##### What are Numbers?

Mar 14, 2006Plato claimed that numbers exist in some mind-independent abstract heaven. Nominalists claim that there is no such heaven.

Paradoxes

Sunday, February 3, 2008

A paradox is a persuasive argument that something, which we judge must be false, is true. Zeno's Paradox, for example, is a convincing argument that it's impossible to move. Paradoxes are valuable in philosophy because they help us become aware of forms of argument that are deceptively convincing yet logically fallacious. John and Ken are joined by Roy Sorensen from Dartmouth College, author of A Brief History of the Paradox, to consider what we can learn from paradoxes.

What is a paradox? People use the word 'paradox' in many ways, for example to denote something very surprising. But these aren't genuine "philosophical" paradoxes. In a genuine paradox, one starts with seemingly true premises, performs reasoning thought to generate true conclusions from true premises, and ultimately derives a conclusion that's apparently false. Here are four paradoxes:

1. Consider the statement 'This statement is false.' If it's true, then it's false. But if it's false, then it's true. So which is it? True? False? Both? Neither?

2. To pass the finish line, a racer must first get halfway there. But to get halfway there, she must first get a quarter of the way. Yet to get a quarter of the way she must get an eighth of the way. And so on, ad infinitum. But then, to pass the finish line, a racer must complete infinitely many tasks. And this seems impossible, for no matter how fast she is, she can only do finitely many things in her limited lifespan. So it would seem that the racer can never pass the finish line! But clearly she does. What gives?

3. One million grains of sand is, of course, a heap of sand. And, intuitively, what remains after removing a single grain from a heap of sand is itself a heap. But then, contrary to intuition, a single grain of sand is a heap! Where does this reasoning falter? Is it false that a million grains of sand constitutes a heap? If so, how many grains does it take to make a heap? Or does removing a grain from a heap not always produce another heap? If so, what is the smallest heap---that is, the heap such that removing one grain produces a non-heap?

4. Let RED be the set of all red things. Presumably, RED is not itself red (since sets, one assumes, are colorless). So RED is not a member of itself. Let BLUE be the set of all blue things. Again, BLUE is presumably not blue, and so BLUE is not in BLUE. More generally, for every quality x, let X be the set of all things that exhibit x. Now consider the set S which contains RED, BLUE, and all sets X that are not members of themselves. Is S in S, or not? If it is, then S is not a member of itself. If it's not, then S is a member of itself. Both results are absurd! But the foregoing reasoning seemed perfectly clear and valid. What does this tell us about our methods of argument and our notions of quality, set, and so on?

With guest Roy Sorensen of Dartmouth College, John and Ken discuss these paradoxes and others. Are they irresolvable, or do they disappear once one thinks about them deeply enough? And either way, do they have theoretical or practical significance, or are they just plain mind-blowing fun?

**Roving Philosophical Report**(seek to 5:55): Zoe Corneli interviews Palo Mancuso of UC Berkeley about the history of Russell's paradox, sketched in (4) above. The story revolves around Gottlob Frege, an unpopular and ambitious German mathematician who in the late nineteenth century tried to reduce all of mathematics to logic, and Bertrand Russell, a young and aristocratic English philosopher who discovered a fatal flaw in Frege's attempted reduction.

**60-Second Philosopher**(seek to 49:35): In this peek at the "paradox philosophy" of Joe Hunt, Ian Scholls shows just how distorted popular notions of paradox can get. Hunt, once the ringleader of a famous Ponzi scheme called the Billionaire Boys Club and now serving a life sentence for murder, held that what something is depends on how you look at it, and that nothing is real except what you want. After "accidentally" killing a con artist who allegedly swindled millions from the Club, Hunt even went so far as to deny that the man was dead! Paradoxical? You decide.