"He wanted to show that fate ruled people's lives, and that those who interfered with it did so to their sorrow..."
See in text (I.)
This line neatly encapsulates one of the story’s main themes. The expressed purpose of the paw is to show that people’s fates are predetermined, and any attempt to change fate will result in something terrible. Though the Whites don’t quite know the gravity of this statement yet, what’s fated will come to pass, one way or another.