 As we noted (§1.2), Frege had developed formal logic to account for our apparently a priori knowledge of mathematics. It is worth dwelling on the interest of this problem. It is arguably one of the oldest and hardest problems in Western philosophy, and is easy enough to understand: ordinarily we acquire knowledge about the world by using our senses. If we are interested, for example, in whether it’s raining outside, how many birds are on the beach, whether fish sleep or stars collapse, we look and see, or turn to others who do. It is a widespread view that Western sciences owe their tremendous successes precisely to relying on just such “empirical” (experiential, experimental) methods. However, it is also a patent fact about all these sciences, and even our ordinary ways of counting birds, fish and stars, that they depend on often immensely sophisticated mathematics, and mathematics does not seem to be known on the basis of experience. Mathematicians don’t do experiments in the way that chemists, biologists or other “natural scientists” do. They seem simply to think, seeming to rely precisely on the kind of “armchair reflection” to which many philosophers also aspire. In any case, they don’t try to justify their claims by reference to experiments, arguing that twice two is four by noting that pairs of pairs tend in all cases observed so far to be quadruples.