Memory: The case of Alexander Aitken
Alexander Aitken was a mathematician from New Zealand, and he’s particularly interesting to me for his cognitive abilities and reports on them. I am trying to design natural techniques for memory and inference, so reports of what it is like and how it works from a person who was quite capable in this regard are quite curious. He provides an interesting contrast to known memory and calculation techniques, so I’ll be musing on this and similar as I compile interesting-to-me bits of information from him and close sources.
Born in 1895, he was not interested in calculation until a good bit later. This is quite nice, as it means he has a better idea of the mechanics. Savants that learn and practice from early childhood are less interesting to me because of this.
Arithmetic in primary school, since I recall hardly anything about it, must simply have bored me. Possibly, I wasn’t taught well. Maybe I simply accepted what the teacher said and did it. When I went up to secondary school, with a scholarship from primary school, I was disappointed at my arithmetic mark. I found that it was only 143 out of 200. I lost 57 marks and if I’d had these I’d have been very high up. As it was, I got a scholarship in spite of that.
After a lesson in algebra, he was suddenly quite curious about calculation, resulting in a great deal of practice in subsequent years.
The master chanced to say that you can use this factorization to square a number: a² - b² = (a + b)(a - b). Suppose you had 47—that was his example—he said you could take b as 3. So (a+b) is 50 and (a-b) is 44, which you can multiply together to give 2200. Then the square of b is 9 and so, boys, he said, 47 squared is 2209. Well, from that moment, that was the light, and I never went back.
And so from the age of what might be 13½ years, up to 17½ when I left that school, I underwent what can only be described as a mental Yoga. I tried harder and harder things until, in the end, I was so good at arithmetic that the master didn’t allow me to do arithmetic.
Quite interestingly to me, the practice was of a more natural sort than simply doing a great deal of arithmetic as many mental calculators do. Instead, there was a good bit of abstraction, strategy, comparison of methods, etc.
This certainly happened: but repetitive drill was incidental to adventurous, persistent, and diversified intellectual exploration of both numerical relations and of his own calculative abilities; so there was also discovery of new and enlarging routines and of progressively higher-order relationships among numbers, and among classes of problems.
This suggests a much more natural and intelligent method of calculation than, say, the mental abacus. The abacus is certainly faster, perhaps in large part due to the sheer quantity of optimization pressure pointed at the method and perhaps in part because it is a nicer representation for our brains in some deep way, but I personally don’t care about doing arithmetic fast specifically, I care about all sorts of structured mental inference.
The sort of calculation and approach that Aitken was doing is very interesting to me because of this. In the course of typical thought, there is a great deal of strategy, attempted speedups, and noticing of more and more abstract relationships between content and routine. I mainly want to be better at that! His mental representations are also very nice, as I will describe shortly.
It would also be nice to understand why the abacus is faster. Better understanding of what mental representations are good for what and why is potentially a very large speedup and quality boost to cognition. My favorite example of this is William Thurston, who learned to visualize high-dimensional shapes through a generalization of the inference of depth in stereoscopic vision. His eyes could not focus together, and so he had to infer depth manually in his imagination; this made it easy for him to generalize to high-dimensional shapes.
Bill had a congenital case of strabismus and could not focus on an object with both eyes, eliminating his depth perception. He had to work hard to reconstruct a three-dimensional image from two two-dimensional ones. Margaret worked with him for hours when he was two, looking at special books with colors. His love for patterns dates at least to this time. As a first-grader he made the decision “to practice visualization every day.” Asked how he saw in four or five dimensions he said it is the same as in three dimensions: reconstruct things from two-dimensional projections.
But I digress, so back to Aitken. I’ll speak more on Thurston another day, as he’s a particularly interesting case for improving the quality and breadth of cognition in natural yet surprising ways.
For basic measures, his memory span for random spoken letters (2/sec) was 10, for auditory numbers 13, and visual numbers 15. It tends to be the case that proficient mental calculators have very large memory spans for their domain, so this is not surprising, and in all honesty not very informative. From the same source, though, we have some points that may give insight into how he did what he did.
Aitken’s memory was intimately linked with his ability to discern multiple properties that were interwoven into distinctive patterns. His discernment could work rapidly to produce an unusually rich, densely structured gestalt of properties; and so many things, that would seem chaotic to a bystander, were, to him, embodiments of multiple properties that meshed into an interesting, memorable pattern.
