The second chapter of my forthcoming book The Annotated Turing is entitled "The Irrational and the Transcendental" and innocently begins: We begin counting 1, 2, 3, and we can go on as long as we want. That's not true, of course. "We" simply cannot continue counting "as long as we want" because "We" (meaning "I" the author and "you" the reader) will someday die — probably in the middle of reciting a very long (but undoubtedly finite) number. What the sentence really means is that some abstract ideal "somebody" can continue counting, but that's not true either: Counting is a temporal process, and at some point everybody will be gone in a heat-dead universe. There will be no one left to count. Even long before that time, counting will be limited by the resources of the universe, which contains only a finite number of elementary particles and a finite amount of energy to increment from one integer to the next. We tend to accept simple statements such as the one that begins Chapter 2 Read More...