The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

Notes

Preface 1. Popper 2002, 38. Chapter 1 1. From the preface to Kant 1969. The original is Die menschliche Vernunft hat das besondere Schicksal in einer Gattung ihrer Erkenntnisse: daß sie durch Fragen belästigt wird, die sie nicht abweisen kann; denn sie sind ihr durch die Natur der Vernunft selbst aufgegeben, die sie aber auch nicht beantworten kann; denn sie übersteigen alles Vermögen der menschlichen Vernunft. 2. It is not clear if Einstein ever actually said this. However, Horgan (1996, 83) quotes a similar statement by John Archibald Wheeler: “As the island of our knowledge grows, so does the shore of our ignorance.” Friedrich Nietzsche uses the same metaphor in The Birth of Tragedy (2000, 97): “But Science, spurred by its powerful illusion, speeds irresistibly toward its limits where its optimism, concealed in the essence of logic, suffers shipwreck. For the periphery of the circle of science has an infinite number of points; and while there is no telling how this circle could ever be surveyed completely, noble and gifted men nevertheless reach, e’er half their time, and inevitably, such boundary points on the periphery from which one gazes into what defies illumination.” 3. This statement needs a little justification. One must distinguish between craft or technique, which does build on itself, in contrast to art and creativity, which do not build on themselves. In fact, creativity demands that the art be different from previous generations. It would be hard to claim that literature progresses when it can be argued that the greatest literature was written centuries ago by writers like Dante and Shakespeare. The Holocaust, genocides, and the wars of the twentieth century are counterexamples to any claims of an improvement with regard to human morality. Notes 356 Notes 4. This chessboard-and-dominoes puzzle was cribbed from Gardner 1994, where it is called the “Mutilated Chessboard.” The puzzle is, however, much older. 5. From Quine 1966, 3. 6. I am not equating human thought and human language. The latter is far more organized, coherent, and codified than the former. Whereas human thought does not have to be intelligible to any mind other than the thinker—in fact, it usually is not—human language is an attempt at making human thought understandable to other human minds. This is true for spoken language and even more so for written language. For the written word, even more codification and organization are required. A great piece of writing will be codified and clear enough for many other minds to appreciate. In contrast, literary theorists describe written language that “descends” to the level of human thought as “stream of consciousness.” Examples of such literary works are James Joyce’s Finnegans Wake and T. S. Eliot’s The Love Song of J. Alfred Prufrock. Most people find these works unreadable. The relationship between human thought and language was dealt with by the Russian psychologist Lev Vygotsky in his book Thought and Language and in the later works of Wittgenstein . Nevertheless, both thought and language are prone to contradictions. 7. Einstein 1936. Chapter 2 1. Wovon man nicht sprechen kann, darüber muß man schweigen. 2. Alas, I was unable to ascertain if Yogi Berra actually said this. 3. Some analysis shows that Epimenides’ declaration is not really a paradox. For one thing, we are subtly assuming that every sentence that a liar utters is a lie. This is false. A liar is someone who lied at least once. We have all lied at least once in our lifetime and hence we are all liars. Furthermore, there is something wrong with the logic in deriving a contradiction . Assume for a moment that Epimenides is telling the truth. This would imply that he is a liar and that the sentence is false. But a false sentence is not a contradiction . In contrast, assume that Epimenides’ sentence is false. That means that not all Cretans are liars and that a Cretan exists who is not a liar. Such a pious truth-teller can be anyone on the island. (What if Epimenides were the only person on the entire island?) If this truth-teller were Epimenides, then he is telling the truth and the sentence is true. That would be a contradiction. However, the truth-teller need not be Epimenides and could be someone else on the island. So there would be no contradiction if we simply accept that Epimenides was stating a falsehood. There is one last interesting idea to point out. We have...