress, i.e. collaboration and rapid sharing of results. If you spend your time in academia worrying about people stealing your work and not trusting anyone, it's going to be a fairly miserable experience.
My advice is to toss the corporate enterpreneur model in the garbage bin.