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

10. Beyond Reason

10 The eternal silence of these infinite spaces fills me with dread.1 —Blaise Pascal At the present time I seem to be thinking rationally again in the style that is characteristic of scientists. However this is not entirely a matter of joy as if someone returned from physical disability to good physical health. One aspect of this is that rationality of thought imposes a limit on a person’s concept of his relation to the cosmos.2 —John Nash The meaning of the world is the separation of wish and fact. Wish is a force as applied to thinking beings, to realize something. A fulfilled wish is a union of wish and fact. The meaning of the whole world is the separation and the union of fact and wish.3 —Kurt Gödel We have come to the end of our journey. It is time to sum up some of our findings and try to make sense of our explorations. In section 10.1 I categorize the different types of limitations that we have considered. Section 10.2 discusses the definition of reason. I conclude by looking at what, if anything , is beyond the limitations of reason. 10.1 Summing Up Every chapter of this book has discussed a different subject and its limitations . There are, however, other ways to categorize the myriad limitations we have found. Here I give another classification of four types of limitations on reason. Beyond Reason 340 Chapter 10 Physical Limitations The simplest type of limitation is one that shows reason does not permit a certain physical object or physical process to exist. The very first limitation we met (in chapter 1) was with the chessboard and dominoes. This is an example of a physical process that cannot exist. There is no way to place the dominoes on the chessboard with the two black corners removed. The barber paradox of section 2.2 also demonstrates that a certain isolated village with a particular rule cannot exist. The same section discussed a reference book that cannot exist. At the end of section 3.2 we saw that the time-traveler paradox shows either that time travel is impossible, or that even if it were possible, some actions by the time traveler will not be permitted. The universe simply will not permit a contradiction-causing process. All of chapter 6 and section 9.3 show that certain physical computers or algorithmic processes cannot exist. There are tasks that simply cannot be performed in this world. And finally in section 7.2, we talked about the possibility that quantum mechanics is inherently nondeterministic . In that case, no physical process can anticipate a quantum mechanical outcome. In all these examples we see that the physical universe is constrained by the dictates of reason. Mental-Construct Limitations A second, more subtle type of limitation states that a certain idea or mental construct cannot exist. I consider language a mental construct that is used to describe a mental state or a part of the universe. When discussing the liar paradox in section 2.1, I showed that certain sentences are neither true nor false. If a sentence is true then it is false, and if it is false then it is true. The mind cannot give such sentences any meaning. Similarly for other linguistic paradoxes—like the heterological paradox of section 2.2 as well as the interesting-number paradox, Berry’s paradox, and Richard’s paradox of section 3.2—Zeno’s paradoxes of section 3.2 deserve some thought. They are not physical limitations since the slacker will get to the door and Achilles will win the race. Rather, they demonstrate a problem with the descriptions of certain actions. The descriptions are faulty because they seen to demand an infinite process. In contrast, the actions are perfectly legitimate. Zeno’s paradoxes show that there are problems with certain mental and linguistic descriptions of basic movements . Similarly, our discussion of vagueness in section 3.3 showed a limitation of this type. Deciding whether a certain collection is a heap, and whether someone is considered bald, is a mental and/or a linguistic Beyond Reason 341 problem. We showed there were certain problems with such vague predicates . The inability to prove or disprove statements like the continuum hypothesis and the axiom of choice (section 4.4) demonstrates limitations of our logical ability. Similarly, most of the mathematical limitations...