Reaching One's Limits

Several times in my life, I’ve hit my limits. Two times have been on tasks of manual dexterity, but they felt quite different. Two times have been in abstract maths.

I

The first time was when I was learning the piano in school. I took all the standard grade exams, and passed without too much difficulty, and then went beyond. I learned two more pieces:

My school had a concert, once a year, in which some students would get to choose, learn, and perform a movement from a concerto, with our school orchestra accompanying their solo. There are no easy piano concertos. I chose the first movement of Grieg’s piano concerto, which is generally considered one of the more approachable ones.

Here’s a video of a person who is not me performing the whole thing. The first movement, the part I learned, goes up to about 14:30.

I was not as good as him. And I certainly was not as calm and collected as him. In fact, I barely remember playing it at all: I remember sitting down, and standing up at the end, and feeling nothing but adrenaline before and after.

By all counts, I played excellently. Some told me I was the best performer in the concert. But I did fuck up a couple of the difficult bits, just slightly, barely enough to notice. I could not go any further.

II

I also learned Un Barque Sur L’Ocean by Ravel.

I could not play it quite like that. My playing was harsher, more aggressive, which was interesting in its own way, but it wasn’t my choice. When you play the piano, your fingers have to make two kinds of movements: they move sideways when travelling from one note to the next, and vertically when playing the note they’ve reached. This sounds obvious, but it is important. Moving more quickly in the sideways direction—so you get from one note to the next in less time—means playing the piece faster, but moving more quickly in the vertical direction—so that the hammers hit the strings with more speed—means playing the notes louder.

What makes this piece so difficult is that you need to play lots of notes in quick succession and have those notes be quiet, or even worse, have them precisely vary in volume from note-to-note. This means putting a lot of energy into your sideways movements, but a very precisely controlled, small amount of energy into your vertical movements. There’s a level of precision that this takes, and I didn’t have it. When I played Un Barque, I hammered the notes out.

There are pieces much harder than Un Barque. They’re in the same book! I didn’t stand a chance against them. That was my limit. I wasn’t too upset; I was at the end of my school career, and going to university, where I expected not to play much classical piano. I ended up playing jazz for most of my time at university.

The second time I hit my limit was learning cryo electron microscopy. I was studying my PhD, and I needed to take some nanometer-scale images of some nanostructures I’d built. “Normal” electron microscopy dries the samples out, and only gives you an outline of their shape. This wouldn’t do: I needed them as they were in solution.

I had to learn a dreadful technique called cryo EM, which involves freezing samples so rapidly that they vitrify: rather than forming crystalline ice, they form a disorganised kind of ice; the water molecules cool too quickly to line themselves up into a crystal, instead they just freeze in place. Liquid nitrogen sits at -196 degrees C, when it’s boiling. It won’t do for this job: the Leidenfrost effect makes it cool down the sample too slowly (when the sample touches the liquid nitrogen, the nitrogen starts to boil off, and this creates a layer of nitrogen gas which insulates the sample from touching any liquid). You need liquid ethane, cooled far below its boiling point in a little bath-in-a-bath of liquid nitrogen.

Once you’ve vitrified your sample, you need to keep it from ever touching air (water vapour will freeze onto it, coating it and making it impossible to image in the microscope) or getting above -120 degrees (the ice will de-vitrify, turning into crystals, which can’t be imaged). You need to keep it in the liquid nitrogen as much as possible, and pass it from bath to bath in the layer of ultra-cold gaseous nitrogen above the liquid.

Then you must put it into a sample holder, and pass that through an airlock into the microscope. You have about thirty seconds between removing the holder from liquid nitrogen, and the sample being destroyed. If you’re wondering “How on earth can you operate an airlock in that time?” I don’t know. I never got it working.

Being good at Cryo EM might have meant being on half a dozen papers, but I just couldn’t get it to work. The technician told me “Yeah, some people never learn.” I was less happy about this limit.

The other two times have been in self-study of material. One was category theory, the other was QFT. I read a book on elementary category theory, which gets to a result called the Yoneda Lemma. The book says something like

“If you think you might be struggling, make sure you can prove this by yourself. If you can’t you’re not ready to move forward.”

I couldn’t prove it. I really did try, but I just couldn’t do it. I gave up. Something similar happened when I tried the end-of-chapter exercises in parts of Quantum Field Theory for the Gifted Amateur.

Some say that you don’t quite hit a wall, things just get exponentially harder. I can agree with this. I expect if I’d had a category theory (or QFT) lecture series, with a real teacher, and in-person discussions, and spent 10h a week on it, I’d have been able to push a little further. But to me, spending 1-2h a week without any instruction, it might as well have been a wall. I moved on to other subjects.

This post was written as part of Doublehaven

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