Propuesta de Salud Ocupacional en Foxter s.a

Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed. Blaise Pascal My case study of interiorization of actual infinity was suggested by a recent paper by Weller et al. [139].2_{The authors analyze stu-} dents’ approaches to the resolution of a classical paradox:

Suppose you put two tennis balls numbered 1 and 2 in Bin A and then move ball 1 to Bin B, then put balls 3 and 4 in Bin A and move 2 to Bin B, then put balls 5 and 6 into Bin A and move 3 to Bin B, and so on without end. How many balls will be in Bin A when you are done?

Instead of trying to resolve it on the spot—either as a mathe- matical or pedagogical problem—I suggest to make from it three new problems (with, possibly, quite different solutions). Please no- tice that I do not care about solutions for the new problems; their only purpose is to shed some light on the old one.

The first problem deals withindistinguishableballs: Suppose you put two tennis balls in Bin A and then move, at random, one ball to Bin B, then put two new balls in Bin A and move a random ball from Bin A to Bin B, and so on without end. Will Bin A be empty when you are done? [?] Still, this is a

good problem: try to solve it.

I love this form of the paradox because it can be shown that every individual ball ends up in Bin B with probability1. [?] But Why?

does that mean that Bin A is empty?

The second formulation is a continuous version of the indistin- guishable balls problem (we first think of balls as molecules of wa- ter, and then, as physicists do, ignore the molecular structure of water and think of it as a continuous matter).

6.2 From potential to actual infinity 119

Suppose you have two tanks A and B, you pour water in tank A at constant rate; meanwhile, water leaks from A to B at smaller constant rate, and so on without end. How much water is in tank A when you are done?

The third problem replaces the balls with quantum particles, say electrons. Then not only individual electrons are indistinguish- able, but their location cannot be specified, so that we can talk only about the expected numbers of particles in Bin A and Bin B.

If these three problems still do not provide enough food for thought, we may recall that we have a number of natural questions which need to be addressed before we put the paradox to students. For example, the Bin Problem, in its original formulation, amounts to computing the value of the expression

{1,2}r{1} ∪ {3,4}r{2} ∪ · · ·

Have not we told the students that infinite expressions, like the sum

2−1 + 2−1 + 2−1 +· · ·

are meaningless unless it is explicitly defined what they mean?3 As frequently happens in mathematics, in order to make a con- cept intuitive, we have to jump to the next level of abstraction; you cannot expect that from students, but a teacher can and should do that for them. Instead of trying to help our students to encapsulate one iterative process which produces a potentially infinite stream of elements into the set ofallelements in the stream, we can start confidently talking aboutarbitrary sets, ignoring as irrelevant the process which produced them.

It frequently happens in mathematics, that, in order to make a concept intu- itive, we have to jump to the next level of abstraction.

The analysis of the examples above makes it abundantly clear that what really matters is that sets are composed of distinguishable el- ements, with every element having an identity of its own. To understand sets asobjects, we need some tools for manipulating them; an object is not an object if we just look at it and do

nothing. Tocompare sets, we need to postulate theAxiom of Exten- sionality(sometimes also called theVolume Principle):

Two sets A andB are equal if and only if each element of either set is an element of the other.4

Notice that the word “infinity” is never mentioned in the definition of equality of sets; the actual process (quite possibly, infinite) of checking, element by element, that they belong to B, is also not mentioned. The Volume Principle is a great example ofabstraction by irrelevanceas discussed in Section 7.8.

This is an exceptionally deep principle: infinity creeps into it through the back door. For example, (potentially) infinitely many

possible definitions of the empty set lead to the same set, THE unique empty set! As Brian Butterworth [155] nicely put it in words,

Although the idea that we have no bananas is unlikely to be a new one, or one that is hard to grasp, the idea that no bananas, no sheep, no children, no prospects are really all the same, in that they have the same numerosity, is a very abstract one.

To which I add that “no bananas” and “no prospects” do not just have the same numerosity, they ARE the same, the equality of their numerosities is a mere corollary.

After that, the question

Is the final set of tennis balls in BinAempty? has a simple answer: YES!

Being brought up in a Hegelian philosophical tradition, I see paradoxes of actual infinity as manifestations of the Hegelian di- alectical transition; as all dialectical contradictions, the chasm be- tween finite and infinite is relative and could —and should—be sublated, removed by change of a viewpoint, the same way as we remove the mould from the cast or scaffoldings from the finished building.

As a teacher, I see my task not as encouraging students to bang their heads against the wall but as providing alternative view- points which eliminate the paradox.

Finally, I wish to mention a remark by Gregory Cherlin: it is likely that comprehension of actual infinity is easier for people whose native tongue makes a clear distinction between verbs for complete and incomplete action.