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...

The problem you raise in your first paragraph is called the measurement problem : What happens when a measurement takes place? Most physicists would not agree with your statement that "Surely the present moment consists of an infinity of phenomena which, with the benefit of Quantum hindsight, may be seen to have actually been certainties." The way most physicists interpret quantum mechanics, the uncertainty about the outcome of a measurement is not "only in the mind of an imperfect observer," but rather in the world itself. For example, before you measure the position of a particle, it simply doesn't have a position. It's not just that we don't know its position, it's that it doesn't have a position. This interpretation seems to be forced on us by experiments like the famous two-slit experiment. In this experiment, particles are fired at a barrier with two slits in it, and then their positions are recorded when they strike a screen behind the barrier. These positions form an...

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 possibility that the world is black and white, which you mention in your question, has been discussed by those great philosophers Calvin and Hobbes .

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...

Space is not expanding "in" anything else. The distances between points in space are increasing, but not because they are moving through some "superspace" that contains space. Mathematicians distinguish between two different approaches to defining geometric properties of a space: the extrinsic approach and the intrinsic approach. The extrinsic approach involves relating the space to some larger space that it sits inside; the intrinsic approach makes use of only the space itself, and not some larger space that it sits inside. For example, suppose we want to study the curvature of the surface of the earth. One way to see that the surface of the earth is curved is to image a flat plane tangent to the surface of the earth at some point. We can detect and measure the curvature of the surface of the earth by noting that the surface deviates from the tangent plane, and measuring the size of this deviation. But this deviation takes place within the 3-dimensional space that the surface of the...

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...

There are two aspects of your argument that worry me: 1. If I understand your argument correctly, you are saying that the liar sentence expresses two different propositions, namely: (a) This sentence is false. (b) "This sentence is false" is false. But don't (a) and (b) mean the same thing? Isn't that, in fact, what your argument shows? So is this really two different propositions? 2. You say that you think the liar sentence is false. But aren't you worried that that would make it true, since it asserts--correctly, in your view--that it is false? By the way, a common first reaction to the liar paradox is to think that the problem is caused by the phrase "this sentence". There is a wonderful example, due to Quine, of a sentence that achieves the same effect as the sentence "this sentence is false" without using the phrase "this sentence". (The example makes use of the same mechanism for achieving self-reference that Godel used in his proof of the Incompleteness Theorem.) ...