The Infinitely Small

May 24th, 2009 • Related • Filed Under

The first article on infinity looked at infinity N the natural numbers and the type of infinity most often associated with the infinitely large. This second article is going to view the infinitely small and the problems and paradoxes associated with it.

The infinitely small has some parallels with the infinitely large and also some parallels with the third type of infinity which will be discussed in the final article. First of all the mathematical expression: The infinitely small is represented by the class of numbers known as the Rationals, which I shall call Q.

Simply, a rational number is one which can be expressed as a fraction, so that it is one natural number divided by another. So the Rationals include for example ½, 1/3 as well as the Natural numbers themselves which can be expressed as 2/1, 4/1 etc.

The Rationals are associated with infinite smallness and it is the constant search for the smallest size which goes on and on which makes this type of infinity similar to the infinitely large. We can take some object – assume a line – divide it in half, divide in half again and again until we complete the infinite series and then enquire what it is we are left with. Just how small can things be?

We could also divide the line into thirds or fifths or sevenths or any fraction (and we will do infinity itself in article three!) and the process, though not the numbers would be the same.

The infinitely large gave us great difficulty both in definition and conception in its mathematical and metaphysical form, is the infinity of the small any different? Can we make a coherent mathematical definition of the infinitely small and if so can we find some correlation of the mathematically infinitely small with some feature of reality?

The difficulty of creating or arriving at the infinitely small has parallels in the process of arriving at or creating the infinitely big. Each step, or downsize makes the object smaller, but no matter how small a thing is we can always envisage the object being a half its size and so the process goes on. No matter how many cuts we make there is always smaller still to come. If we make infinite cuts and end up with infinitely small pieces in infinite quantity then we have once again, as with the infinitely large, used the term infinity to create the infinitely small. We have defined the infinite by using the infinite.

Conceptually then we run into the same type of problem we have with the infinitely big. A logical definition is impossible and conceptually we can only imagine a never ending process which is potentially infinite rather than ever becoming actually an infinite.

When we try to put the infinitely small into some kind of real world situation the conceptual problems become even more acute. For example, imagine the problem known as the divided stick. Divide it in half again and again an infinite number of times and what is it we would be left with? It seems we would have an infinite number of pieces of stick infinitesimally thin. But what can that mean?

Do the pieces of stick have width or not. If they have any width at all then an infinite number of them would reassemble to have an infinite length, but if they have zero width then no number of them, including an infinite quantity could reassemble the stick into something with any length. Once again we find our intuitions stretched. Just as we have problems truly comprehending the mathematics of the infinitely large so we find similar conceptualizations problematic with the infinitely small. It seems we can know what it is we are discussing yet fail each time we attempt to define it or clarify what it is we know.

Many have claimed the infinite is unknowable, yet it seems clear that we know what it is we do not know. This is another paradox of the infinite, the paradox of thought about the infinite. We think we know it, but we don’t. We can see that we don’t really know it, but we can’t really give it up. We are in a position where we can acknowledge the infinite but have to accept that we cannot do anything with it. Any attempt to define the infinite is doomed to fail because given our own limitations we can only grasp those concepts which are suitably limited themselves.

But this just gives an even deeper paradox. If we cannot come to know the infinite, then more seriously we cannot come to know that we cannot come to know anything about the infinite. Yet this is what we have just done. We have defined the infinite as that which is beyond definition. This is a contradiction in itself and a limiting factor on what is essentially the unlimited.

Aristotle claimed that the infinite was central to the scheme of things. It would seem as though our own finitude is as much a part of the study as a study of the infinite.

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