[The first in a sequence of retrospective essays on my five years in math graduate school.]
My favorite analogy for graduate school is the gravity turn: the maneuver a rocket performs to get from the launch pad to orbit. I like to imagine a first-year graduate student as a Falcon X rocket, newly-constructed and tasked with delivering a six-ton payload into low Earth orbit.
Picture this: you begin graduate school, fresh as a rocket arriving at Cape Canaveral and bubbling with excitement for your maiden voyage. Your PhD adviser, on the other hand, is the Hubble Space Telescope. Let's call her Dr. Hubble (not to be confused with the astronomer of the same name). Dr. Hubble is ostensibly the ideal guide for your first orbit insertion. After all, she is famously good at staying in orbit - she’s been up there since 1990.
But problems quickly arise as you probe Dr. Hubble for advice on how to approach the launch. Namely:
- She left Earth more than thirty years ago, and space technology has since been completely revolutionized.
- She states all advice at an extremely high level with birds-eye-view detachment, observing, as she is, from a vantage point a thousand miles overhead.
- Most fatally, the Hubble Space Telescope vessel does not include the lower-stage rockets that brought her into space. In fact, she doesn’t include large engines of any kind. Her thirty years of experience free-falling in orbit will do you very little good until you break out of the stratosphere.
The problem is even worse than this, however. It is not that Dr. Hubble, despite her best intentions, gives outdated advice. It is not even that Dr. Hubble cannot consciously articulate all the illegible skills she’s reflexively performing to stay in orbit. The problem is that even if you could perfectly imitate what Dr. Hubble is doing right now, you would likely still crash and burn.
What I didn’t understand going into graduate school is that academic mathematicians are often working in a state akin to the free-fall of orbit. The Hubble Space Telescope remains in orbit around Earth because it travels horizontally so quickly that, even as it’s continuously accelerating towards the Earth, it continually misses. The laws of physics have arranged it so that it is not possible - barring deliberate sabotage - for her to fall back into a sub-orbital trajectory.
Similarly, a successful research professor is embedded in an intricate system that, as surely as Newton’s laws, keeps her in a state of steadily producing new research. Many of her ground-breaking papers are not one-off productions - they produce sequels, variants, and interdisciplinary applications year after year. She has cultivated dozens of long-time collaborators of the highest level who freely share ideas and research directions, and has the reputation to find more at will. She attends conferences every other month that keep her updated on the leading edge of the field. Every year her research group grows, as if by clockwork, adding a couple graduate students and postdocs to whom she can delegate projects with only the gentlest supervision. As a result, the careers of many other people depend on Dr. Hubble to continue producing research at a steady rate. Every incentive is aligned for objects in motion to stay in motion, and it would take deliberate sabotage to bring Dr. Hubble out of her successful research trajectory.
This is not to say that academic researchers all start cruising in free-fall after they leave graduate school or make tenure. It is perfectly normal for a spaceship that reaches orbit to proceed onto its next adventure after some rest, continuing on to visit another planet or leave the solar system altogether. The best researchers I know are similarly courageous, taking on more responsibilities and pushing past their comfort zones time and time again. I’m merely remarking that once one reaches a certain horizontal velocity in space, it is actively hard to fall back down from the sky.
Contrast this to the sorry state of Dr. Hubble’s new graduate student stranded on the launchpad under the blistering Florida sun. He has no prior publications producing continuous dividends, no access to brilliant and dependable collaborators, no knowledge or intuition about what problems are within reach, no students to farm ideas out to, and no reputation to trade off for any of the above. Above all, nobody else really depends on him, so his motivation to succeed is mainly shallow self-interest. This is particularly hard on him, as there are many things he would do in a heartbeat for someone else that he can’t work up the energy to do for himself. The singular advantage he has over his adviser is youth - a finite amount of extra fuel that he must burn quickly and judiciously like a first-stage booster rocket in order to reach her altitude.
There is a paradox inherent to orbit insertion: rockets launch straight up, while orbit is all horizontal. For some diabolical reason, a spaceship must spend its initial phase accelerating in a direction completely perpendicular to its desired velocity. That reason is called the atmosphere: in order to avoid continuously paying the toll of air resistance, a rocket spends a period of time flying straight up. But any additional vertical motion past the upper atmosphere is wasted motion, so at some point (and sooner is better than later), the rocket starts turning smoothly towards the horizon and accelerating towards orbit. Thus is birthed the smooth quasi-hyperbolic curve known as the gravity turn, the ideal orbit insertion trajectory.
