In another thread this issue has come and I already given some of my views on it. But as I feel that it is somewhat marginal in that context I decided to start a new thread.
Ever since Galileo we know (loaded term so bomb away at this notion that I'm willing to argument on it) that the right way to do physics is with mathematics. There is no denying, there is no discussion and there is no single shred of controversy in this point. If you think you do, once again just bomb away that I'm willing to discuss things in this point too.
The problem is that a lot of people that this is an elitist, snobbish approach to Physics and so try to contest it. Most of these people, from my personal experience, are armed with the very dangerous weapon of pop science and/or some courses physics trimmed down to people that won't follow a career in physics. Since they are relatively smart in their area of expertise, and with good probability they got good grades in those two or three trimmed down versions of physics they feel they got a feel for the subject.But the hard truth is that they don't have a feel for the subject (I'm not counting highly exceptional people on this analysis).
Physics is basically modeling real world events (this is an oversimplification to which I don't fully agree but it's enough for our current discussion I think. But if you feel like arguing about this too I'm all for it.) and this modeling is done in 99.999999999999% of the cases using differential equations. Ordinary, partial, linear, nonlinear, scalar, vectorial, tensorial are the kinds of equations that appear and if you do have a feel for physics you should have a feel for those types of equations (this is a necessary but not sufficient affirmation). So if you can't understand solutions of ordinary differential linear equations you certainly don't know much about physics. Of course you can have some notions, and solve some textbook exercises of physics (nevermind the fact the real world isn't very much like the textbook world) but in what concerns real physics you are very much lacking if you can't even do that highly simple thing: to interpret the solution of an ordinary linear differential equation. As an example I leave this:
Given y=y(x) solve py''=-qy where y'' denotes the second derivative of y in order of x and p, and q are constants.
Can you see what that problem is about? Can you solve that problem even without knowing what it is about? Of course you can. Just consult any elementary book on ODEs. After arriving at the solution can you understand the why of the solution?
Answer truthfully to those questions and then ring me a bell. When I propose this I'm not proposing it with an elitist flavor (at least not that I'm aware of), but I do it to urge people to have a knowledge that is informed, formed, responsible, and self-critical. Of course I know we don't have time to do everything we like and some things just have to be done in a more superficial way, but what I'm arguing for in here is that at the very least people do have the conscious of superficiality of some areas of their knowledge.
To finalize I'll just leave three links that stress the need of Math in Physics:
Siegel's page also have some other interesting articles: