For the answer to the question about 0.999..., see Question 181 . Mathematicians try to ensure the correctness of math by never accepting a mathematical statement as true without a proof. Of course, it's always possible that a mathematician will make a mistake when writing or checking a proof, so even if a mathematician has proven a statement and the proof has been checked by other mathematicians, there is still a small chance that there is a subtle mistake somewhere in the proof. (It has occasionally happened that flawed mathematical proofs have been accepted for years before someone finally spotted the flaw.) So if you're looking for an absolute guarantee of correctness, I don't think you're going to find one. But even if we ignore the problem of careless errors, there are other questions one could raise about whether or not a proof of a mathematical statement guarantees the correctness of the statement. Usually a proof of one mathematical statement makes use of other mathematical statement...

I think I need to know a little more about what you're doing that keeps leading to the number 5. Perhaps you are putting two rabbits in a box, and then putting in two more, and then counting how many rabbits are in the box, and by the time you count them they have already reproduced. In that case, your method of computing 2+2 is flawed--perhaps you should switch to using only male rabbits. Or perhaps the way you count is "one, two, three, five, six, ..." In that case, I'd say that you're using the word "five" in the way that the rest of us use the word "four". That doesn't change the mathematical facts--2+2 is still 4--it just means that you're expressing that fact in a confusing way. (For more on this, see question 216 .) By the way, I assume that your story isn't true--you haven't really been adding 2+2 and getting 5, you're just saying that to make a philosophical point. The reason I'm making that assumption is that there is nearly universal agreement among different people about the...

Yes, people have studied statements that contradict the Axiom of Choice. One of the most widely studied is the Axiom of Determinacy (AD). There is a Wikipedia entry that will tell you more about it. The question of whether or not AD is consistent with the Zermelo-Frankel (ZF) axioms of set theory is a bit tricky. Of course, by Godel's Incompleteness Theorem, we can't even prove that ZF is consistent (assuming it is), so we certainly can't prove that ZF + AD is consistent. It is not even possible to prove that the consistency of ZF implies the consistency of ZF + AD. If you're willing to make a stronger assumption (the consistency of ZF + the existence of certain kinds of large infinite cardinal numbers), then you can prove the consistency of ZF + AD.

There's a part of your question that I think requires clarification. You say that if you keep adding one to a number, then the number "becomes infinite". I don't think I would say that. The number keeps increasing, and it will eventually exceed one million, or one billion, or any other number I might choose. But it is always finite; it never actually becomes infinite. Similarly, if you keep generating people (or chairs, in Alex's example), then the number of people keeps increasing, but it is always finite. (As Alex said in his example, "at any given time there are actually only a finite number of chairs.") However, you could talk about the collection of all people ever generated by this (infinite) process, and that would be an infinite collection. Thus, it is important to distinguish between the collection of all people in existence at any particular time, and the collection of all people ever generated by the process. The former is always finite, but the latter is infinite. (By the way,...

The answer is that the work of Shakespeare would almost surely come out. More precisely, the probability that at least one of them will type Shakespeare is 1. This doesn't mean that it is absolutely certain to happen. It means that if you did the experiment repeatedly, and kept track of how many times at least one monkey typed Shakespeare and how many times none of them did, you would expect that the fraction of the time that at least one monkey typed Shakespeare would approach 1. It could occasionally happen by chance that no monkey typed Shakespeare, but if you kept repeating the experiment you would expect this to happen so infrequently that in the long run, the fraction of the time that no monkey typed Shakespeare would approach 0. To simplify things, let's assume that a monkey types 18 random characters, and each character is either one of the 26 letters or a space. One 18-character string that the monkey could type is "to be or not to be", but of course there are many others. The number...

It is true that we can imagine a universe in which when you walk forward 2 meters, you end up 3 meters ahead of where you started. However, I would say that in that universe, 2+2 is still equal to 4, but addition does not describe how one's position changes when one walks forward. Inhabitants of that universe might invent a new mathematical operation, "walk-addition", to describe how one's position changes when one walks forward, but that would be a different operation from the operation of addition. In fact, in our universe we have done something similar with geometry. Einstein discovered that Euclidean geometry does not accurately describe our universe. We didn't conclude that Euclid was wrong, we just concluded that we needed a different kind of geometry to describe our universe. The point is that mathematical objects and operations are not defined in terms of their applications to the physical world. They are abstractions, and although those abstractions may be motivated by things we...

