Outer Limits of Reason by Noson S. Yanofsky Read Free Book Online Page A

every single Richard phrase. The phrase does describe a number if and only if it does not describe a number. What to do? 9
    These last two paradoxes can be seen as self-referential paradoxes. In a sense, they can be summarized by the following two descriptions:
    â€¢ “the Berry phrase that is different from all Berry phrases”
    â€¢ “the Richard phrase that is different from all Richard phrases”
    From this point of view, they are simple extensions of the liar paradox. Self-reference is very common and we must be careful with it.
    Further Reading
    Many of the paradoxes can be found in places such as Quine 1966, Hofstadter 1979, 2007, Barrow 1999, and Poundstone 1989. Sorenson 2003 is a clear and well-written introduction to paradoxes. Chapter 5 of Sainsbury 2007 covers the liar paradox and other forms of self-reference. Chapter 3 of Paulos 1980 provides a humorous look at all self-referential paradoxes. Yablo’s paradox is found in Yablo 1993.
    A formal version of self-referential paradoxes can be found in Yanofsky 2003, which is derived from Lawvere 1969.

3
    Philosophical Conundrums
    Moreover, although these opinions appear to follow logically in a dialectical discussion, yet to believe them seems next door to madness when one considers the facts. For indeed no lunatic seems to be so far out of his senses.
    â€”Aristotle (384–322 BC), On Generation and Corruption , 325a15
    All are lunatics, but he who can analyze his delusion is called a philosopher.
    â€”Ambrose Bierce, The Collected Works of Ambrose Bierce
    It depends on what the meaning of the word “is” is.
    â€”William Jefferson Clinton
    Long before modern scientists took up the task of investigating the limits of reason, philosophers were analyzing the complexities of our world and our knowledge of it. In this chapter I explore some of the ancient and contemporary philosophical aspects of reason’s limitations.
    In section 3.1 , I begin by discussing some very fundamental questions about concrete and abstract objects and the way we define them. In section 3.2 , the very nature of space, time, and motion are analyzed using some of Zeno’s paradoxes. The section ends with a short discussion of time-travel paradoxes. Section 3.3 is concerned with vagueness. Section 3.4 is centered on the very notion of knowing and having information. These sections are independent of each other and of the rest of the chapters. They can be read in any order.
    3.1  Ships, People, and Other Objects
    In ancient Greece, there was a legendary king named Theseus who supposedly founded the city of Athens. Since he fought many naval battles, the people of Athens dedicated a memorial in his honor by preserving his ship in the port. 1 This “ship of Theseus” stayed there for hundreds of years. As time went on, some of the wooden planks of Theseus’ ship started rotting away. To keep the ship nice and complete, the rotting planks were replaced with new planks made of the same material. Here is the key question: If you replace one of the planks, is it still the same ship of Theseus? This question about a mythical ship is the poster child for one of the most interesting problems in all of philosophy, namely the problem of identity. What is a physical object? How do things stay the same even after they change? At what point does an object become different? When we talk about a certain object and say that “it changed,” what exactly is “it”?
    What happens if you change two of the ship’s planks? Would that make it somehow less of the original ship than after one plank is changed? What if the ship consists of a hundred planks and forty-nine of the planks are changed? How about fifty-one changed planks? What about changing ninety-nine of the hundred planks? Is the single plank at the bottom of the ship enough to maintain the original lofty status of the ship? And what if all of the planks are changed? If the