The Theory of Forms

Draw a circle. However carefully you do it, the line wobbles, the ends do not quite meet, the radius varies. Now notice what you just did without thinking: you judged it. You know it is not a perfect circle. But if every circle you have ever encountered is imperfect, where did you get the standard you are judging against?

This is the question that produced the most influential idea in Western philosophy, and Plato's answer is that the standard is real and is not in the drawing. There exists, somewhere other than the physical world, the Circle itself: perfect, unchanging, and grasped by reasoning rather than by looking. He calls such things Forms. Every drawn circle is a flawed copy, and what makes it a circle at all is that it participates in the original.

The argument does not depend on circles. Take two sticks you would call equal. Measure them finely enough and they are not equal; nothing physical ever is. Yet you understand Equality perfectly well, well enough to know the sticks fall short of it. The understanding cannot have been abstracted from the examples, because every example is a failure. Something else must be supplying it.

Plato applies this beyond geometry, and that is where it starts to bite. There is a Form of Justice, and every actual law and verdict is an imperfect copy of it. This is not decoration; it is the whole answer to the sophists, who taught that justice is simply whatever a city says it is, and that a skilful speaker can therefore make anything just. If Justice is real and unchanging, rhetoric cannot manufacture it. Your city can be wrong about justice in the same way a shaky drawing can be wrong about a circle.

Notice what this commits him to. The Forms are not concepts in anyone's head: that is the most common modern misreading, and it destroys the argument. If Justice were just what people think, it would vary with what people think, which is exactly the view Plato is trying to defeat. The Forms have to exist independently, unchanged whether anyone contemplates them or not. That is a large claim about what exists, and Plato knew it.

He knew it well enough to attack it himself. In the dialogue Parmenides he raises an objection that has come to be called the third man. If a particular man and the Form of Man are alike, then their likeness needs explaining, which seems to require a further Form that both share in, and that one needs another, forever. He also cannot say clearly what participation actually is. He puts these objections in writing, under his own name, and does not answer them. Whether that means he abandoned the theory or thought it needed rebuilding is still argued about, which tells you how seriously the difficulties were taken.

The idea survived anyway, because the problem it addresses will not go away. Working mathematicians routinely talk as though they discover objects rather than invent them, and defending that talk means saying something Platonic. So does anyone who claims a law can be unjust, which requires a standard the law itself does not set. So, in a smaller way, does a design team appealing to a system that no single screen fully implements, or a lawyer appealing to the spirit of a statute against its letter.

Each of those is the same move: a standard that no instance perfectly embodies, and against which every instance can be measured and found wanting.