Galton's numbers

Happening to re-read reference 1 the other day, in connection with the ongoing Church story of reference 2, I noticed that reference 8 (here reference 3) appeared to have come adrift. It turns out to have been the source of the snap about the mental arithmetic abilities of one Flinders Petrie, said to rest on an unusual ability to visualise the process with something akin to a slide rule. Specifically from the section on mental imagery, from where I strayed onto the section on number forms which follows. Galton himself is to be found at reference 4.

The start of this section is snapped above any my interest was caught because while I think of myself as as a person both reasonably imaginative and reasonably numerate, fond of numbers and of counting things, I did not think that I thought of numbers in the way described here.

Galton does not claim this visualising ability for himself and relies on the reports of his correspondents, many of them prominent in their own fields. While the details vary, the general idea is that people who have this ability have always had it, that the details do not change with time and that particular numbers occupy definite positions in the visual field, positions which can be related to those of real objects. Visualisations of numbers which could coexist with real objects. 

They and he plot the positions of successive numbers, getting all kinds of interesting shapes and patterns. Shapes which are not necessarily visible in themselves but which are tracked as the subject runs through suitable numbers.

A chunk of Plate I, including some of these forms, is snapped above. Forms which do not appear to include negative numbers and which presumably come unstuck with large numbers. Which all seems rather bizarre, but Galton has been careful and seems quite sure of himself. He started with a small number of correspondents reporting this sort of thing, he then published a report in Nature and a lot more people write to him, gradually coming to the present summary. He estimates that one in 30 men and one in 15 women have this ability, see numbers in this way. All in all, it seems improbable that these reports, that these number forms, are just the inventions of idle minds; they must have some basis in subjective reality.

But is the prominent role of the number 10 reasonable? I thought at first not, but now, given that our counting is organised that way from a very early age, I am more comfortable with it.

Then yesterday, it so happened that I was stuck inside a scanner for half an hour or more, immobile and eyes shut. It was not a noisy scanner and so it seemed a good opportunity to try visualising numbers for myself.

My visualisation of numbers was not particularly strong, but Arabic numbers were stronger than Roman numbers and they were quite florid, perhaps something like the snap above. On any one occasion they were in a particular place in the visual field, usually somewhere centre right, but the position changed from occasion to occasion.

If I wanted to do successive numbers, it worked better if I moved along a bit. If I tried to do one number on top of another, they seemed to interfere with one another, suggesting to me that whatever was going on involved a part of the brain which mapped onto the visual field in a more or less straightforward way. Sometimes the impression was of one of those bulbs which contains filaments for all the numbers, arranged in a block, one in front of the other, with the operator choosing which filaments to activate. For me, such filaments light up red.

Moving along a bit often involved a zig-zag, of which there are some examples in Plate I above. So a number form of sorts, but not a form which persisted from one occasion to another.

A quick try this morning with eyes open, in daylight, does not seem to work for me at all.

A detail

An page 80, Galton says that the eyes are turned inwards and upwards during sleep. I don't know how he arrived at this observation, but it does not recognise the modern distinction between REM and non-REM sleep and does not appear to accord with the work with cats at reference 6, which  I reached from reference 5, turned up by Bing. There are tendencies, but not those suggested by Galton. To be fair, quite a long time ago. Furthermore, in the margins of this paper, there is a suggestion that the movements of the eyes in REM sleep are not correlated with contemporary visual events.

Conclusions

I am clearly not one of Galton's elect. But it is, nevertheless, something to try from time to time. Will my abilities develop?

It will make a change from the alphabet game which kept me busy for a bit, seemingly last noticed properly all of three years ago, at reference 7.

PS 1: a relic of the days when plenty of thoughtful people had quality time on their hands, it not all being drained away into rushing around, television and social media?

PS 2: further to reference 8, I have already noticed at the end of reference 10 our Prime Minister's relatively humble background and at the end of reference 9 his relaxed attitude to presents, perhaps to tickets to football matches or to pop concerts. And I was sorry to read again today of his relaxed attitude to clothes, both for himself and for his wife. It is true that he is not paid much by the standards of bankers or lawyers, that he is not even paid as much as his own chief of staff (which I think bizarre): nevertheless, given that according to recent statistics - very accessible I might say - from the ONS, he earns roughly four times as much as the average (working) voter and given that a lot of his living expenses are paid for him - I don't think that taking all these presents plays well. A leader of the Labour Party, possibly with pretensions to being a socialist, should live modestly, as should his wife. Buy their clothes from M&S and John Lewis. Tinned pilchards in tomato sauce in preference to caviar. All of which led me to wonder whether this sort of greed was an occupational hazard among the wives and partners of important men: Mrs. Starmer is by no means the first to get into the news on this account. But I don't suppose Mrs. Atlee did.

PS 3: checking today that both Mr. and Mrs. Atlee came from families which were comfortably off by the standards of the day. Prosperous if not rich.

References

Reference 1: https://psmv5.blogspot.com/2024/07/raynes-park-on-sea.html.

Reference 2: https://psmv5.blogspot.com/2024/08/church-one.html.

Reference 3: Inquiries into Human Faculty and its Development - Francis Galton - 1883. 

Reference 4: https://en.wikipedia.org/wiki/Francis_Galton.

Reference 5: https://www.physoc.org/magazine-articles/where-do-we-look-while-sleeping/.

Reference 6: Tonic and phasic phenomena underlying eye movements during sleep in the cat - Márquez-Ruiz J & Escudero M - 2008.

Reference 7: https://psmv5.blogspot.com/2021/10/the-alphabet-game-revisited.html.

Reference 8: Keir Starmer faces backlash over early failure to disclose £16,200 in donor-funded clothing: The then opposition leader did not originally specify nature of donation from Waheed Alli - Anna Gross, Jim Pickard, Financial Times - 2024.

Reference 9: https://psmv5.blogspot.com/2024/02/trolley-640.html. The background.

Reference 10: https://psmv5.blogspot.com/2024/07/victorian-haydn.html. The gifts.