How is graduate school like a gravity turn? For one, it is an enormous error in a gravity turn to try to directly imitate the velocity vector of a ship in space while still at sea level. Regardless of its power, a rocket launched horizontally will quickly nose-dive into the Atlantic. Similarly, a student can rarely succeed in graduate school by solely imitating the activities of established researchers. The student must engage instead in certain activities, such as studying fundamental background material and actively networking, that are mostly orthogonal to a research professor’s day-to-day.
For another, it is an equally enormous error to dip your nose cone towards the horizon too late, and spend too much fuel accelerating vertically. Once you break the atmosphere, all excess vertical velocity is wasted motion. At some point during graduate school, the student must transition away from activities that only grant temporary altitude. Becoming knowledgeable gets you to a great place to start doing research at a higher level. But spending too much time studying without attempting original research renders you a mere encyclopedia. Taking classes, networking, applying for fellowships, and going to student summer schools all follow the same principle - there is an appropriate amount to do, past which they increasingly approach wasted motion as far as getting into orbit is concerned. (Of course, if you enjoy any given activity intrinsically, then by all means continue to do it as much as you want.)
An additional consideration is that, while the gravity turn is the most technically fuel-efficient method of orbit insertion, not everyone who arrived in orbit took this most efficient path. In every department there are superstar students who were outfitted with nuclear reactors in place of conventional rocketry, and these folks get to space by pointing their nose cones in any old direction and blasting off. If you’re such a person, just blast off; calculating the optimum gravity turn curve might be the real wasted motion. Also, many of your professors will likely have fallen in this rarefied category in their own graduate school experience, so their advice on efficient gravity turns will be entirely theoretical in nature.
It is worth remarking though, that even a nuclear rocket might learn something useful from practicing the gravity turn maneuver. Just because you have an easy time leaving Earth’s atmosphere and have no need of finesse, doesn’t mean your travels won’t land you on Venus someday. And breaching that monstrous atmosphere will take every ounce of efficiency you can muster.
A natural question remains: if many graduate school activities only count for temporary vertical altitude, what constitutes horizontal motion that is useful for permanently entering orbit? Examples include:
- Producing good research, as every nice paper you write continues to pay dividends year after year.
- Becoming an attractive collaborator, partly by acquiring enough reputation that people are willing to work with you, and partly by being productive and pleasant enough that they stick around.
- Learning to support the research of others, as much of your potential impact lies not in personal contribution, but in the network effects accumulated from being a positive community member.
This last skill begins at the very start of graduate school, where the biggest immediate impact you can likely have is facilitating your adviser’s and other collaborators’ research.
I will close by reminding the reader that the gravity turn maneuver is not a truth delivered from up high that holds for all time across all circumstances, but an engineered solution to an inelegant and ever-varying practical problem. Launching from a moon base, for example, does not require a gravity turn at all because the moon has no atmosphere to fight against. There, you could comfortably reach orbit by blasting off almost horizontally from the lip of a crater. Only you know exactly where you’re launching from and the thrust-to-weight ratio of your vessel. Adjust your gravity turn accordingly.
I hope it is a comforting thought that free-fall is possible: that one day through all the striving of graduate school you may reach a position where the system propels you forward in your research and all you have to do is sit back and relax. I hope that on that day you continue to strive anyway.
I'm reminded of the career advice page on Terry Tao's blog. When I first found it many years ago as a student, I wondered why someone like Tao would bother to write about stuff like "work hard" and "write down what you've done" and "be patient" and "learn and relearn your field". Wasn't this for "mere mortals" like me who have to do the best we can with the (relatively) limited brains we've got, instead of prodigies who win IMO gold medals at 13 and get PhDs from Princeton at 21 etc? But for whatever reason this particular nuclear rocket practiced the gravity turn maneuver pretty seriously; and (at least in math circles) we know how he turned out.
I really, really liked this idea. In some sense it's just reframing the idea of trade-offs. But it's a really helpful (for me) reframe that makes it feel concrete and real to me.
