Indulge me in a little thought experiment. Imagine a rocky planet a long way from here, orbiting a star rather similar to our own sun, at just the right distance for water to remain liquid. Let this planet have an atmosphere containing all the elements needed for interesting biochemistry to take place – carbon, hydrogen, oxygen, nitrogen, phosphorus, sulphur and so on. Now imagine that on this planet lives a single lifeform – one composed of trillions of tiny plant-like organisms amassed in one huge layer that covers the entire planet. This lifeform quietly converts the sun’s energy into food, and while it does so, it thinks. What kinds of thoughts might it ponder?
Let me tell you more about this lifeform. It has lived on this planet for a very, very long time. It is blind, as it has no need for sight. It cannot hear, or touch, or taste, as those senses serve no purpose for its survival. It cannot move, as it has nowhere to go.
But it does grow, and it senses the sun’s light on its surface, creating the food that sustains it. This feeling gives it pleasure. And it can think too – not quick thoughts, like our own, but slow, deep thoughts of an alien plant-like nature. It has no brain, but the enormous distributed interconnectedness of its body makes it rather like a brain in its structure.
What kind of mathematics does this being think about? It knows nothing of numbers, because numbers are a tool for counting, and this creature has nothing to count. It does not think about objects, because it knows of none. It has never imagined addition or multiplication or set theory or things like that. It has never conceived of things, and would be quite unable to grasp the concept of things.
It does know about the passing of time, however. It knows about change too, as it senses the sun’s rays slowly moving across its exposed body as day turns to night and back. It feels the cycles of the day, of the seasons, of the years. It knows about surfaces – not the simple, flat triangles and circles of Euclid, but the smooth, undulating, complicated surfaces of its own form. It knows about volumes as well, but not cubes or pyramids, or anything so childishly simplistic and unreal. It knows nothing of lines or points, as these do not exist in its world.
So it understands geometry and form; magnitude and change. It can do mathematics! And so this is what it does. After all, it does not know words or images or communication – things that humans spend most of their time thinking about.
Yet how does it measure the world without numbers? It must conceive of magnitude in some kind of continuous way. Not fractions or decimals, which are a crude way of approximating an analogue world with digital measures. But somehow it must invent an entirely new way of thinking about metrics.
What would this mathematics look like? That is my question for today.