Wednesday, March 2, 2016

Escaping the Paradox: Heaps, Pirates, M. C. Escher and Language

There is an ancient paradox that goes something like this:

Imagine you have 10,000 grains of sand. If you had that much sand you would call it a heap of sand. Now imagine you remove one grain of sand so that you have 9,999 grains left. Is it still a heap? Yes, it still is a heap.

Remove one more grain of sand. Do you still have a heap of sand?

Keep removing single grains of sand. Each time you do you still have a heap of sand. When you are left with only two grains of sand is it still a heap? No? At what point did the heap of sand stop being a heap? Was it with three grains? or more?

That is the paradox. 10,000 grains of sand are definitely a heap, and if you take away one it is still a heap, but if you keep taking away single grains of sand when does it stop being a heap?

This paradox has plagued philosophers and students for over 2000 years and it keeps discussions going in introductory philosophy classes, which provides much employment to professional philosophers. But before we resolve this paradox I wanted to write a little about the art of M. C. Escher, because some of his most famous art can also be paradoxical. Below is his famous drawing entitled "Waterfall".
Image from www.mcescher.com.
What is paradoxical about this image is that the water at the "bottom" appears to flow "up" until it reaches the "top" where it falls "down" to begin the process all over again. Additionally each bend in the stream appears to be directly over another part of the structure, thereby creating an apparently impossible structure. This paradox, or optical illusion, how ever you want to call it, both confounds and delights all who see it.

While most people consider the work of M. C. Escher, appreciate it, perhaps hang a copy in their house or office, very few stop and take the time to consider why it is a paradox and ultimately escape the paradox.

If we just consider a single part of the image, say just the waterfall part, or one of the bends, by themselves there is no problem, nor a paradox.


Removed from the larger context these constituent parts are not paradoxical. So how did these individual parts go from non-paradoxical to paradoxical when put together?

The answer is that the paradox only exists because we assume more than there is in the image. The image itself is only a two dimensional collection of lines and shading that all together we interpret as a waterfall, a stream, and a brick structure with columns. The structure does not exist in three dimensions. The collection of two dimensional lines and shading create an image of what we interpret to be a three dimensional structure. If the structure really was three dimensional then it would defy the natural order of the universe, but it is not, so it does not.

The paradox only exists because we take each individual part, the waterfall, the bend in the stream, and we can imagine a real three dimensional structure like that, but when we try to fit the imaginary three dimensional parts together, we fail, and thus we have a paradox.

But if we remember that we are only looking at two dimensional lines and shading which only imply flowing water, columns and a brick structure, the paradox does not create a problem, and definitely does not trigger an existential crisis. If we do not make the leap from representation to actuality what we are left with is an interesting picture that does not break the laws of physics and geometry.

So now we can return and resolve the heap paradox. The reason why it creates a paradox is because each individual part is logical and non-paradoxical. There is nothing illogical about considering either 10,000 or 9,999 grains of sand to be a heap of sand. So if we have 10,000 grains and take away one we still have a heap. Much like a single bend in the stream in "Waterfall", it does not create a paradox. But it we then group a series of individual bends together we are left with a paradox.

With the drawing the paradox was created by mistaking a 2D representation for a 3D reality. In the heap paradox the mistake is extending words and language beyond their representations. In this case extending the definition of the word "heap", which is by definition inexact, to mean an exact value. Yes 10,000 grains of sand can make a heap because 10,000 grains of sand would be hard to count and thus for all practical purposes we cannot distinguish between 10,001, 10,000 or 9,999 grains. Hence we use the inexact term "heap".

The heap paradox only remains a paradox if we commit an equivocation and alter the definition of the word to mean an exact number. An exact number implies an exact boundary between "heap" and "not heap", which did not exist in the original definition.

So should we insist on the eradication of all paradoxes from our language? Heavens no. These paradoxes, much like the drawings of M. C. Escher add richness to our language and are the basis to our humor and entertainment. But if we forget the nature of language we might be confronted with a paradox and conclude that the nature of reality is broken, when it is only our understanding that is limited. We must remember that our paradoxes are rooted in a misuse of language. If we remember that then we can escape the paradox and it can be humorous and entertaining, but if not, then, like Frederic in The Pirates of Penzance, we will be slaves to a misuse of language.