The ease with which he learnt and remembered anything, and indeed whether he learnt and remembered it at all, depended squarely on the meaning and interest it had for him. Thus, whenever something interested him deeply, he was typically able, later, to recall many details of it despite his having had not the slightest conscious intention of committing anything to memory.
This is quite nice in my opinion. Users of memory techniques report that in order to recall something, they have to actively employ the technique. Aitken’s case seems much more natural, with the memory flowing from strong interest in the material, and somewhat dependent on the relationships discerned between the incoming content and what you already know. This is much closer to what I want; the mental overhead of managing and executing a particular technique seems annoying, I’d much rather just have at hand whatever I have been interested in!
And even more curiously, it seems that the experience of committing to memory is extremely and explicitly relaxing, quite unlike the reported effortfulness of memory techniques.
I discovered that the further I proceeded, the more I needed relaxation, not concentration as ordinarily understood. One must be relaxed, yet possessed, in order to do this well. Sometimes one enters a bookshop and there, displayed on the stalls, are various interesting books. One selects, dips, reads, becomes intent, until the stage is reached when all surroundings are forgotten. Afterwards, one leaves the shop and enters the street again, blinking at the light and at the people as if one had come out of an anaesthetic. And so it is here. The one requisite is that a live interest in the subject should fix an undeviating attention. … Interest is the thing. Interest focuses the attention. At first one might have to concentrate, but as soon as possible one should relax. Very few people do that. Unfortunately, it is not taught at school where knowledge is acquired by rote, by learning by heart, sometimes against the grain. The thing to do is to learn by heart, not because one has to, but because one loves the thing and is interested in it. Then one has moved away from concentration to relaxation.
This is also a property that I would quite like for my own memory; effort tends to be annoying, I’d much prefer something smooth. And very interestingly (and conveniently), the state of being “relaxed, yet possessed” is characteristic of classic meditation techniques such as the jhanas. Thankfully I am already passable at the jhanas, so experimentation to figure out what he is doing here will be immediately quite tractable.
In fact, I’d argue that this characteristic of being “relaxed, yet possessed” is pretty close to the key reason why the jhanas are a natural state. I’m even somewhat displeased with Jhourney’s pedagogical materials and public comms, as they seem to mainly emphasize that jhana is a feedback loop of positive emotion, sort of “the opposite of a panic attack”. I think this misses the mark in some very key ways, to the point where there are many states that I’d describe as a feedback loop of positive emotion that are not jhana and are missing some of the properties that make the jhanas genuinely nice rather than fancy wireheading (derogatory). More on that in another article though, I could go on about this for a while. For now, a coarse summary of my position is that the jhanas are wireheading in the sense that they are about good wire management; the positive emotion feedback loop is a (contingent and optional) consequence of the core goodies, not really the main event.
Now back again to Aitken.
One last introspective report is worth mentioning. When he attains the answer to a problem with rapidity, good timing and a feeling of ‘all correct’, then he cannot easily say whether he calculated this answer or recalled it—especially if he definitely knows that he has done this particular calculation before. In other words, the activity may lack experiential characteristics enabling him to apply to it one or other of two mutually exclusive labels. This report is a reminder of a familiar fact: people classify one leading sequence of activity as ‘thinking’ and another as ‘recalling’ and do so by criteria which are not entirely hard-and-fast: these two broad categories, useful though they are, shade into each other. This being so, it is not surprising that Prof. Aitken should sometimes find it impossible to distinguish between recalling and thinking. What is striking is that, in his case, the vague borderline between recalling and calculating should straddle a complexity of achievement which, for most people, so evidently implies hard thinking.
This is quite something. I’d naively expect that the clear distinction between calculation and memory would be quite important to maintain in order to achieve high skill because they seem to be quite basic and different things, natural categories. The surprising shading of memory into calculation and vice versa in this case reminds me of a recent study on how the hippocampus is used for the organization of generic state transitions/computations.
One issue that I have with existing memory techniques is that they load content with meaningless association in order to make it easier to integrate. The PAO method, for instance, requires associating an arbitrary person, action, and object to all 2 (or 3, or whatever) digit numbers, and the linking method encourages making a narrative-ish chain to link together arbitrary content. I don’t like creating meaningless associations in my brain, and much less remembering and depending on them. This is part of why Aitken is so interesting to me. He has the ability and some propensity to associate, but it is not used (at least consciously) for memorization. In the following, he is asked to describe whatever associations he can with the number 7.
