HERE ARE SOME MYSTERIOUS and strange things, as well as pointers to further reading.
1. Mathematics is crawling with strange things. How many people do we need to gather together in a room to be 100 percent sure that two of them will have the same birthday? 366. Right . . . ok, good. How many people do we need to gather together in a room to have a 50 percent chance that two of them will have the same birthday? That’s right, 23. . . . ! . . . Not 183, which is one-half of 366.
It is somewhat hard to find books explicitly on mathematical weirdnesses because mathematicians live and work in an ethos of tough-minded, no-nonsense theorem-proving. But one good book is Havi, Nonplussed. I also recommend Kaplan and Kaplan, Art of the Infinite.
2. Almost everything about quantum mechanics shocks the sensibilities and leaves one baffled. A good book for laypeople is Al Khalili, Quantum. Another good book is Smolin, Trouble with Physics. Smolin’s book comes at the universe’s strangenesses from another direction. His book is an attack on string theory (strings are suppose to be the fundamental stuff of which the matter in the universe is made). He claims it is bankrupt as a physical theory. String theory itself is crawling with weird conclusions and predictions, but in a way, its conclusions and predictions aren’t weird enough. Smolin’s point is that physicists settled too quickly for the strangenesses that they can mathematically model while ignoring the fact that their failures pointed to yet deeper mysteries. Smolin suggests that we have to face up to the fact that, as of now, we lack any fundamental theory that unifies both quantum physics and gravitational physics. Once we face this, we can set off afresh in search of a new, better theory.
3. Logic contains my favorite weirdnesses. There are a lot of logics. This is somewhat strange in itself, since logic is supposed to be foundational, especially for mathematics and computer science. If it’s foundational, why is there more than one of them? “Different subjects require different logics” is the obvious answer. So, for example, artificial intelligence uses one kind of logic and mathematics another one. The problem with this glib answer is that there are way more logics than subjects that rely on them. And most subjects, for example, set theory and artificial intelligence, can use incompatible logics as their foundation. So, again, why are there so many logics?
Probably the most famous example of a profound strangeness in logic is Kurt Gödel’s proof that arithmetic contains truths that cannot be proved. (A somewhat useful if controversial place to start is Goldstein, Incompleteness. I recommend Kennedy’s article at http://plato.stanford.edu/entries/goedel.) In 1931, Gödel proved that there are true arithmetical statements which can’t be proved within arithmetic, but one can know they are true anyway; indeed, one can prove they’re true. This might seem contradictory. What of it? The real shocker here (though this isn’t Gödel’s idea) is that there are logics that allow some contradictions to be true (the contradictions are false, of course, but they are also true). Even stranger: many of the contradiction-friendly logicians insist that true contradictions actually exist in the real world, and are prepared to name names. I refer the intrepid reader to Priest, Beyond the Limits of Thought and In Contradiction, and to his excellent short story “Sylvan’s Box.” Finally, and though it is somewhat technical, my paper “The Bishop and Priest” holds the reader’s hand, guiding him or her to actually seeing a true contradiction. This paper is available on my website.
By the way, Gödel was a font of exciting mysteries. He apparently derived a solution to Einstein’s relativity equations that shows that time is circular. See Yourgrau, A World Without Time.
4. The mysteries inundating philosophy cut right to its core. Indeed, philosophy itself is mysterious. Philosophy has made no progress since Aristotle, a genius who lived twenty-three hundred years ago. Every major problem Aristotle worked on is still with us to this day, although each problem has been updated and reformulated to take account of the advances in science and practical morality. Here’s an example of practical morality. We know now that owning slaves is morally wrong; this is only practical knowledge, however, because philosophers cannot agree on why owning slaves is wrong. They cannot agree if owning slaves violates the requirement to strive to produce only happy consequences, or if owning slaves violates our duties to other living beings (see chapter 7 for a discussion of these two approaches to morality). For more on philosophy’s lack of progress, see my paper, “There Is No Progress in Philosophy.” The two best books on philosophy’s inherent mysteriousness are McGinn, Problems in Philosophy, and Nagel, The View from Nowhere. Warning: McGinn looks for a mystery-dismissing solution. Another good book is Sorensen, Brief History of the Paradox.
5. Miscellaneous. Did you know that there is a very old, currently undecipherable text complete with detailed colored drawings sitting in Yale University’s Beinecke Rare Book and Manuscript Library? The author, purpose, and meaning of the text are unknown. Attempts to decode it or translate it defy all modern techniques used by linguists and cryptologists. It’s called the Voynich Manuscript. Most scholars believe it was written sometime in the fifteenth or sixteenth century, but no one is sure (recent carbon-14 dating does put the date of the paper in the fifteenth century). The language the manuscript is written in, if indeed it is a language, is completely baffling. Detailed statistical analysis of the symbols making up the manuscript, however, leads most scholars to believe that it is in fact written in some language, just not one used on planet Earth by any known culture or people.
This and other wonderful mysteries are detailed in Poundstone, Labyrinths of Reason. Poundstone discusses scientific confirmation, meaning in language, metaphysics, logic, and epistemology. All of these fall within the boundaries of philosophy. So, this book could have been placed in the philosophy section. But I’ve stuck it here because, besides the Voynich Manuscript, Poundstone looks at some weird aspects of computer science and discusses some of the works of Jorge Luis Borges.
Another good general book is Brooks, 13 Things. It is an informed and enlightening discussion of such things as cold fusion (which stubbornly won’t go away), the placebo effect, and the perhaps already successful search for extraterrestrial intelligence. These three cases and nine more rest on real, public evidence that Brooks lays out clearly for the reader. I say “nine” because as of this writing, one of Brooks’s thirteen has been explained, it seems. I leave it as an exercise for the reader to find out which one.
Lastly, I recommend Cartwright, Dappled World. Here, a distinguished philosopher of science argues that the actual world is a lot messier than the world of scientific laws and scientific theories. This is a good book for explaining how science really works.