I'd long been familiar with "the expert blind spot" — the issue where experts will forget what it's like to see like a non-expert and will try to teach from there. Like when aikido teachers would tell me to "just relax, act natural, and let the technique just happen on its own." That makes sense if you've been practicing that technique for a decade! But it's awful advice to give a beginner.
This post extended my thinking about the expert blind spot. I hadn't noticed that this would apply to things like the tradeoffs involved in an academic career. I remember encountering these pitfalls and getting weird advice about how to navigate them.
Thinking in terms of the gravity turn helped a lot of this click together for me.
It's such a simple, clear metaphor.
I also found it an engaging read. Perhaps because I related to it so well from my own academic background. But as style goes, I think it's solid.
My only criticism is that as a visual piece, the meat of this post comes across as a wall of text. It might have been nice for the author to find ways of breaking it up a little more. Modern online audiences aren't used to reading books anymore!
But that's pretty minor in the scope of things. I think it's basically great as is.
This seems like it's good advice for someone trying to become a career researcher, but is it really best to have so many career researchers? The prototypical physics grad student (more than a couple of my friends are those so I may just have a biased perspective) starts off with courageous ideas about how he's going to push science forward and restructure physics. But then he encounters the rigamarole of the whole process you describe in your post and it stops him from doing what he originally dreamed. He needs to get published. He needs to do original research. He needs to help his advisor and other professors do their research. He needs to do all of that because otherwise he won't be respected enough to actually have a career in physics research. But doing that kind of work isn't why he got into physics in the first place!
So the typical grad student either realizes that accomplishing his goal of restructuring quantum mechanics isn't in line with the practical necessity of having a career or he gets shunted out of academia because there's 100 other students who optimized their behavior towards becoming researchers and they all look better on paper than him. If none of the grad students optimized towards becoming career researchers and instead really focused on what's important to them this problem wouldn't exist, but the incentives are misaligned and it takes just a few defectors to force everybody else into defecting too.
The method you analogize to a gravity turn is highly optimized to turn grad students into career researchers, but it isn't optimized at all to push science forward in any meaningful way. The gravity turn analogy romanticizes the whole career researcher situation. Playing the game: becoming a respected researcher so you can earn a paycheck, have the respect of your colleagues, and occasionally do some effective work, that's not the dream.
I'm confused about the claim that the academic process is at all misaligned with his original dream. Isn't doing original research and getting published the clearest path - though perhaps not the only one - on the way to the goal of restructuring quantum mechanics? Isn't helping his advisor and other professors do their research one of the best ways of learning the ropes in the meantime? Isn't acquiring the respect of your colleagues exactly the path to having a whole community and field at your back to effect those paradigm-shifting breakthroughs, instead of going it alone?
They may or may not be instrumental in achieving the original goal, but they're not the goal and certainly not the envisioned process. That's regardless of whether the original process was ever realistic.
In particular the process toward "having a whole community and field at your back" is about 95% politics, not research, and requires a very different mindset and skill set than actually doing research.
Is "politics" here the mind-killer, or does it just mean getting to know people and participating in the research community? Because unless you're going to be a hermit and produce jewels that the world will spontaneously beat a path to your door for, the latter is an integral part of not just a career, but of a life.
I'm referring to political skills. While many of these are also generally useful in life, many (often the same ones) are primarily of value for negotiating moral mazes in ways that are detrimental to interpersonal relationships and sometimes even society in general.
It is often true that the latter skills are an integral part of life, but that doesn't make them any less distasteful to someone who wants to make the world a better place in some way.
No I don't think the academic process is aligned with making paradigm-shifting breakthroughs. Scott Alexander wrote a good piece that address this question. His purpose was to rebut the notion that modern scientists are way less efficient than their historical counterparts. I generally agree with his conclusion that the modern academic research apparatus isn't hampering scientific advancement in any way that would affect the trendlines. Yet I think he also cites a lot of good evidence which rebuts the opposite notion: that academic research has done anything positive for scientific advancement. Although Scott himself doesn't come to that conclusion.
Most of the examples of paradigm shifting work I can think of came from giving people who were very smart a large stipend of money to live off of and allowing them to research what they wanted (Newton, Liebniz, even Einstein counts as working as a patent examiner essentially gave him a stipend and an office where he got to do thought experiments). The other similar effective method is getting a lot of smart people working together, give them a bunch of money of course, and also give them a goal to accomplish within a few years (i.e. Manhattan Project, cryptography protocols).
