Cocoa, the brain and convergence

Am I the only one who remembers being instinctively put off from a chocolate drink right after suddenly getting the classical reference (apparently correctly)? Why is that so? What will a great ancient athlete do that many anonymous contemporary athletes on green cans won’t? Or is it the juxtaposition of chocolate and malt powder with a name you suddenly found in Aristotle (or a book on Aristotle and friends…)?

At any rate, I recently saw again this:

at the museum, and it brought the entire matter back. Milo: te hace un ex-campeón envejecido quien, para probar sus fuerzas, trata de abrir un tronco con sus manos y es devorado por lobos (in the inevitable long term). This has to be said in a very particular voice to work as a commercial slogan.

* * *

A recent event called to mind the classic The Man with the Shattered World. (Note: my best wishes and hopes for recovery for the surviving victim.) It is a case study of a soldier who survived a severe brain wound in WWII – in fact, it is built around excerpts from the patient’s notebooks on his condition, with commentary by his neurologist, A. R. Luria.

Commenting on this in a logic textbook (A course in mathematical logic, 2nd. ed, Springer, 2010, pp. 35-36), Yu. Manin quotes:

Zasetsky lost the ability to interpret the syntactic devices for organizing meaning: “In the school where Dunya studied a woman worker from the factory came to give a report.” What did this mean to him? Who gave the report – Dunya or the factory worker? And where was Dunya studying? Who came from the factory? Where did she speak?

This is a fairly difficult example composed by Professor Luria, but here is what Zasetsky himself writes:

I also had trouble with expressions like: “Is an elephant bigger than a fly?” and “Is a fly bigger than an elephant?” All I could figure out was that a fly is small and an elephant is big, but I didn’t understand the words bigger and smaller. The main problem was I couldn’t understand which word they referred to.

What attracts our attention is the complexity of Zasetsky’s metalinguistic text describing his linguistic difficulties. The subtlety of the analysis seems incompatible with the crude errors being analyzed. This could be explained by the retrospective nature of the analysis, but the following even more complicated description was written concurrently with the experience of the mental defect being described…

There is also a point here that should be clear to any mathematician, since it is a common part of our working experience; I am surprised Manin does not comment on it. It would seem that Zasetsky has most likely chosen a toy example: that is, a very simple situation that he can examine easily and completely, even in his damaged state, yet which does contain the essential difficulty that he is trying to grasp and explain.

Incidentally, since I did allude to a book that everybody’s philosopher friends like to put down, I recall that it comments at some point (Russell, op. cit., p. 161) that Aristotle was sometimes dealing with philosophical non-problems caused by the fact that his implicit analysis of language ignored or misunderstood relation words. It would be interesting to get a reference to a more detailed argument in support of (or against) this position. Further on Plato (Russell, op. cit., p. 129):

Plato is perpetually getting into trouble through not understanding relative terms. He thinks that if A is greater than B and less than C, then A is at once great and small, which seems to him a contradiction. Such troubles are among the infantile diseases of philosophy.

Zasetsky again (as quoted in Manin, op. cit., p. 36):

I know that when you rearrange the words, the meaning changes. At first I didn’t think it did, it didn’t seem to make any difference whether or not you rearranged the words. But after I thought about it a while I noticed that the sense of the four words (elephant, fly, smaller, larger) did change when the words were in a different order. But my brain, my memory, can’t figure out right away what the word smaller (or larger) refers to. So I always have to think about them for a while…

The text before (“All I could figure out was that a fly is small and an elephant is big”) makes the parallel clearer, if anything.

* * *

Since nearly everybody who knows enough mathematics and reads English can also read mathematical French, I feel I can safely switch to French at this point.

Soit a_0+a_1+a_2+… une série qui ne converge pas au sens usuel. Il y a plusieurs méthodes pour assigner une valeur à certaines séries divergentes : la moyenne de Cesàro, la sommation de Borel, la sommation d’Abel… Bien entendu, il y a des séries qui convergent selon certaines méthodes et pas d’autres.

Question: est qu’il y a des exemples de séries à qui une méthode de sommation utilisée dans la pratique donne une valeur, et à qui une autre méthode (aussi utilisée dans la pratique) donne une autre valeur ? Il devrait y en avoir: ce type de desaccord est une des raisons pour lesquelles on parle de nos jours d’assigner une valeur à une série divergente, plutôt que (comme l’aurait dit Euler) de la trouver (comme si elle existait à priori). En même temps, l’existence de théorèmes taubériens impose des restrictions: deux méthodes liées par un théorème taubérien ne peuvent pas se contredire.

Alors? D’un point de vue historique, il y a la série 1-1+1-1+… (1-1+1-1+… = 1+(-1+1)+(-1+1)+… = 1+0+0+… = 1 ou 1-1+1-1+… = (1-1)+(1-1)+(1-1)+… = 0+0+0+… = 0?), mais tous les procédés utilisés aujourd’hui paraissent lui donner la valeur 1/2. Il y a aussi des exemples très artificiels comme celui dans G. H. Hardy, Divergent series, p. 73, où l’une des deux méthodes considerées ne serait jamais utilisée dans la pratique. Existe-t-il des exemples plus réalistes?


About valuevar

I am a number theorist with side interests in combinatorics and group theory.
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