Department of Mathematics and Statistics

A Necessary Balance: Alec and Harry Aitken 1920-1935
P.C. Fenton


<<     < >     >>

20.   Mental calculation and memory

A letter from Aitken to his student J.T. Campbell in March 1932 contains a passage that exemplifies his use of non-trivial mental calculation in his research. Setting aside the context, he wanted to know how many terms in the series for $e^x$ are needed to get $\frac{1}{2}e^x$, more or less, for any positive integer $x$.160

Now I give you the following remarkable empirical fact, which I cannot prove. I have no tables by me, but I think we have

$\begin{array}{llll}e^1= 2.7,\quad & e^2=7.4,\quad & e^3=20.1,\quad & e^4=54.6,\\e^5=148.3,\quad & e^6=403,\quad & e^7=1095\quad &{}\end{array}$

or thereabouts, and the halves of these are 1.36, 3.69, 10.05, 27.3, 74.1, 201.5, 547.5 about. Now I note the marvellous facts:

. $\begin{array}{rclc}{}&{}&{}&\mbox{as against}\\1 + \frac{1}{3}1/1! &= &1.33\quad\quad &1.36\\1 + 2/1! + \frac{1}{3}2^2/2! &= &3.67\quad\quad &3.69\\1 + 3/1! + 3^2/2! + \frac{1}{3}3^3/3! &= &10.0\quad\quad &10.05\\1 + 4/1! + 4^2/2! + 4^3/3! + \frac{1}{3}4^4/4! &= &27.2\quad\quad &27.3\\1 + 5/1! + 5^2/2! + ··· + 5^4/4! + \frac{1}{3}5^5/5! &= &74.1\quad\quad &74.1\\1 + 6/1! + 6^2/2! + ··· + 6^5/5! + \frac{1}{3}6^6/6! &=& 201.6\quad\quad &201.5\\1 + 7/1! + 7^2/2! + ··· + 7^6/6! + \frac{1}{3}7^7/7! &= &547.6\quad\quad &547.5\end{array}$

Thus, take the series $e^x$ as far as $x^x/x!$, but retain only $\frac{1}{3}$ of that term. The sum is, to a high approximation, $\frac{1}{2} e^x$.

Similar passages, in which complex calculations are used to support a conjecture, occur in his correspondence – less frequently in published papers – and display an empirical quality rarely found in mathematics. Three days later he told Campbell that he had discovered (‘as I had imagined might be the case’) that Ramanujan had proposed the same result.161 No evidence that he tried to prove the conjecture has survived, if indeed he did.

Aitken’s memory and mental calculation had made him something of a celebrity. He gave demonstrations to a students’ society as early as 1927, in one of which he reeled off the first 707 decimal places of $\pi$. Others followed, delivered with a beguiling combination of reticence and showmanship.162

In May 1933, J.D. Sutherland, a lecturer in psychology at the University of Edinburgh,163 subjected Aitken to a range of tests, mainly of memory. Sutherland was moving into psychoanalysis at that time and, though he spoke about the experiments at a meeting in 1937,164 seems to have lost interest in them and the results were never published. Much later he commented:

[M]emorising seems to get linked up with deep unconscious satisfactions, i.e. the activities seem to tap what the psychoanalysts would call primary process rather than belonging to what we more ordinarily think of as conscious cognitive activity. I am on rather delicate ground here because my chief evidence was a statement that Aitken made to the effect that he had bouts of compulsive factorising and memorising. Thus, when walking along the platform in the station he said at times he had to make an effort to stop remembering or manipulating the numbers of carriages and/or the engines. I felt early on in my contact with Professor Aitken that this whole activity of his was linked up with many processes which could only have been illumined had he been the subject of psychoanalytic study. His own emotional state has been fairly brittle at times and there is, of course, an obvious exhibitionistic element in his performances.165

The experiments and what Aitken recalled of them were explored 30 years later by another lecturer from the same department, Ian Hunter, who met Aitken in December 1960. Hunter’s interests were in memory and cognition and in Aitken he’d stumbled on a gold mine. Reciprocally, Aitken recognized in Hunter one who could appreciate the rarity of his gifts.

Hunter went on to undertake a deep investigation of Aitken’s mental processes but at this first meeting they talked mainly about Sutherland’s memory tests: a certain list of words; a passage devised by F.C. Bartlett (1932) concerning two men of Egulac who travel to Kalama; and the identification of 100 short passages randomly selected from Bach’s Well-Tempered Clavichord, Beethoven’s piano sonatas and Chopin’s piano works (for this last test the answers were composer, work, movement, approximate place etc, and he was given a week to brush up).166

On 10 December, a few days after their first meeting, Aitken wrote to Hunter.