Money and smart people seems to be a good baseline for what's required for scientific advancement. Academic research has a lot of money and smart people that's for sure! But it also has a lot of other features, the features you describe in your post, and it's not clear to me that they actually do anything. Based on historical evidence it seems that if we gave research grants to smart and personable university graduates and gave them carte blanche to do with the money what they wished that would work just as well as the current system.
This thought is not unique to you; see e.g. the French CNRS system. My impression is that it works kind of as you would expect; a lot of them go on to do solid work, some do great work, and a few stop working after a couple of years. Of course we can not really know how things would have turned out if the same people had been given more conventional positions,
I believe that you are correct in your observations regarding graduate level education. Allow me to offer my experiences in undergrad engineering:
In my undergraduate experience, I was of the opinion that a lot of professors simply didn't bother to put in any effort. Countless times I would visit a professor's office to be met with the physical manifestation of executive functioning deficits. Frankly, I expected better from professors who required that their students stay on top of 18 credit hours of engineering courses every semester. I sometimes would hope that this double standard was, in fact, malicious. At lest then I could still maintain that these professors were intelligent, albeit aloof (a phenomenon which I may discuss in a future post.) Unfortunately, I don't think this is the case. I eventually settled on an approach of attending lectures, not to listen, but rather to study and complete homework on a fixed schedule. As somebody with diagnosed (mostly) non-social ASD, I eventually adapted to this environment by developing strategies to overcome my own executive functioning difficulties, so why couldn't professors do the same? In several STEM courses, it took 3-6 weeks for professors to return graded exams, at which point it was far too late to actually implement fixes for my shortcomings on that particular exam.
- I might add that a lot of these experiences were relegated only to the engineering department. I took several elective courses in political science and international affairs. Much to my confusion, these courses offered everything that I was hoping to experience in the engineering program. I won't bother to speculate as to why these areas of study had competent, kind, and intelligent professors. Perhaps somebody else (a liberal arts STEM double major?) could shed some light on this phenomenon.
It seems as though undergraduate professors claim to do two things: teach courses and publish research. When I looked into the latter, I found that most (all but one) of the faculty in my department hadn't published anything since their dissertation (with one exception, a professor who "co-wrote" some 15 papers a year - with no obvious contributions, I might add.) So if STEM professors aren't spending significant time on lectures (it seems to be common practice to recycle lecture slides that were concocted circa 2000, usually by other professors) and aren't publishing research, what the hell are they doing? I think that your post makes a very good analogy for this "coasting" phenomenon that is so prevalent in academia. Imagine my shock when 90% of undergraduate professors were previous graduates of my university who floundered in a professional environment and decided to phone it in for the rest of their careers as tenured faculty...
Frankly, I'm not sure how to remedy this state of affairs. I am now of the opinion that professorship (in STEM, at any rate) is reserved for washed up and failed academics, a complete waste. I don't necessarily think that this is a wholly systemic failure, mostly because I had some good experiences with professors. Is this a case of some bad apples that clutter the university system? If so, what hiring practices could dissuade these bad apples from joining the system? I hold the opinion that tenure is severely detrimental to the university system, since the safety that it provides does the opposite of encouraging professors to excel, not to mention how it is held over the heads of non-tenured faculty and used to take advantage of adjunct faculty.
This reply has turned into a bit of an off-the-rails rant, but perhaps it could be of use to students who are currently in or considering engineering or STEM as a course of study. My advice? talk to students at your prospective institutions and scour ratemyprof (or whatever the modern iteration is called) to get a general impression of your departments faculty (its a lot more accurate than professors would lead you to believe.)
Exploring academia a bit (4 institutions for my bachelors, then 2 more doing boring but useful stuff in labs surrounded by PhDs), my rough impression is that the best teachers are adjuncts at low prestige colleges.
They tend to love their subject and relish the opportunity to teach smart people who are hungry to know things in their area. The fact that they are wasting their life and talents on a job that pays relatively little, and is insecure, and isn't very high status... doesn't bother them much because they just love teaching the subject.
We are using the word "coast" differently - what I meant by coasting is that many of the professors I know would have to actively sabotage their own research groups and collaborators to not produce ~five nice papers a year (genuine though perhaps not newsworthy contributions to the state of knowledge).
Of course, the state of affairs seriously varies with the quality of the institution.
