When learning or doing mathematics, we quite frequently have to create mental images of mathematical objects with eidetic qual- ities as close to that of the images of real things as possible. (If we are duped, as a result, into the belief that mathematical objects exist in some ideal or dream world, this happens only because we want to be duped.) Sometimes such images are easy to create (the concept of “triangle” has for me the same eidetic intensity as that of “chair”, say), sometimes it is more difficult; using a computer tech- nology metaphor, we are frequently reduced to moving the icons for graphic files around the screen instead of seeing the contents of the files.
When it comes to the concept of infinity, how do we control it?
5.3
The Sand Reckonerand potential infinity
There is no smallest among the small and no largest among the large; but always something still smaller and something still larger. Anaxagoras We have to distinguish between thepotential infinityof an iter- ative process which could be continued on and on, and theactual infinity of the output of this process being imagined as completed, encapsulatedand made into an object.4
The mother of all iterative processes is counting, and the poten- tially infinite set of natural numbers is the mother of all potential infinities. A word of warning is needed: we have to distinguish be- tween iteration and repetition. The sun rising in the morning is, of course, the mother of all repetitive processes; but the sun is the same, yesterday, today and tomorrow, while numbers are alldiffer- ent. However, for the bulk of its history, mathematics was a branch of astronomy; without doubt, the extreme precision of repetition of many astronomic phenomena very much influenced the develop- ment of the culture of mathematical rigor. Let us do a small thought experiment: imagine that atmospheric conditions on Earth were, for the last 5000 years, slightly different: a light haze obscured the stars in the night sky (without adversely affecting the climate and conditions for the development of agriculture, etc.). How would mathematics develop? Would it ever reach the stage beyond basic arithmetic and purely procedural geometry, without proofs?
The crucial importance of the potential infinity of natural num- bersas a mathematical problemwas realized early on, and Archimedes wrote a fascinating book,The Sand Reckoner, bringing the poten- tial infinity of natural numbers home to his contemporaries. To that end, he developed an elaborate terminology to describe the num- ber of grains of sand in bigger and bigger spheres, with the radius reaching the Sun and growing further. We have to take notice that
the potential infinity of natural numbers required demonstration; the book was deemed, over centuries, important enough to be saved and copied, so that the text survived. One of the happiest moments in my teaching life involved passing around the audience, in a cal- culus class, a copy ofThe Sand Reckoner(in translation).The Sand Reckonerwas probably the first book ever in the genre of “popular mathematics”—and remains a masterpiece even if judged by mod- ern standards.
The opening line of the book is wonderful:
Some people believe, King Gelon, that the number of sand is infinite in multitude.
Archimedes needs just two more sentences to complete the set- up of the problem:
I mean not only of the sand in Syracuse and the rest of Sicily, but also of the sand in the whole inhabited land as well as the uninhabited. There are some who do not sup- pose that it is infinite, and yet that there is no number that has been named which is so large as to exceed its multitude. Thus the first paragraph of the book already contains the key idea of the solution: we need names. Regarding indefinitely devel- oping processes, as long as we can give names for some of their intermediate instances, which are spread, like milestones, all over the process, we feel comfortable, we are not afraid that we may run out of names.
Archimedes describes an iterative process of building bigger and bigger masses of sand; however, he feels that he is in firm control of the potentially infinite process because, in modern terminology, • The individual instances produced in the process are related by
linear order (magnitude).
• Some instances (objects) are measured (or “counted”) by natural numbers, and this measure is compatible with the order: bigger objects have bigger measures;
• The linear order satisfies what is now known as Archimedes’ Axiom: for every object there is a bigger measured one.5
Nowadays the acceptance of the po- tential infinity of natural numbers is just part of common, everyday culture.
These are the most basic, “atomic”, types of potential infinity, and they are the easiest to handle, for reasons that we have already discussed in Section 4.2: the mental images of po- tential infinity are built on the basis of pre-existing hardwired structures of our mind: order and numerals. For example, we see the potential infinity of time through the potential infinity of the calendar.
Nowadays the acceptance of the potential infinity of natural numbers is just part of common, everyday culture, and children
5.3 The Sand Reckonerand potential infinity 97
(like little Jojo) absorb it at a very early age. In any case I never encountered a student who would question the existence of poten- tial infinity; the issue is whether our students are able to control its simplest manifestations; for example, can they compute the100-th term of the sequence
1,3,5,7,9,11, . . .?
I suspect that, in a young child, a healthy scepticism about the possibility of counting indefinitely might be more a sign of potential mathematical abilities (because she may need to check for herself that numerals, indeed, do not get out of control) than the readiness to accept, already at the20th term, that the dull routine will drag on forever.
In a young child, a healthy scepticism about the possibility of counting indefi- nitely might be more a sign of potential mathematical abilities than the readiness to accept, already at the20thterm, that
the dull routine will drag on forever. We frequently forget, however,
that the potential infinity of natu- ral numbers is already an abstrac- tion, the result of encapsulating the necessarily finite process of count- ing as part of an idealized infinite process. In common teaching prac- tice, the intermediate steps in the abstraction are skipped or taken for granted. To illustrate our careless- ness, consider the following thought
experiment (taken from the bookMathematical Aquarium[288] by Victor Ufnarovski where it is formulated as a “competition style” problem).
Assume that we are given an extremely reliable computer with an eternal source of electric supply. The computer is programmed to print, via an external printer with a un- limited supply of paper and cartridges, consecutive natural numbers:
1,2,3,4,5,6,7,8,9,10,11. . . .
(However, it cannot read its output.) Prove that the com- puter will sooner or later fail.
Indeed, it will fail for reasons of its intrinsic limitations: the computer has only a finite number of internal states (since it has only finitely many memory cells, for example); printing a new num- ber requires a change in some of these states. The computer work- ing indefinitely, some of its states reoccur, say, at moments of time T1 and T2; but then it will print, from time T1 on and from time T2on, the same sequence of numbers, which means that it cannot do its job properly. (Notice that this argument can be viewed as an application of the Pigeon Hole Principle, Section 4.1.) In short, the computer will fail because of the eventual overflow of memory. Of course, the answer would be different if the computer could read its own output and use the paper tape as an external memory device.
Our brains are finite state machines, and we cannot count for- ever not because we are mortal but because our brains are finite.
Of course, you may wish to try to circumvent the problem by inventing ever more elaborate and compact abbreviations, but all successful ones will require recursion (reference to abbreviated names for previous numbers); if recursion is accepted, the count- ing becomes reduced to repeating, again and again, “the previous number plus one, the previous number plus one, the previous num- ber plus one”, which, we have to admit,isthe most compact form of description of natural numbers. This is, by the way, why a famous programming language is called C++: it was developed from the languageC, and in both languages the operation of incrementing, in recursive procedures, of a valuenby1was deemed to deserve a special notation:n++.
Still, we have an in-built facility for thinking about processes and sequences of events, and as long as potential infinity appears as a linear process or a sequence of events (and is interspersed by a “counted” subsequence labelled by numerals), we usually have no trouble in interiorizing it. The sprawling infinity of hyperbolic tessellations (Figures 5.2 and 5.4) is much less intuitive because we soon lose the control of the intermediate steps: even if you know that the pattern of adjacency is the same, it is somehow hard to believe.