The Parable of the Cube

This is an extract from the book, showing the relevance of mathematics to the argument.

It is a perennial problem with theology that I say one thing and you say another and… what do we do next? The things we say can’t both be true, so what is it to be? Compromise, anathema or despair? Here is a down-to-earth, practical example from mathematics to show how sometimes two eyes can be better than one. It is best done with a friend, but if you are on your own then you only need to use a little imagination.

  • On the table in front of us there is an object.
  • You pick it up and hold it up to the light and you say, ‘It has four sides.’
  • I pick it up in my turn. I hold it up to the light and I say, ‘It has six sides.’

Where do we go from here? How many sides does the object really have? You say four, I say six, so which of us is wrong? Who wins and who loses?

If you leave the maths out of it and think about disagreements in general, you will know the kinds of answers that people give at this point. Some will say we ought to be respectful of diversity and I am being needlessly confrontational in talking about ‘right’ and ‘wrong’ at all. Politicians might encourage us to split the difference and agree that the object has five sides. A broad-minded churchman of a certain kind might tell us that in a very real sense four and six are the same thing.

You and I both listen politely to these answers, but none of them makes sense. We may not agree on everything, but we do definitely agree that four is not six, six is not four, and neither of them equals five. We may have got hold of opposite ends of the stick, but at least we know that it is a stick, and the same stick, and that sticks matter. These easy ways out are not ways out at all: on that, we most definitely agree.

There is another way of dealing with this disagreement without sinking into meaninglessness. It is not a way out, but a way through. It will enlighten us both.

In my initial statement of the story I carefully withheld one vital fact. Here is the story again, with that fact put back in.

  • On the table in front of us there is a cube.
  • Pick up the cube and hold it up to the light with its face towards you. Squint at it with one eye shut, and you will see a square silhouette against the light. That makes it four sides.
  • Now hold the same cube at a different angle, this time with one corner pointing directly towards you. When you squint again, you will see a hexagon: six sides.

We are both right, and we have both won. That is not because we have given in to ‘four equals six’ or ‘they both equal five’ or whatever other nonsense the spectators have been urging on us. We have won not because we both see exactly the same but because our observations, though different, are observations of the same thing. It is a thing that cannot be summed up in one single silhouetted view. A cube, which is a three-dimensional object, makes one shape if you project it one way onto two dimensions and another shape if you project it another way.

Mathematically, this enlightenment has come from discovering that there is a geometry beyond the two dimensions of silhouettes and shadows. The ‘four’ and ‘six’ of our individual experiences are only shadows of the true cubical reality, which is a solid object, not a bare outline. It is a solid object with six faces, eight corners and twelve edges.

The moral of the story is not just mathematical. What it has just told us is this: No eye has ever seen a cube and no eye ever will. That is not a paradox, but the strict and precise truth. The eye cannot see the cube in itself, the cube in its full glory of cubicality. All the eye can see is shadows, not realities.

Given those shadows, our minds can put together what two eyes have seen, to give us knowledge of something the eye alone cannot see. Or, to put it another way, the eye without the brain is as blind as the brain without the eye. Or to put it another way still: we see the world best, in its solidity, in its reality, when the left eye and the right eye see it in different ways.

As with mathematics, so too with theology and with the Creed. If in this book I show you something from a direction that you don’t expect, I am not saying that the direction you are seeing it from is wrong, and I am not claiming for my own point of view anything more than the status of a shadow. All I claim is that my shadow is a truthful one. I am saying that if we can somehow perceive the truth from two directions at once, we have a chance of perceiving it better. ‘Four sides’ and ‘six sides’ are both equally wrong, but they both equally lead us towards the truth.