All exact reasoning, especially mathematical reasoning, rests on the supposition that a precise statement is either true or false, and cannot be both true and false. Consequently mathematicians have in recent years been forced to give a lot of attention to certain cases in which simple and apparently flawless reasonings can be used to establish contradictory conclusions; for the situation which arises is as uncomfortable to the mathematician as it would be if he could prove that two sides of a triangle were together greater than the third side, and also prove that they were together less than it.