Curated. There are several insightful pieces in this post. Of particular relevance to broader intellectual progress is the insight that academic professionals can be regarded in a state of freefall due to the systems they've built themselves into. Building such systems is the goal of LessWrong, and this post points starkly to such a system in the wild in a way that's quite helpful.
I get the sense that this gravity turn analogy is applicable and useful in contexts beyond academic research (in mathematics). However, other applications aren't coming to my mind right now. If anyone has ideas, I'd love to hear them.
This was just excellent. Excellent prose, excellent insights.
It reminds me very much of the situation confronting a baby lawyer, as well.
I love it. Nice analogy. I considered forwarding it to my son (17). But his interest is more in business and I wonder what the proper analog is there.
To me the analogy still holds in business context.
I feel like during high school I mostly went straight up and got lucky that a spaceship (company) passed by, snatched me and gave me horizontal velocity. And now I'm on a pretty stable orbit. I could get thrown out the airlock but my horizontal velocity would most likely be enough to grab onto another spaceship.
In retrospect there were things I could have done to start getting some horizontal velocity and making my falling down back to earth less likely.
I'm also thinking that our being in orbit makes it harder to give good advice to our kids just like Dr. Hubble is not the best source of advice for a new graduate student.
Was your trajectory like mine ? Or more horizontal at first ? Or totally different ?
I got snatched by a spaceship in low earth orbit when I made a short hop that was not intended to get me into orbit. Part of that spaceship later crashed but I got out and into another one passing by via a line from a crewmate. I later switched to higher orbit spaceships on my own based on lots of spare fuel I had. I think I could have made better use of my fuel but for me, the sky looked pretty chaotic. I think I give my kids better advice than I had (which is basically none). A lot of that advice is to practice hops early to not worry too much about the perfect start as the rocket technology and the spaceships seem to change fast. Think about it: My current job didn't even exist when I was their age. I adapt a lot of Paul Graham's advice too.
Very nice analogy. Also this bit: "even as it’s continuously accelerating towards the Earth, it continually misses" sounds like Douglas Adams' (Hitchhiker's Guide..) explanation how how to fly: aim to fall to the earth, but miss.
Really well written, would recommend crossposting to the EA forum if it isn't already.
I'm responding to your comment here:
Empirically, it seems like, if your goal is to restructure a field, maybe getting a day job working in a patent office is better than trying for tenure straight out?
I grant that these stories might have observational biases, because the whole outsider/underdog thing is just so picturesque, but...
If you've applied for tenure track spots without success for two years, maybe spend 3 years in a patent office writing up and polishing a solid paper or four about what you actually think is important and true? Then keep working at the patent office till the universities come begging you to share some of your halo with them in exchange for cash?
If someone that you don't respect at all says something true, and grounded in data, and so on... you should probably believe it even in SPITE of them having never done any favors for you.
In the same way that argument screens off authority, in good clean reasoning, it also screens off "respect".
Another example: maybe Kariko just tried to go relatively "straight up"... and then no one really funded her, so then in 1995 she basically fell back down "out of orbit", and her career would have ended if she had just left (like a lot of people who bomb out of academia)... but instead she kept plugging away as an adjunct (helped by a soft landing she found in the lab of Drew Weissman). Then just 10 more years of working in relatively obscurity until a breakthrough in 2005.
This paper reported something very important, despite the fact that its lead author never got tenure and had relatively little "respect"... doesn't make the paper not important.
(Even so... after that point, Kariko just needed 8 years more to find a stable place in industry. Then just 7 more years until a global medical emergency comes along where her vaccine technology can save the day, and then she started to win a pile of awards (but still no Nobel?) and drop quotes like:
I'm not saying you can't get quite a bit of prestige and gubmint money from gladhanding your way through academia and juking your education stats with MPUs and other tricks.
I'm just saying that if you optimize for respect in a domain funded by a government bureaucracy you're likely to get respect in the short run... but (to me) it isn't obvious that people who make big intellectual contributions are also optimizing in this same way.
An ideal market can be characterized by economists, and they can prove theorems about its overall idealized properties.
An ideal academia has NEVER been well characterized by meta-scientists, and the best take on this I know has made progress by postulating that science makes rational progress mostly through a judiciously diverse mixture of the insane mistakes of its scientists who follow through on their private hunches even with they individually rationally selfishly... should not.