This is a puzzle created by the mathematician Augustus De Morgan. He offered this as a "proof" that 2 = 1, which is (as he understood) absurd, and dependent on sophistical reasoning. Can you spot the flaw? 1. Let x = 1. 2. Clearly,then, x 2 = x. 3. So x 2 - 1 = x - 1. Now, divide both sides of THAT equation by x-1. That 'should' turn it into the following 4. x + 1 = 1. But since we started by saying that X equals 1, we have now concluded that 5. 2 = 1. QED. That plainly cannot be. But where is the sophism? The answer begins with remembering a critical point in number theory: don't divide anything by zero. Ever. Any chain of inference that involves the division of anything by zero at any point is invalid. The first proposition above tells us that x = 1. Later we are instructed to divide both sides of a certain equation by x-1. This...
Posts
This was my answer to the question whether Plato's account of Socrates can be relied upon: Plato’s account [or accounts] of Socrates has [have] to be considered in connection with other evidence, especially in my view the comedy THE CLOUDS, written by Aristophanes during Socrates’ lifetime. That is the TL;DR answer. For more, bear with me. Aristophanes portrays Socrates as a sophist, and in fact as the leader of the sophists. And he portrays sophism in general as what we might call a cult, the purpose of which is to make bad arguments seem good, and good arguments seem bad. Now this is odd on its face. Plato after all portrays Socrates as an anti-sophist. In PROTAGORAS especially, and even in the early going of THE REPUBLIC (the exchange with Thrasymachus), the Platonic Socrates devotes himself largely to showing the sophists — or to showing impressionable youngsters — the errors and arrogance of sophism. And Plato’s Socrates seems to object to s...