Point

The Elements opens with twenty-three definitions, and the first is four words in Greek and seven in most English translations: a point is that which has no part. Nothing is added. Euclid does not say what a point is made of, where it is, or how many there are; he says only that it cannot be divided.

That is a stranger claim than it looks. Almost every other object in Book I is built, and Euclid tells you how to build it. The point is not built. It is assumed, placed, or found where two other things cross. This makes it the only primitive in the whole system that the two classical tools cannot produce on their own, because both tools consume points rather than making them: a compass needs a centre and a radius, and a straightedge needs two points to join. Constructions begin with points already given.

Where new points do come from is intersection. A circle crossing a circle, a line crossing a circle, a line crossing a line: these are the three events that generate every fresh point in a classical construction, and the algebra of exactly those three cases is what later proved certain figures impossible to draw. The whole theory of constructibility is, at bottom, a theory of which points can be reached from the ones you started with.

A point has no length, no breadth and no interior, so it has no size at all. It has position and nothing else. Mathematicians after the nineteenth century largely abandoned the attempt to define it, and treated point, line and plane as undefined terms fixed only by the axioms relating them. That is the modern position: a point is whatever behaves like one.

None of this is evasion. It is the recognition that the starting element of a system cannot be explained in terms of the system it starts.

Because the point has almost no mathematical content, what has been said about it is mostly philosophical, and traditions that reason about origins have found it useful for exactly that reason: it is the smallest thing that can be said to exist at all.

The Pythagoreans called the unit the monad and treated it as the source rather than as a member of the number series. One is not so much the first number as what numbers are made of. Later Greek writers extended the idea into geometry: the point generates the line, the line the surface, the surface the solid, so the whole of extension unfolds from something unextended. The scheme reaches its most influential form in Neoplatonism, in the idea of an ultimate One that is beyond number and from which all multiplicity proceeds.

In Hindu tantra the bindu occupies the centre of the Sri Yantra, the point on which the nine interlocking triangles converge. Commentators in the Shri Vidya tradition describe it as the concentrated seat of the deity and the condition before duality, and meditation on the diagram is described as a movement inward toward it. The same word names the dot marking nasalisation in Indic scripts and the mark worn on the forehead, and the shared vocabulary is not incidental to how the tradition talks about it.

A saying circulated widely in Sufi literature has Ali declare that he is the dot beneath the letter ba, the point that opens the Quran's first formula. Its attribution is not established and scholars generally treat it as later material, but the image itself, all revelation compressed into a single mark, shows how far the figure has been pushed.

What these have in common is not a shared doctrine. It is that a tradition wanting to speak about an origin which is not itself made of parts has very few images available, and the point is the obvious one.