Fig. 5.4.A pattern of angels and devils based on M. C. Escher’sCircle Limit IVand the hyperbolic tessellation{4,5}. Rendition by Douglas Dunham [28], reproduced with his kind permission.
5.2 From here to infinity
Straszne, ze wiecznosc sklada sie z okresow sprawozdawczych. What really scares me is that eternity is made of deadlines. Stanislaw Jerzy Lec I have seen the future and it is just like the present, only longer.
Kehlog Albran Mathematics, as we know it, is possible only because the lan- guage processing system in our brain provides us with powerful mental tools for dealing with potentially infinite sequences of men- tal objects. At the interface with mathematics, we can identify two reasons for that.
The first one is that our natural language is potentially infinite. We discover infinity not when we look at the stars in the night sky;
Fig. 5.5.The underlying tesselation of M. C. Escher’sCircle Limit IVis the hyperbolic tesselation of type{6,4}: it is made of regular hexagons (6- gons), with4hexagons meeting at each vertex. In this picture, if one cuts every hexagon by diagonals into6triangles (thus separating devils from angels), one gets a hyperbolic mirror system corresponding to the set of palindromes in the Coxeter language with the alphabet{a, b, c}and basic equivalences
aba≡bab,acac≡caca,bcbc≡cbcb.
(Equivalence classes of synonymous) words of the Coxeter language can be used to label the angels and devils, giving us control over the sprawl- ing chaos.
The tessellation of type{4,5}in Figure 5.4 consists of hyperbolic squares (or regular4-gons), with5squares meeting at each vertex.
Rendition by Douglas Dunham [28], reproduced with his kind permis- sion.
not when we see railway tracks merging at the horizon; mathemat- ical infinity is usually first encountered when a realization dawns on us that the purely linguistic exercise of reciting numerals pro- duced by certain fixed and entirely linguistic rules, will apparently never end. I remember how excited and shocked was my son, then four years old, when he found himself on this endless numerolog- ical treadmill. Recently, I relived these memories when I watched the charming French documentaryEtre et Avoir [422], where in a similar scene, the teacher, Monsieur Lopez, nudged the puzzled and somewhat sceptical child, little Jojo, into counting on and on.
The second reason is that the language processing modules of our brain are built to deal comfortably with potentially endless lan- guage inputs:
5.2 From here to infinity 93
. . . nor the ear filled with hearing. . .
Even more importantly, language processing is predictive, in that we subconsciously try to guess the next word. This is why all kinds of limits at infinity, completions and compactifications are easier to comprehend when they are represented by sequences or words. The examples are abundant; we have all encountered infinite decimal expansions like
π= 3.1415926. . . , and even trickier ones:
1 = 0.9999. . .
If you compactify the integers in the2-adic topology, you come to2- adic integers which can be conveniently represented by sequences of binary digits infinite to the left:
−1
3 = 1 1−4
= 1 + 4 + 42+ 43+· · · (by the formula for a geometric progression) = 1 + 100 + 10000 +· · · (in binary notation)
=. . .1010101
Or you may wish to use continued fractions to represent the golden section as the limit of an iterative algorithm:
1 +√5 2 = 1 + 1 1 + 1 1 + 1 1 + 1 1 +· · ·
Mathematics is about the reproducibil- ity of our mental constructions, hence aboutcontrol.
My own first clash with infinity happened at the level of grouping objects in counting, which is, I now suspect, something very similar to bracketing or parsing. I remember my extreme discomfort when, as a child, I was taught division. I had no bad feelings about dividing10apples
among 5 people, but I somehow felt that the problem of deciding how many people would get apples if each was given2apples from the total of10, was completely different. (My childhood experience is confirmed by experimental studies, see Bryant and Squire [154].) In the first problem you have a fixed data set: 10 apples and 5 people, and you can easily visualize giving apples to the people, in rounds, one apple to a person at a time, until no apples were left. But an attempt to visualize the second problem in a similar way, as an orderly distribution of apples to a queue of people, two
apples to each person, necessitated dealing with a potentially un- limited number of recipients. In horror I saw an endless line of poor wretches, each stretching out his hand, begging for his two apples. This was visualization gone astray. I was not in control of the queue! But reciting numbers, like chants, while countingpairs of apples, had a soothing, comforting influence on me and restored my shattered confidence in arithmetic.2
As soon as I started to consult the literature, I discovered that some of my observations had been made before. In this particular case, Frank Smith had already used the expression “mathematical chant” in his bookThe Glass Wall[129].
As I have said on many occasions in this book, mathematics is about the reproducibility of our mental constructions, hence about control. We want to control the mathematical objects we create, we want to be able to deal with them as with real life things, we want to performactions.
The mother of all iterative processes is counting, and the potentially infinite set of natural numbers is the mother of all potential infinities.
I remember that, at the age of three or four, I (like many children of that age) had a so-called eidetic imagination: I could close my eyes and see things at will, with all their details and colors, almost indistin- guishable from real things in real life. I could see a car and I could open the door in this car—and it opened as a real door in a real car. Later this disappeared: my brain learned to save resources and compress the images (with loss of data, as al- ways happen in compression) into more manageable, compact and easy-to-store formats.
Anna, my wife, told me that her last eidetic episode was a age of 10: her parents sent her to bed and did not permit to finish the book she was reading. Anna glanced over the last two pages without having time to read a single word. In her dream that night, she read the two pages. In the morning she checked the book: her reading in sleep was correct, including a word she had never encountered before, but guessed its meaning.3
Anna Borovik nee Vvedenskaya
aged 10
The nature of eidetism still remains a mystery. I was alarmed to read in Lorna Selfe [224, p. 112] that
Eidetic imagery was once thought to be a normal stage of development in all children and therefore to be related to other facts of early cognitive development, such as sensory rather than verbal modes of encoding experience, and con- crete rather than abstract modes of thought. However, re- cently the question has been raised as whether eidetism is a normal phenomenon with adaptive significance or whether it is essentially maladaptive and a direct manifestation of brain pathology.
It was written in 1977, and I would like to know whether the med- ical assessment of eidetism has changed.