I would give a slightly different moral to Peter's story. Mathematicians could have defined 5 divided by 0 to be infinity--one of the wonderful things about mathematics is that we can define things however we want. However, what Peter's proof shows is that if you define division by 0, then some of the familiar algebraic laws aren't going to work anymore. (It is an interesting exercise to identify the algebraic law used in the proof that would stop working if we defined division by 0.) So it would actually be quite inconvenient to change the usual definition of division, according to which division by 0 is undefined.

If I interpret your question as a straightforward mathematical question--Is there a number whose square is 2?--then the answer is of course yes. The numerical value of the number is approximately 1.41421. But perhaps that isn't what you meant. Perhaps your question is about the sense in which mathematical objects like numbers exist. Do they really exist (whatever that means), or are they just, in some sense, figments of our imaginations? You might want to look at the answers to question 139 , and also the links to the Stanford Encyclopedia in those answers, for a discussion of two different ways that people have interpreted mathematical existence, namely platonism and intuitionism. It is perhaps worth mentioning that both a platonist and an intuitionist would agree that the square root of 2 exists, but they would mean different things by that. The platonist would mean that the square root of 2 is one of the objects in a world of mathematical objects that exists independent of us and our...

Yes, lines in mathematics go on forever, and such things most likely don't exist in our universe. So why do mathematicians study them? There are most likely only a finite number of elementary particles in the universe. Should mathematicians say that the positive integers end at some finite number? The study of the positive integers would actually be a lot more difficult (and a lot less attractive) if we placed some bound on the numbers to be studied, based on the number of particles in the universe. It is easier (and more interesting) to study the infinite collection of all positive integers, even if most of those numbers will never be used for counting objects in the physical universe. In general, mathematicians don't think of themselves as studying things that exist in the physical universe. Rather, they study abstractions, like infinite lines or the positive integers. These abstractions may be motivated by things in the physical universe, such as the lines we draw on paper or the process of...

This is a very difficult question, for two reasons: 1. It is difficult to say exactly what "random" means. 2. There are unresolved questions in the foundations of quantum mechanics that are relevant to your question. Consider, for example, flipping a fair coin. This seems random, in the sense that we don't seem to be able to predict the outcome. Half the time the coin comes up heads and half the time it's tails, and we don't know which it's going to be until it lands. But in another sense, it doesn't seem random at all: If you knew the speed at which the coin was spinning, its exact position above the table, the air currents in the room, etc., then the laws of physics should allow you to predict how it will land. If you think of randomness as being about our lack of knowledge of how things are going to turn out, then the coin flip seems random. If you think of randomness as being about some sort of indeterminacy in the world, independent of our knowledge, then the coin flip doesn't seem...

It is tempting to think that the phrase "the successor of the largest number anyone has ever thought about before next Tuesday" unambiguously defines a number. After all, it seems that we could compute the value of "Steve" as follows: Wait until next Tuesday, and then make a list of all the numbers anyone has ever thought about (a finite list, given the finite history of humans thinking about numbers), find the largest number on the list, and add one. But what counts as "thinking about a number"? From your question, it appears that you want to count thinking about a number by means of a description of that number, including descriptions like ... the definition of Steve! So the computation of the value of Steve isn't as simple as it sounds. What will happen next Tuesday when we sit down to compute the value of Steve? Well, you and I have been thinking about Steve, so when we go to make our list of all the numbers anyone has ever thought about, Steve will be on the list. This means that as part of...

I'd say we're speaking another language. In this language the numeral "2" means 3 and the numeral "5" means 6, so "2+2=5" means 3+3=6, which is true. But 2+2 is still equal to 4. This isn't really a fact about mathematics, it's a fact about language. If we changed the names we use to refer to people so that "John Kerry" referred to George W. Bush, then in that new language the sentence "John Kerry is president of the U.S. " would be true. But it wouldn't change the facts about who the president is. George W. Bush would still be president, we'd just be using a different name to refer to him. If you want to overturn the results of the last election, this isn't the way to do it.

I have a minor quibble with one of Richard's statements. He says that an intuitionist "can prove that not every number is either positive, negative, or equal to zero." I don't think an intuitionist would claim to be able to prove that. Rather, an intuitionist would say that he is unable to prove that every number is either positive, negative, or equal to zero. It's a small point, but there is a difference between being unable to prove that something is true and being able to prove that it is false. For more on this, see the entry in the Stanford Encyclopedia of Philosophy on Weak Counterexamples . Notice that conclusion 2 in that entry is "we cannot now assert that every real number is either positive, negative, or equal to zero," which is different from saying "we can now assert that not every real number is either positive, negative, or equal to zero."

See the answer to question 181 .

Yes, it is true that .9999... = 1, and there's nothing paradoxical about it. But to see why that is, you need to think about the meaning of decimal notation. Consider a decimal number of the form: 0.d1 d2 d3 d4 ... where each of d1, d2, d3, ... is one of the digits from 0 to 9. Of course, d1 is in the tenths place, d2 is in the hundredths place, and so on. What this means is that the number represented by this decimal notation is: d1/10 + d2/100 + d3/1000 + ... But if the list of digits goes on forever, then this summation goes on forever, so now we have to ask what an infinite summation means. You can never finish adding up infinitely many numbers, so we can't just say that this is what you get when you finish adding up all of the infinitely many numbers. Here's how mathematicians define this infinite sum: Start with d1/10, then add on d2/100, then add on d3/1000, and so on. The process never ends, so you will never actually get to the answer. However, as you add more and more...

Consider a small finite set--say, the set of members of your immediate family. You understand this set by knowing a list of all the members of the set, and being familiar with all of them. Now, the point of your question is that that approach clearly won't work for infinite, or even very large finite, sets. My grasp of the state-space for chess, or the collection of all integers, cannot involve being familiar with all the elements of the collection. So what does it involve? You hint at a couple of possibilities in your question: In the case of the state-space for chess, you say that "the rules of chess describe the boundaries" of the state-space. So your understanding of that set consists in knowing the rules that determine what is in the set and what isn't, even though you don't have a complete list of the elements. You also refer to the fact that "we can so easily instantiate well-formed members" of these collections. For example, I can generate as many positive integers as I want by counting,...