When I left you and Dr Collins that day, I thought of two things at any rate. – those obscure proceedings at Egulac and Kalama, and the list of words which Dr Sutherland gave me to memorize quickly. (I did it in two glances; and he took away the sheet.) Well, I thought about this list as I walked up Chambers Street from the Staff Club, and at once got part of the list
 ... make, woman, friendly, bake, ask, cold, stalk ...
I realized that this was in the middle, and must be preceded by perhaps seven words and possibly followed by ten words or so. At the top of Chambers Street I had three of the immediately preceding words – as I thought,
 ... dead, long, ship, make, woman, etc
But I had a lingering idea it might be
 ... head, long, ship, make, woman, ...
“Dead” or “head”, thought I? Why this doubt, quite apart from similarity of sound? At that very moment the focus shifted towards the other end, and I had this:
 ... dead, long, ship, make, woman, friendly, bake, ask, cold, stalk, dance, village,
 pond, sick, ... angry, needle, swim, go
and in the sound of the last three I recognized the end of Dr Sutherland’s list. I took a bus now to George IV Bridge and went to the top deck. No sooner seated, I had that late gap filled, thus:
 ... dead, long, ship, make, woman, friendly, bake, ask, cold, stalk, dance, village,
 pond, sick, pride, bring, ink, angry, needle, swim, go.
As the bus went down the Mound, passing New College, I suddenly got the beginning, and saw why there had been that hesitation between “dead” and “head”. The first word in the list was “head”! Now I had it:
 head, green, water, sing, dead, long, ship, || make, woman, friendly, bake, ask, cold, stalk, ||
 dance, village, pond, sick, || pride, bring, ink, angry, needle, swim, go
(notice the caesurae which I have marked, ||); and feeling this at once to be 7 plus 7 plus 4 plus 7 = 25, and remembering from 1933 that almost my first observation was that there were 25 words, I felt the complete certainty, which I have at this moment, that I had the list, and in proper order. I do not doubt it.
Well, we are now at Hanover Street ....

One marvels at the longevity and accuracy of Aitken’s memory. The dramatic structure of the letter, so that what might be a dry recitation of fact is transformed into a living narrative, is similar to the mathematical narratives he developed in his letters to Schlapp and Edge, and serves the same function, of humanizing an abstract process. As a subtext there is his evident conviction that what he chooses to say is important, down to what might be considered extraneous personal detail. It is extraordinary that in the course of this intense experience of recollection he could correlate his progress on the two paths he was pursuing, the one through memory, the other through the streets of Edinburgh.

Perhaps most interesting however is the doubt he expressed at the end of the letter, concerning the version of the Egulac-Kalama story Sutherland had given him. His memory of the passage, which he transcribed into the letter, differed in ways he could not reconcile with the version he found in Hunter’s book (Memory: Facts and Fallacies).

The later part of the story mystified me a good deal. I have no recollection that it was seals they were fishing for – as I now see in your book it was. I have a strong geographical sense, having been in many lands, and I distinctly remember wondering where Egulac and Kalama were, whether they were real or fictitious places. I decided that if they were real, these people might be Eskimos, or bordering on that region; perhaps Alaska, or even across the Behring Strait. Now “seals”, I think, would have given such a clue that I should not have forgotten it; but I have no recollection of seals. Is it possible that Dr Sutherland gave me a slightly shortened version?

Hunter doubted it but promised to check with Sutherland. As it turned out Aitken’s confidence was vindicated. What Sutherland had given him was actually the response of a student of Sutherland’s who had taken the same test, but from Bartlett’s original passage, and it contained no mention of seals.167


160  The series in question is $e^x = 1 + x/1! + x^2/2! + x^3/3! + ...$. He needs to know what $m$ should be so that $1 + x/1! + x^2/2! + x^3/3! + \dots + x^m/m!$ is approximately $\frac{1}{2}e^x$.
161  Actually a slightly more refined version of it. Aitken gave Campbell references to Ramanujan and to a partial solution by Szegö on 26 March 1932. See Berndt, Bruce C. and Rankin, Robert A. (eds) Ramanujan: Essays and Surveys (AMS and LMS, 2001), 246-247, for a discussion of this ‘ultimately famous’ problem. A complete solution was published in 1995. (I am indebted to Professor Berndt for this reference.)
162  ACA to I.M.L. Hunter, 2 Sept 1961, 6 Dec 1960.
163  Later Director of the Tavistock Institute, London.
164  See p.458 of I.M.L. Hunter, Large cognitive accomplishments: Aitken revisited, in Problem Solving and Cognitive Processes (1995), 447-462.
165  J.D. Sutherland to I.M.L Hunter, 15 Jan 1962.
166  His score was 95, ‘two blanks and three near misses’. ‘One that escaped me altogether was a bar, or bar and a half, from Bach’s fugue in G sharp minor, Book II of the Well-tempered Clavichord. I was chagrined at this, because at that time I thought I knew the 48 Preludes and Fugues so well from memory that if they had been destroyed, I could have written them all out from memory. As it is, I could no longer compass such a test as this, which I think exceeds any I ever did. The memorizing of π to 2000 decimals is slight by comparison with it.’ ACA to Hunter, 12 Dec 1960.
167  This episode is analysed in detail in I.M.L. Hunter, Large cognitive accomplishments, op.cit., 456-459 and 461-462.
<<     < >     >>