How do we know that modern mathematics is correct? Thanks, Ryan.

Indeed it seems that we don't even know what it means to say that modern mathematics is correct. Does it mean that it contains no hidden contradictions, that it accurately describes 'the world of numbers', that it supports empirical science, or what?

If we define the science as the study of reality to find the relations among objects through scientific method, is mathematics a science? Or, it's something like logic which can be used in a wide range of sciences, philosophies, and even daily usages.

I don't much like the expression 'scientific method', because it suggests that scientific practices come down to one unitary thing, and because it suggests that that thing is fundamentally different from everyday forms of investigation. But your question still stands: is pure mathematics a science, alongside physics, biology, psychology, etc, or is it fundamentally different? Mathematics is of course used in other sciences, but that is not distinctive. Physics is used in chemistry, and chemistry is used in biology, to give just two other examples. Nor is mathematics clearly different on the grounds that it does not study real objects, since some philosophers and mathematicians believe that mathematical objects and relations are real, though abstract. But many (though by no means all) philosophers would say that mathematics is nevertheless fundamentally different from the (other) sciences, because it is not empirical: we do not need to test its results by means of observation and experiment.

Do people invent equations, or do they discover them? Examples of the sorts of things I am thinking of are Newton's laws of motion, or Mandelbrot's sets.

It helps to begin by distinguishing laws of nature from our hypotheses about them. Then the first question is whether there really are laws of nature out there. I'm one of those philosophers who believes there are, though just what it takes to be a law is hard to say. For example, is a law just an objective pattern of properties, or does a law have a special kind of necessity? But some philosophers would deny that even the patterns are fully out there, because they hold that the structure of properties is something scientists impose on the world: the world does not come pre-carved into natural kinds. So the answer to your question is a little complicated. Even if the laws depend on our own scheme of classification, it would probably be misleading to say we invent them: it is not as if we can just make them any way we like. And even if the laws of nature are fully objective and out there independently of us, it is still up to us to think up the hypotheses that are supposed to describe them. In any...

If a certain concept is seen as indefinable, such as the concept of infinity, how can it be said to exist?

I'm not going to comment on infinity, but on the general point. First of all, which existence is in question: a concept or a thing? If a thing, it would seem possible that there could be something that exists but that we cannot conceptualise. So here there would be no question of a definition, yet the thing exists. If a concept, well, one in fact argue that some concepts must be undefined, or else the chain of definitions would have to go on for ever. (Though circles might be another possibility.) But if it is OK, indeed obligatory, that some concepts are undefined, then maybe it is OK for a concept to be indefinable. There is a great deal of philosophical excitement over the question of what it is to grasp a concept, but I think that it is widely agreed that definition is not the only way in.

5 divided by 0? Personally, I believe that it is infinite based on the idea that division is just repeated subtraction just like multiplication is repeated addition. For example, in 4/2, it's pretty much like saying how many times can you subtract 2 from 4 before you get to 0.

I'm going to answer this question indirectly, by means of a simple algebaic argument that you may already know. Suppose we begin by assuming that A = B Now consider the following argument from that assumption: A 2 = AB (Both sides multiplied by A.) A 2 - B 2 = AB - B 2 (B 2 substracted from both sides.) (A + B)(A - B) = B(A - B) (Each side rewritten.) (A + B) = B (Both sides divided by (A - B).) (B +B) = B (A replaced by B, since assumed equal.) 2B = B (Left side rewritten.) 2 = 1 (Both sides divided by B) Pretty neat, eh? But there had better be something wrong with this argument, since the assumption is fine and the conclusion is crazy. If you want to figure it out for yourself, stop reading now; otherwise continue below. * * * * * * * The fallacy occurs in the line where I divided both sides by (A-B), and...

Is mathematics independent of science? And, vice versa.

I agree with Alex about the way mathematics is independent of science. Einstein proposed that space is curved and hence non-Euclidian, but this didn't undermine Euclidian geometry, because that geometry is about an abstract space defined by the axioms of the system, not about physical space. So Euclidian geometry turns out not to apply to physical space, but it has not been refuted by physics. There is however another way in which science and mathematics are not independent. Mathematicians may choose which problems to work on with an eye to what kind of mathematics might be particularly useful in science, and even more frequently scientists choose which problems to tackle by reference to the mathematical tools that are available to them.

if it's zero degrees out and tomorrow it is going be twice as cold, how cold will it be?

If a stick starts out being two feet long and then it becomes twice as short, it becomes one foot long. Since there is such a thing as absolute zero, I would have thought that twice as cold as temperature T Kelvin is T/2. (But I'm assuming that the Kelvin scale is linear...) So work out what the zero you have in mind corresponds to in degrees Kelvin and, as they say in England, Bob's your uncle.