Q:

How can we depend on the mathematical axioms that break down at the nano-level?

Mathematics "makes sense" at our scope of view. For example, gravity seems to obey the fact that it accelerates at a rate of 9.8 meters a second^2. However, when an object drops, the force of gravity is too weak to overcome the electrostatic force - so it never reaches 0. 
Could we say then, we "discover physics"? We describe certain equations for the physical phenomena that is happening? Couloub discovered his equations that led to the above conclusion.
Also, why don't we see a bright sky at "night"? Because of physics (universe expansion, etc).
My question is, how can we depend on mathematics 100% in physics if it leads to obstacles like these? Instead, if we took it with a large grain of salt, it could work.

A:

Mathematics is just a systematic way of stating facts about the world.  It is only useful inasmuch as it is internally self-consistent.  The latter fact means that there is nothing to "assume" about mathematics.  It is a relationship between axioms and conclusions that enables one to succinctly summarize many observations.  
Something like Galileo's gravitational experiments in Pisa is necessarily logically prior to a conclusion like "in the absence of air resistance, all objects near to the surface of earth will accelerate downward with an acceleration of $9.8\,\,{\rm m/s^{2}}$."  All the latter statement does is summarize the experiments.  And all mathematics gets you is having the power of having that summary turn into an assumption for a later experiment, which can then be checked against other things.  
In the end, you could, in principle, do physics without mathematics.  It would just be infinitely harder to understand any of it.  Mathematics is just a language that we can use to focus our thinking.

