Remembering the Things That Fade

A few years ago, I became convinced by a friend that I’d been neglecting pure mathematics and that engaging with it might be fun, so I started reading Terence Tao’s Analysis I, an introductory textbook on analysis for maths students. Tao is, by all accounts, a superb mathematician, and I liked the book right away. His terminology and definitions made the text feel distinct and the material much clearer than it otherwise would have been. Before long, I was understanding various concepts like continuity, limits, and set theory in a much deeper way than before.

But, as often happens, after a while, I took a break from reading the book, planning to return to it soon. A few months later, I picked it back up and, to my dismay, I had a familiar experience: almost everything had vanished. All the rigorous definitions, axioms, and proofs I had previously worked through had simply disappeared from my memory. Use it or lose it, as they say, and I had not been using any of this knowledge in my daily work. Ideas like Cauchy sequences and the axiom of choice rarely appeared in my research, so they faded from my mind as well.

Still, I wanted to understand the material, not just in the moment but in a way that would actually last. Without a solution to this problem of forgetting, I put the book down again and turned to other things.

This is part of a general problem. People often need knowledge that is not immediately useful but is important to keep in the long term. Medical students want to know human anatomy throughout their careers rather than only for a couple of weeks. Language learners want to retain the vocabulary they acquire instead of forgetting it a few months later. And mathematicians, physicists, and other scientists want to remember specific ideas, definitions, models, and theorems with clarity whenever they need them.

But human memory works on the use-it-or-lose-it principle. Ideas that are not being used tend to be forgotten. And from the point of view of memory, a forgotten idea is an idea that might as well never have been learned.

Space repetition

Fast-forward a year or so from the disheartening discovery that I had forgotten most of Tao’s book. Another friend, Paul Raymond-Robichaud, introduced me to spaced repetition, software for creating and reviewing flashcards. In particular, he showed me Anki. Language learners use it predominantly, but Paul’s innovation was to apply it to mathematics. Ideas, definitions, and proofs that would ordinarily be reviewed only once are reviewed repeatedly. When you start forgetting a card, the software notices and shows it more often until it sticks.

I have been experimenting with the software a bit more than a year at the point of me writing this, and my way of engaging with textbooks has changed dramatically. I used to read books in a very global way. I would follow the main arguments and derivations, knowing that the details would likely fade and that the overview would have to be enough.

There are two issues with this. First, it is difficult to retain the overview of a topic without remembering at least some of the details that give it structure. Without those, the overview becomes too vague to be useful. Second, we usually read textbooks because we want the specifics, the subtle arguments, the precise definitions, and the detailed results. Yet at the same time, going over detailed derivations feels pointless if they are destined to be forgotten.

A new way of reading

Reading textbooks with spaced repetition resolves these pain points entirely. Any idea you liked enough to turn into a card will be remembered because the software ensures it does not fade. It turns knowledge acquisition into a sort of ratchet: you can move forward but not backwards. The software will notice when an idea has been forgotten (because you cannot recall the card during a review) and will show it more often until it stabilises. Ideas that are understood well will rarely appear, which keeps the process efficient.

Moreover, the ideas can be highly specific. Entire derivations can be captured in cards, or in sets of cards, and remembered without much effort. After reading a book this way, you do not just remember the overview. You remember the ideas you found most important in great detail. Even while you are still reading, the experience becomes clearer and smoother than without spaced repetition, because the key ideas (like definitions and axioms) are always at your fingertips.

Having solved the forgetting problem, I have been reading Tao’s book again, along with several other textbooks simultaneously, another benefit of spaced repetition. I still take long breaks from each of them. Yet when I return to a book now, it feels as though I had set it aside only moments earlier. The ideas have remained active in my mind through the daily reviews provided by Anki. The old pattern of coming back to a book only to discover I had forgotten everything has gone. It is the forgetting, rather than the ideas, that has vanished.