Of Math and Memory, Part 1
Memory is not sexy in mathematics.
“Rote memorization” is the most degrading slur you can fling at a math class. “Reciter of digits of pi” is the most awful caricature of mathematicians in the public eye. In grad school, the cardinal sin is to read a paper with a focus on memorizing names and results: we are bombarded with exhortations like if you learned the Arzelà-Ascoli theorem deeply, it would be impossible to forget. Apparently, if you really understand mathematics, everything (down to the accents on the names of 19th century Italian mathematicians) would be so natural as to render rote memorization completely unnecessary.
All these attitudes can be quite detrimental to the young mathematician who, at the end of the day, needs to memorize an enormous amount of arbitrary data in order to get up to speed in their field. In this post, I will tell some archetypal stories about how memory, especially short-term working memory, is perhaps the scarcest resource in mathematical work.
In a future post, I will attempt to provide some solutions to address this scarcity.
Have you ever tried to copy a phone or bank account number from one place to another, without the benefit of Ctrl-C? You stare at the number for ten seconds, repeating it back to yourself in a rap-like rhythm. That sick beat, you hope, will help you remember an extra digit or two.
Conjure up that feeling of impending doom as you repeat those numbers back to yourself, knowing full well you can’t move 10 digits in one go. That’s the feeling of not having enough working memory. It’s the same feeling in each of the following scenarios.
These are not technically true stories, but they are all pieced together from literally true events.
A brilliant analytic number theorist is half-way through a riveting talk on the distribution of low-lying zeroes of L-functions. About three-quarters of the way through the blackboard space, the speaker finally switches gears from giving motivation and carefully treads into a long, technical calculation. Every Cauchy-Schwarz application and Fourier transform is clearly explained and surprisingly simple, until –
The speaker reaches the bottom of the blackboard and begins erasing. You can almost hear the collective sigh of despair as most of the listeners think the same thought.
We’ve reached the end of the line.
After that half of the calculations are erased, only a handful of senior mathematicians who know the subject inside and out follow the rest of the talk.
Three mathematicians are throwing around ideas in a meeting. One is suddenly struck by inspiration, and starts explaining how to carry out a tricky change-of-variables. Another joins in with excitement, quickly catching on and offering a crude approximation which simplifies things significantly. All of this is happening in the air, so to speak. Writing things down would severely hamper their progress.
The third person, a younger graduate student, has a number of questions about the equations everyone’s keeping in their heads. The first time they ask for clarification, they are reminded gently that all calculations are in characteristic . The second time, they are informed of a standard fact about eigenvalues of random matrices, and given a minute to catch up.
The third time, they can’t remember whether was defined in the Fourier domain. They don’t ask.
In the next meeting, there are only two mathematicians.
I’m out for lunch, and need to attend a seminar talk afterwards. My weekly meeting with my PhD adviser is two hours away, and I haven’t made any progress this week. Away from pen and paper, I rack my brain and scrape the bottom of the proverbial barrel for any stray thought that might be worth presenting to him.
By some miracle, a casual remark during lunch sets off a series of revelations. I begin methodically working out the details in my head, getting more and more excited that I’m onto something. I completely ignore the seminar talk, running back and forth over the calculations in my mind. I get more and more confident that it works.
I walk into my adviser’s office and try to explain the idea to him, only to realize that I’d forgotten an essential intermediate step and mixed up two important variables. I get up to the board and attempt to work things out from the beginning, but I’m so flustered by this point that I keep forgetting what I’m doing.
We spend the hour going back and forth on minor technicalities, trying to see if there’s anything to my idea. In the end, my adviser becomes pessimistic that there’s anything at all and gently shoos me out for his next meeting.
When I get back to my office afterwards, I pull out pen and paper to try to salvage the idea.
I figure out all the details in fifteen minutes.
It is difficult to collaborate with someone with significantly more or less short-term memory. Someone with more will appear to skip ahead three steps at a time, and you will continually feel in their debt for asking them to explain details. Conversely, someone with less will often ask you to rewind and write ideas down that you find inessential.
It’s difficult to read a mathematical paper without a good short-term memory. A reader who needs to keep referring back to the statement of Lemma 4.3(a) does not have the mental capacity to think about the big picture. If the paper is improperly structured, introduces clumsy notation, or is liberally sprinkled with abstruse citations, trying to follow it can feel like taking a forgetful random walk. How many times will I flip back to the conventions section before I remember the difference between and ?
It’s difficult to either follow or give a mathematical talk without a good short-term memory. An audience member can get lost by zoning out briefly and losing track of an important definition or theorem statement. A speaker who doesn’t remember the contents of their slides constantly reads off them and has no attention to pay to the audience. Next, an audience question about a previous slide breaks the artificial flow of the talk and causes a minor catastrophe.
People often worry that they cannot do mathematics because they are not clever enough. This is a very serious worry, because as far as we know everyone is born with a certain amount of clever and nobody really knows how to get more.
I think people should instead worry they cannot do mathematics because their memories are too poor. And I think this is very good news, because memory can be trained, and deficiencies in memory can be optimized around.
To be continued…