“Here is the response to ‘7’: ‘The line of poetry “They passed the Pleiades and the planets seven”—mysteries in the minds of the ancients—Sabbath or seventh day—religious observance of Sunday—7 in contrast with 13 and with 3 in superstition—7 as a recurring decimal “.142857” which, multiplied by 123456, gives the same numbers in cyclic order—a poem on numbers by Binyon, seen in a review lately—I could quote from it.’”
But these are not employed for memory! His method seems to be something else. In fact, he quite distrusted mnemonic methods for reasons similar to me.
Some years ago Prof. Aitken memorized this decimal to a thousand places. He did it as a kind of intellectual tour de force and acknowledges that it ‘would have been a reprehensibly useless feat had it not been so easy’. He ranged the digits in rows of fifty, each fifty being divided into ten groups of five. He then found that if he read over the digits in a particular rhythm and tempo, he could fairly easily memorize a group of fifty digits. ‘The learning was rather like learning a Bach fugue.’
Apart from this rhythm and tempo, there was no ‘interpretation’ of the digits, no devising of mnemonic relations. ‘Mnemonics I have never used and deeply distrust. They merely perturb with alien and irrelevant association a faculty that should be pure and limpid. Our present civilization, not only urban but rural, full of noise and interruption as it is, offers every hindrance to that relaxed meditation upon which the strength of memory thrives best.’
As a result of this learning, and periodic ‘brushing up’, he is able to recite the first thousand digits of Pi and also able to pick up the digits at any given point within the sequence without having to start at the beginning. The writer has tape-recorded this forward recitation, checking it against a typed copy of the digits, and the following report is based on this recording.
The random access is quite curious to me as well, and seems very much like the sort of thing that would be actually useful to have for whatever intellectual work. Existing mnemonic techniques aren’t as good for this, typically requiring a traversal of a mental path of some sort.
Apparently there is no visualization or really any sort of particular sensory representation of the digits while encoding.
In subsequent discussion, the procedural difference between forward and backward recall emerges. In forward recall, no visual imaging occurs. ‘Seeing would put me off. If I were forced to visualize [either here or in calculation], I would be much slower. The strong auditory-rhythmic pattern carries almost like bars of music. It’s this funny faculty of neither seeing or hearing.’
A funny faculty of neither seeing or hearing is an odd description and probably significant to how his method actually works. This is also a nice quality to have; imagining sensory representations takes effort and bandwidth that could be used for other things, or may result in interference.
Like most skills, there was an initial period of conscious effort and deliberation required for his calculations, followed by a more unconscious mastery and fluency. Of particular note is the cumulative hierarchically organized progression; this seems much more natural to me than the development of the mental abacus or similar. It also seems likely to be more useful in transfer to other domains, as the hierarchical structure developed in such a case seems to be quite typical of any interesting domain.
Although Prof. Aitken cannot recall with certainty the details of his early calculative development, there was clearly that cumulative, hierarchically organized progression which so pervasively characterizes the acquisition of any comprehensive skill, e.g. learning to use a verbal language, to use telegraphic language, to typewrite, to play chess. ‘In the process of constantly extending my ability, during the period when I was doing that, I had to think very much of what I was doing. I found that the gains were cumulative and, so to speak, stratified, in the sense that they formed a deposit sinking deeper and deeper into the subconscious and forming a kind of potential upon which, in certain states, I made drafts at astonishing speed.’
It’s also worth mentioning that his memory was not just good for arbitrary stuff; he typically used it for memorization of music, literature, mathematics, and other intellectual content. Certainly he was capable of recall of meaningless data, but the domains in which he practiced and used his memory were actually interesting in themselves. He memorized Virgil’s Aeneid during his time in high school, which is somewhat unfortunate for me, as he does not give much account of the development of his memory as he does his calculation abilities.
From his memoir, there are a few points specifically about his early memory and development.
“I found that gains were cumulative and, so to speak, stratified, in the sense that they formed a deposit sinking deeper and deeper into the subconscious and forming a kind of potential which, in certain states, I make drafts at extraordinary speed. I discovered, too, that the more I used relaxation, not concentration as ordinarily understood; for concentration, the dinning in by force, is the surest way to put a clamp on your brain, to stop its easy functioning and to lose all your gain. Memory naturally played a great part, and reciprocally memory itself was strengthened in all other departments of acquired learning as well. For example poetry, especially Latin poetry, and music became easier to learn at a few readings.”
The point about the reciprocal development of memory and other skills is especially notable to me, as this seems like a potential basis for transfer and generalization, which seems required for a natural and useful form of memory. It seems he is claiming that this was a key mechanism in his development; strong general capability certainly at least describes the outcome of his development.
