This will be a short exposition on the research philosophy of one of our PhD supervisors — a philosophy we've since taken and applied to many areas in life, including but not limited to research. The analogy itself is not perfect, but the underlying idea is one that I wish I had learned earlier, and one that I think many who are just starting out in math and/or physics (or really any research-heavy endeavor) could benefit from.
A Quick Aside On Quantum Gravity
I found math by way of physics, and my initial introduction to physics was through the whole "quantum gravity" saga. For those that don't know, the problem of quantum gravity is truly one of the most incredible intellectual dramas that humankind has ever seen. Tomes have already been written on the subject, so for now I'll try summarizing it in a few sentences in case anyone doesn't know about it.
Humankind has been able to synthesize all our understanding of fundamental physics into two theories — quantum theory and general relativity ("gravity"). The journey to only 2 theories is itself a many-centuries long — and completely riveting — story that we won't get into now. The problem, though, is that 2 is more than 1 (as you're encouraged to check).
Even worse, these 2 theories are deeply at odds with each other (mathematically and philosophically). The problem of "quantum gravity" is to find a single theory to unify these last two pesky models and bring all of humanity's understanding of fundamental physics under a single umbrella.
Pretty Cool, So What's The Problem?
Many individuals have written fantastic books describing this state of affairs, and it captivated (and still captivates) generations of physicists, including myself. I spent years dreaming of the day that I'd be the person to finish the task — to make my mark and uncover some of the deepest truths of our Universe, just like the greats before me.
This is all well and good — you need motivation for staying up late in the library studying — but it does come with side effects that we should be aware of. Namely, any of the actual research that would need to be done to make any progress on such a thing is going to be tiny in scale compared to the big, overarching philosophical problem.
To put it bluntly: I didn't want to work on boring, specific calculations — I wanted to gaze out the window in an old oak library and have the singular flash of brilliance that would catapult me into the hallowed ranks of Newton and Einstein (I was like 16, cut me some slack).
This is where the aforementioned PhD supervisor's philosophy comes into play.
How To Eat A Cake
Imagine a delicious cake sitting on a table. Now imagine you've set yourself the following rule: you cannot eat the cake unless you can fit the whole thing in your mouth at once.
What would happen?
Well, you wouldn't eat any cake.
The point of this analogy is to realize that any big lofty goal is fantastic. Quantum gravity, Riemann's Hypothesis, curing cancer. These are all pretty big and delicious cakes. But one thing that young researchers sometimes struggle with is realizing that anyone who has ever eaten a whole cake (or even just a lot of cake) has done so by first taking a small bite.
In fact, in order to take a small bite, you need wedge your knife in somewhere first. And before that, you need to sharpen your knife, otherwise you're going to make a mess of things.
If you try to eat the cake too fast, you'll get sick and throw it all back up, and then you probably won't even want to eat the cake at all any more. And this is a particularly interesting part of the cake analogy. There are two things that come to mind here.
First, trying to eat too fast and then vomiting and not wanting to eat anymore is a direct reference to burnout. So don't do that.
Second, going too fast and puking is also a reference to not learning your fundamentals well enough (because you're in a rush). Then one day you wake up and realize you can't make much sense of the research papers you're reading, or your upper-level classes. Then, if you want to proceed, you need to go back and eat your thrown-up cake (sorry, this is kind of a gross analogy), which is much less fun than just being patient the first time.
Cake And Research Projects
Cake-eating mistakes can occur even in grad school. A wide-eyed and bushy-tailed grad student who only wants to work on the biggest open conjectures often finds themselves floundering. Don't be afraid to work on the not-sexy stuff. Do concrete calculations, prove tiny lemmas, get deeply into the weeds.
This is the research-level equivalent of sharpening your knife. Once your knife is sharp you'll be ready to cut into a cake and start wiggling it back and forth, carving out room to take your first bite.
Wish I'd Known Earlier
This way of eating a cake is obviously the correct, and only way to eat a cake. So why is it not super obvious when it comes to research and/or mastery more generally? In the case of math, physics, etc., I think it's largely due to how we learn these subjects.
When you read a textbook on general relativity, you first get a nice intro to differential geometry, then some nice thought-experiments about elevators, and then you proceed, step-by-step, chapter-by-chapter, exercise-by-increasingly-harder-exercise, to derive Einstein's equations in a linear, logical manner. A good textbook makes it feel like you could have discovered all of its contents, in an equally linear and logical way, and probably in less time.
This is obviously great — it's fun, it's beautiful, and it's a very efficient way to learn things — but it's not how Einstein discovered general relativity. And it's not how you'll discover your own cake, either. So, enjoy the phase(s) of sharpening your knife, of making your first cut, of wiggling it around and realizing you actually don't like this cake that much, of finding a new cake, of sharpening some more, etc. This is the fun of it, and this is how research is done. And you hopefully won't puke all over the table.