The mystery
Those two gestures encode a philosophical war about the nature of reality - and once you see it, you'll never look at the painting the same way.
The short answer
Plato points upward because he believed that ultimate reality exists in an invisible realm of perfect, eternal Forms - the physical world is just a shadow of that higher truth. Aristotle points downward because he believed reality lives here, in the physical world, observable and measurable. The gestures are Raphael's brilliant shorthand for the deepest split in Western philosophy.
The twist
Raphael likely never studied philosophy formally. He was a painter. Yet he managed to compress 2,000 years of intellectual debate into two pointing fingers.
Common mistake
Some viewers assume the gestures are simply artistic convention - a generic sign of wisdom or speech.