15 March 2005

Backwards Logic

Proofs are discovered by working backwards: we get clear first about what we want to prove; then consider next what we would need to show, to get that result easily; then consider after that what prior step is needed to show this; and so on, until we arrive at something we can show straightaway.

Plato was a mathematician. The Republic seems a developed and single argument. Is perhaps the best way to understand it, then, to read it backwards?

Here's how that would work for books II-IV. We get clear first about the goal: it is to show that justice is like health in being placed in the second of three classes of goods (what is welcomed for its own sake and for what it brings along with it). But health is welcomed for itself because it is a good ordering. But only something with parts can be ordered. But justice exists in the soul. Thus we must show that justice, and the soul, have parts. But the soul cannot be seen. Thus we need some device for representing it (suppose) a city-state. Thus the soul has three parts, corresponding to the three essential parts of a city (market, military, government). (That reason, spirit, and desire can conflict is an ancillary consideration.) Moreover, a city-state reveals the order that should obtain, namely, an order of ordering (which should rule, which should be ruled) or of 'authority'.

Something like that. The details aren't so important now. The key point is to read the thing backwards.