The Duplication of the Square in Plato’s Meno

Shortly before beginning his questioning of the slave-boy, Socrates reports an opinion which he considers to be “Something true, it seems to me, and beautiful”. The reported opinion concludes “For inquiry and learning as a whole is recollection”. To illustrate the conclusion, Plato has Socrates enter into a detailed mathematical discussion. Nowhere else does Plato present mathematics in any detail. Presumably, there was a reason for doing so on this occasion. To what extent did Plato use mathematics as a model for his philosophy? When he speaks of learning as recollection, is this an expression of a metaphysical belief (something true) or is it mystical insight (something beautiful) or would Plato not have seen much of a distinction between the two? In this appendix, we shall not be discussing such questions. We shall merely be presenting some prerequisites that are needed by anyone wishing to engage in the controversies. It is likely that the slave-boy passage is drawn from a core piece of mathematics that would have been recognized, in Plato’s time, by educated readers. It is likely that, in a mundane sense, those readers would indeed have been recollecting the material. To put ourselves in their position, we must try to see the mathematics as they might have seen it. One style of teaching mathematics is first to state a problem, then to present a solution, and then to prove that the solution is correct. The problem: given a square, how can we construct a new square such that the area of the new square is double the area of the original square? The solution: a diagonal of the original square is to be an edge of the new square. The two squares are depicted in the left-hand part of the following diagram. The proof: extending two of the lines, as shown in the right-hand part of the diagram, we obtain five triangles all with the same area. The original square is made up of two of the triangles. The new square is made up of four of the triangles. So the new square does indeed have twice the area of the original square.