ice109 said:
i guess you could say that in math when you solve a problem you literally solve a problem. in physics when you solve a problem you define something( a quantity, a force, a field etc). they're wholly different practices.
I just fully realized that you said this: I should point out that even in mathematics, many problems are solved by defining things. Often, you are faced with a situation where you know the conclusion to a theorem and the problem is to figure out what hypotheses are needed.
One of my favorite examples was a lecture on Bézout's theorem: suppose you have two plane curves
A and
B defined implicitly by the equations
f(x, y) = 0 and
g(x, y) = 0 respectively, where
f is a polynomial of total degree
m and
g is a polynomial of total degree
n. Bézout's theorem states that if their intersection is finite collection, it contains
mn points.
For example, consider two ellipses:
9 x^2 + 16 y^2 - 25 = 0
16 x^2 + 9 y^2 - 25 = 0
Both of these equations have total degree
2, so there should be 4 points of intersection. And you can check that they intersect at (\pm 1, \pm 1).
Now, as I've stated it, Bézout's theorem is false. It turns out that you need to consider points at infinity, complex points of intersection, and count multiplicities.
Today, there are outstanding questions that have this problem: we know what the theorem should be, but we don't know how to set up the hypotheses correctly, and the road to progress on these questions is to discover how to define things correctly.