Do we rely to heavily on mathematics in applience to physics especially in theory ?
Yes we do. When you want to make predictions you have to calculate numbers, and that means you have to have a mathematical model of your physics.
Did you ask "too heavily?" Then no. Mathematics is very important for physics, because mathematics is objective, clear, logical and crosses all national boundaries. It's the best language for the job.
Well the interesting part is that a pure mathematician (no intention to hurt your feelings if you're one) who works on the same set of differential equations for instance, that a physicist does, is concerned more about the mathematical functions which are its solutions. He is normally not concerned with the visualization and the physical meaning of these equations (which become very complex as the level of detail and precision increase).
On the other hand, a physicist would like to see the differential equation and deduce from it the associated boundary values/eigenvalues to reach conclusions about the meanings of the solution in the real world, instead of becoming too absorbed in mathematical analysis that takes him (or her) away from it.
So while existence and uniqueness of a solution may be pure math for the mathematician, it means a lot in terms of a theory or a physical meaning for the physicist.
To come back to your question, mathematics is an integral part of physics and you can--to a good extent--say that physics is the real world+related mathematics+conclusions...
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