If mathematics is just a useful tool we made up, then how can it describe the world with perfect accuracy? And how come as soon as you remove mathematics and mathematical equations from physics, physicists can no longer objectively explain or predict anything?
In: Physics
If you think of how humans explain the world around them as building blocks, Math is the foundation block.
Math is, as you suggested, a tool by which we can enumerate things, and describe stuff, but math by itself is a pretty ethereal concept.
Physics an application of math, using that tool to describe and predict the world around us that we observe. We see a stone fall. But to describe how fast it fell, how hard it hit the ground, how it deformed the dirt on impact we apply mathematics to the science of physics.
And the blocks continue to build from there. Chemistry is an application of physics, really. The ways that chemicals react really boils down to the physical attractions between atomic particles. And by extension requires mathematics to enumerate those interactions and balance chemical equations, etc.
Moving further we get to biology, which is really just applied chemistry. All the metabolic processes occurring within the cellular structure is just a series of chemical reactions.
In all these cases, the basement of the stack is Mathematics.
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