For all anybody can see to the contrary it might just as well be blue. There is nothing inconceivable about its being blue, as there is about two and two making five. Of course, that gold is yellow is no doubt a mechanical necessity, that is, it is determined by causes, and in that sense could not be otherwise. But it is not a logical necessity. It is not a logical contradiction to imagine blue gold, as it would be to imagine two and two making five. Any other proposition in mathematics possesses the same necessity. That the angles at the base of an isosceles triangle are equal is a necessary proposition. It could not be otherwise without contradiction. Its opposite is unthinkable. But that Socrates is standing is not a necessary truth. He might just as well be sitting. Since a mathematical proposition is necessarily true, its truth is known without verification by experience. Having proved the proposition about the isosceles triangle, we do not go about measuring the angles of triangular objects to make sure there is no exception. We know it without any experience at all. And if we {215} were sufficiently clever, we might even evolve mathematical knowledge out of the resources of our own minds, without its being told us by any teacher. That Caesar was stabbed by Brutus is a fact which no amount of cleverness could ever reveal to me. This information I can only get by being told it. But that the base angles of an isosceles triangle are equal I could discover by merely thinking about it. The proposition about Brutus is not a necessary proposition. It might be otherwise. And therefore I must be told whether it is true or not. But the proposition about the isosceles triangle is necessary, and therefore I can see that it must be true without being told.