Arequipa – Perú
EL APRENDIZAJE POR DESCUBRIMIENTO DE BRUNER.
Bennett’s pleas finally work, and Thompson fulfils his wishes by praising the equation. i. Yes. I fear I omitted to acknowledge your difference equation (n+1, m) =
m(n, m) + (n, m−1). But I can assure you I understood, appreciated and admired it! It has (like the 3 per cents) a “sweet simplicity”.45
CHAPTER 3: THE ENUMERATION OF RHYME SCHEMES
ii. It looks to me as though our series was not unconnected with our old problem of the cell-partitions in the dividing cells on the surface of an egg. The numbers don’t agree; but I rather fancy that is merely because the conditions of the cell-problem exclude certain particular cases. However, I have not found a moment to look farther into this.
iii.
. . .
iv. I have doubtless lots more to learn with regard to ACA’s formulae as well as your own. But it seems clear that your (n, m) Table beats ACA’s binomial coefficients easily, — both for speed and simplicity.
. . .
He is here comparing Bennett’s table of Stirling numbers of the second kind to Aitken’s table with binomial coefficients. It is perhaps a bit unfair to choose these two for comparison, at least when looking at speed and simplicity, as this is not Aitken’s fastest or simplest result. His table of extreme differences, Aitken’s Array, is simpler than either of these two.46 It is, however, no wonder that Thompson chooses the table
with binomials for comparison, as Aitken himself believed it to be superior to his other methods.
3.4
Conclusion
The most curious aspect of Aitken’s three letters on the rhyme schemes is not what is actually in them, but what is not. As we have seen, everything Aitken contributes to this discussion, except for identifying the Bell numbers as counts of rhyme schemes, is older material that he has already published, and yet he remains curiously silent on that point. Not even when Bennett appears to have outdone him does he give a reference to his five-year-old article. His only attempt at defending himself consists of implicitly saying that he has seen it all before. By admitting that Bennett has seen one thing that he himself has not, he is indirectly saying that the rest of it is known to him already.
46To produce thenthBell number using Bennett’s equation, one must perform (n−2) multiplications and (n−2)+(n−1)
Aitken’s remarks that the generating function must be found by everyone ‘who direct some consideration upon the matter’ could be read as an attempt to trivialise Bennett’s results, consciously done or not. The contents of the letter from Thompson that Aitken is responding to can only be guessed at, but considering how he wrote to Bennett that he believed Bennett had outdone Aitken, it is possible that this sentiment shone through when he was writing to Aitken as well.
When Aitken chooses not to drag his old paper into the light, this could very well be because he thinks it is too late. He has by now realised that others have written about these numbers before him, if in a slightly different guise. He might also think it unwise to ‘show off’. Doing so would also raise the question of why he did not mention the paper first time around. He might not wish to let Thompson know that he, in a way, had hidden explanations from him when he first wrote on this mystical number sequence. Had Aitken agreed with Bennett’s comment that ‘marvels melt into normality with understanding’, such a presentation would be rather dishonest. More likely, however, Aitken would not have agreed, and presumably did find these numbers exactly as intriguing as he let on, even with the knowledge he already had. Another matter is that he might have feared that confessing to having given a simplified version might cause offence, something he would be keen to avoid, considering his great respect for Thompson. Another possibility is that the paper has been mentioned, but in speech. This is not very likely, as there would then have been traces of this in the letters. If nothing else Thompson would be inclined to mention it to Bennett, and he does not.
As it is, the matter is probably not very important to him. The correspondence starts out as mere amusement to him, and he does not expect it to get so serious. The conclusion to the whole affair is that Thompson and Bennett think Aitken knows less than he actually does. This probably would not affect Thompson’s opinion of Aitken much, as he knows Aitken already, but it is possible that Bennett ends up with a slightly less flattering impression of Aitken than Aitken strictly speaking deserved.
What this correspondence shows is that papers published in the Notes would have held interest to more than just teachers, if they had taken the time to read it. Aitken does not expect Thompson to have seen this article of his, which turns out to be correct. He cannot have expected Thompson to read the Notes at all, at least not in 1933. Interestingly enough, he does expect Thompson to have read the Mathematical
CHAPTER 3: THE ENUMERATION OF RHYME SCHEMES
Gazette. In his letter dated 19th of December 1938 (3.3.2), he asks if Thompson has seen a certain proof appearing in the English journal.47 He does not mention what journal
this proof is found in, he only mentions the author, so he must expect Thompson to recognise it based on that. As it turns out, he is wrong, because he eventually has to furnish Thompson with the full reference, but this does say something about the status of the two journals, at least in Aitken’s eyes. This is of course one man’s opinion, but it bears remembering that Aitken knows the journal very well, after his long service as the Editor. When he seems to think it a lot more likely that an academic in Scotland has read theMathematical Gazette than the Notes, he is probably correct.
The Mathematical Gazette must have enjoyed a certain status that the Notes did not.