# How do we know modern day math is correct? An example would be one is equal to zero point nine repeating. You can divide them both by three, and get point three repeating, but if you times point three repeating by three you can only get point nine repeating... another question could be, where does the rest of it go?

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've been adding 2+2 all day, and I keep getting the number 5 as the answer. Does the number 4 not exist, or do we just perceive it differently?

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...

# Quantum behaviour says that before a phenomenon is observed there may be a number of possible outcomes. Once observed, the number of possible outcomes becomes one; what actually happened? 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. Uncertainty exists only in the mind of an imperfect observer; there’s no such thing as foresight outside of a limited, dry mathematical framework. This leads me to think the following; i) That everything is as it is because it could not possibly have been any other way. ii) All the things in the universe whose extremely improbable existence we marvel at and things which everything else depend on who, if they were any other way, lots of other things wouldn’t work either, were actually (in retrospect), absolute certainties. Is this a gross misunderstanding of Quantum theory, an obvious conclusion, or a line of thinking with some mileage? I can see it leading...

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...

# Are there any contradictions of the Axiom of Choice (AOC) that are consistent with basic mathematical logic? Has anyone tried to develop a non-AOC theory?

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.

# Can something be infinite if there is a definitive number of it? Here's an example: I take a number, the largest I can think of, and never stop adding one to it. The number becomes infinite. Now if you take the number of human beings, and never stop adding to it, is the number of human beings infinite? In contrast, dinosaurs cannot be added to therefore they would not be infinite. Does this make sense?

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,...

# Are you as Philosophers allowed to say that the rock on my desk is red? For we really don't know. We perceive it as red but what if our eyes are not showing us what is really there? For all we know, everything could be black and white.

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 .

# If an infinite number of monkeys were at an infinite number of typewriters, would the work of Shakespeare eventually come out?

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...

# Science states that space is endless, and ever expanding. But, if we are inside the planet earth, the planet earth is inside the galaxy, the galaxy is inside space, then what is space inside? What is it expanding in? And if space is endless, how can it expand?

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...

# Would you agree that numbers are synthetic truths rather than analytic truths? This is because I can imagine a universe, whenever I walk 2 meters foward, space itself 'bends' so I end up 3 meters ahead of where I started. In this universe, when 2 is added you end up with 3. 2+2=5 (or maybe 6).

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...

# Is the Liar paradox, stated as "this sentence is false" false? The Liar surely means, "The proposition expressed by this sentence is false", but this implies that there is one and only one proposition contained within the sentence. If this is not the case then the whole statement is false because "The proposition" must pick out exactly one object. The direct proposition expressed is "This sentence is false", yet surely since the predicate "is false" applies to the sentence in question, "This sentence is false" is false is a proposition that is also logically entwined with the sentence. Since the sentence expresses two propositions, and not one, there is no object which corresponds to "The proposition expressed" and so the whole sentence becomes false.

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.) ...