Math has many applications. To take one of your examples, a physicist might use a linear equation to describe the position of an object moving at a constant velocity. Some areas of mathematics have applications to other areas of mathematics (which may then have applications outside of mathematics). Polynomials may be a good example of this. Polynomials are sometimes used to approximate other, more complicated functions, and such approximations may be useful for solving a variety of mathematical problems (which may then have applications outside of mathematics). But you also ask why math was created. Often when mathematicians create mathematics, they are not thinking about applications, they are just trying to solve a problem because they find it intriguing. You might say that in many cases mathematicians solve mathematical problems for the same reason that mountain climbers climb mountains: because they are there. The applications often come later. A good example of this might be prime...

It depends on what you think a proof is supposed to be. If you think a proof is just a certain kind of abstract object--say, a finite arrangement of symbols that satisfies the rules of logic--then it seems that a computer-checked proof is a proof just as much as a human-checked proof is. But another way of looking at it is to think about the role proofs play. When you read and understand a proof, you become convinced that a certain mathematical statement is true, and the proof is what convinces you. Now, suppose you are convinced by a computer-checked proof--say, the proof of the 4-color theorem. What is it that convinced you that the 4-color theorem is true? Not the proof that the computer checked--you never saw that proof, and it's far too long for you to read yourself. No, what convinced you of the truth of the 4-color theorem is a certain physical experiment: connecting a box full of wires to a source of electricity and seeing what happens. It seems that your belief in the 4-color theorem...

I assume the relationship you have in mind is e iπ = -1. I wouldn't go quite as far as Richard in saying that this is"independent of how we choose to do mathematics," because it doesdepend on how we choose to define exponentiation for complex numbers.Mathematicians were faced with a choice when they were deciding how to defineexponentiation with complex numbers, and there was no one definitionthat was the unique "right" definition. However, the definition thatmathematicians chose is a very natural one, and once you choose thatdefinition the relation e iπ = -1 follows. Perhaps it would be of interest for me to give an explanation, forthose who are familiar with power series, of why the definition ofexponentiation with complex numbers is natural. If you studied powerseries in a calculus class, then you are probably familiar with thefollowing formulas: e x = 1 + x + x 2 /2 + x 3 /3! + x 4 /4! + ... cos( x ) = 1 - x 2 /2 + x 4 /4! - ... sin( ...

Richard makes a good point, but I still think that I had a good point also, although I may not have expressed it as well as I might have. It is often said that Euler proved that e ix = cos( x ) + i sin( x ), but it seems to me this is somewhat misleading. Many (most?) modern complex analysis books present Euler's equation as a definition , not a theorem--it is the definition of e ix . On this view, what Euler did is not to prove that e ix is equal to cos( x ) + i sin( x ), but rather to show that it would be a good idea to define it that way . If you want to regard Euler's equation as a theorem , then you have to take something else to be the definition of e ix . You could do that--you could, for example, take the power series for e x , which was derived for x a real number, and extend it into the complex numbers to define e z when z is a complex number, and then Euler's equation would be a theorem. But this requires...

Euler's equation is important in quantum mechanics. In quantum mechanics, the state of a particle is given by a function, and the formula for that function generally includes a term of the form e ix , which, according to Euler's equation, is equal to cos (x ) + i sin( x ). The appearance of cos( x ) and sin( x ) here introduces an oscillatory term into the formula. This is why quantum mechanics predicts wave-like behavior of particles. There is a geometrical interpretation of Euler's equation. The complex numbers are usually pictured as a plane. (See question 316 for more about this.) In this plane, the number e ix always lies on a circle of radius 1, centered at 0. When x = 0, e ix = e 0 = 1, which is the rightmost point on the circle. As x increases, the point e ix moves counterclockwise around the circle, completing a full revolution every time x increases by 2 pi. Although Euler's equation has applications in science, I'm...