He seems to have been an exceptionally strong student for the effort he put in. From the editor’s note in his memoir:
“On his first day of high school, Aitken was dubbed ‘Swotty’ after memorising a Latin vocabulary at a glance. It takes a certain milieu to produce a nickname like that, but it stuck with some affection and went with him to the trenches. Aitken seems only to have minded that the label was so misapplied: ‘Few pupils at OBHS have ever worked less’, he insisted. ‘I played an inordinate amount of fives, I took some part in athletics, and used to leave preparing my lessons until just before classes, walking up the three hills of London Street.’ Nonetheless, in his final year, he took first place overall in the National University Scholarship Examinations by a considerable margin, having come first in Latin, third in both English and Mathematics, and fourth in French.”
Memory specifically was, by his account, the driver of his arithmetical power. From this I figure that his academic performance probably followed from the same.
He thought this ability to put an answer in storage was what distinguished the calculator from what might be called the man in the street. The man in the street forgot the stages in between.
And, as tends to be the case, his mental power was probably in large part due to genetics. Others in his family were apparently quite sharp (and in arithmetic specifically even!), though less is said about their memories. Interestingly to me, Aitken still very much had to develop the faculty with numbers; it was not something that was as early and spontaneous as his memory. What all this means about the trainability and generality of methods derived from his reports (or in general) is unclear.
“As a child, Aitken’s horizons were not to be encompassed by any classroom. ‘Can you spell cat?’ asked the infant mistress on his first day; he preferred to spell words like ‘Invercargill’. Others in the family had been precocious. An uncle had shown an early gift for mental calculation, but there was no money in the family for education. Aitken’s father also had a pronounced facility with figures, mentally totting up the pounds, shillings and pence in the grocery account books, and requiring his eldest boy to do the same.”
More evidence points towards his memory having been developed early and through little conscious control of his own. He recounts that his memories of the war resulted in illness and insomnia which recurred until his death, though decreasing in intensity with time. Presumably, a more consciously developed faculty would not do this. In response to a letter, Aitken wrote:
“What a nice letter! You have no idea how it heartened me in this state of smashed nerves in which I am, the pavement, or floor, rising or falling under me like the deck of a ship in a swell… . These breakdowns, Sidney, are a legacy of the war, esp. Sept. 27 1916, when we were thrown against a fourth line defended by barbed wire; dash of 1000 yards, under all the machine gun fire, shrapnel, and high explosive in Christendom! I got further than most, 500 + yards; of the 400 or so who started, 387, say, lay on the ground at halfway; 12 and 1 officer got into the trench—to find it empty! The Germans had vacated to shoot at us the better from their support line. This was 2.25 pm. I waited, badly shot in arm and foot, until 8 pm, and crawled 500 yards (4 hours) to our lines. To this I attribute the smashes of 1927, 1929, 1934, 1941, 1948, and now this. This last has not been so bad; 1927 was perhaps worst, 62 days of almost complete insomnia; 1929, 42 days; 1941, 21 days of complete insomnia. In such states the mind works at 100 times (more) normal speed; and my desk is littered with pencilled stuff written at 1 am, 2.30 am etc.”
One silver lining for my interests in developing my own memory is that he seems to at least implicitly regard his memory as a result of practice. Less is said about deliberate development, but still:
“There is much to be said for leaving memory alone, for letting the faculty of forgetting perform its natural diffusive function. There are dangers in an intensive discipline of memory training, and I should not advise anyone to try it. It is not the brain, but the whole personality, even to the remotest parts of the physical and nervous system, that participates; and if ‘Brother Ass’ is ridden too hard, the rider risks a severe fall.”
Hopefully I may figure a way to train my own memory without the downside potential he warns of. I am unsure if the naturalness, comprehensiveness, and ease of his memory inherently goes hand in hand with potential disaster, but I suspect (and hope) not. Perhaps the best of both worlds can be obtained; as a proof of concept, those who use mnemonic techniques have the choice to actively deploy their memory or not. A method or training that still develops naturalness and ease may be tricky, but the optionality available to most mnemonists could at least be a first line of defense against the ills that plagued Aitken after the war.
With that, I’ve about finished. I suspect there is more insight to be had from a complete read of his memoir and his mathematical work, but as far as easily accessible views into how his mind worked, I think I’ve exhausted my search. I have much more speculation to do and experiments to run, but those will remain in my personal notes for now.