Socrates Treats Language like Math
Socrates, in the Republic, assumes that language can be as precise as math. For example, in book four he says if his imaginary city is good then it is “wise, courageous, moderate, and just.” These four words form a reasonable definition of good, although I hesitate to say it’s complete. Socrates’ does not hesitate, and he uses this definition to launch a curious analysis:
Therefore, just as with any other four things, if we were seeking any one of them in something or other and recognized the other three first, this would also suffice for the recognition of the thing looked for. For plainly it couldn’t be anything but what’s left over. With these things too, since they happen to be four, mustn’t we look for them in the same way?
It’s as if Socrates defined good using a summation:
good = wise + courageous + moderate + just
Therefore, we can define just using a subtraction:
just = good − wise − courageous − moderate
This reasoning may work if Socrates’ four-part definition is complete. If it isn’t complete, you’ll have extra terms:
good = wise + courageous + moderate + just + X + Y
After moving the known terms to the right side of the equation, we’d be unable to isolate the just from the unknown terms:
just + X + Y = good − wise − courageous − moderate
Socrates’ approach, besides needing to overcome this difficulty of additional terms, would also require precise knowledge of the good, wise, courageous, and moderate to get to the just. It also assumes the terms of a definition can be manipulated like numbers, which may be true in this case, but certainly is not true in general.
Why does Plato treat language this way? I think he was enamored with the precision and beauty of mathematics, and so he wanted to appropriate this precision and beauty to the ideas he cared deeply about, such as the good and the just.