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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There is a philosophical debate on whether math exist concretely in the world but you answered your own question.
Math is a tool that we can use to model the world. Physics is not this thing that exists as an independent entity, it is the specific form math take when being used to describe physical phenomenon.
Another way to think about it: Physics does not USE math, physics IS math but for the world.
Because everything can ultimately be described in numbers. Size, mass, speed, acceleration, frequency, etc. All of the essential characteristics of everything are ultimately a number or equation of numbers. Even colour can be described as a wavelength.
Scientists have discovered that the universe operates according to extremely predictable rules. Math is a great tool for summarizing these rules, so that if you know the conditions you can predict quite precisely what’s going to happen.
For example, if you want to shoot a cannonball at a distant object, at what angle should you aim the cannon? Physics can give you a specific number, not just an idea.
If a shovel is just a useful tool we made up, why does it move the dirt we find in the world so well?
Because we designed it that way. The same is true for math. The universe does things that correspond to adding and multiplying and cosines and integrals and differential equations, math was invented to describe what the universe does. There are areas in math the are completely abstract and don’t relate to any sort of physical reality, but most math has its origins in trying to describe some sort of physical phenomenon.
There’s debate over whether math was invented or discovered.
It may be that we discovered a process by which the world works and assigned symbols to it.
Imaginary numbers and quaternions (sp?) lead me to believe that it was discovered.
We didn’t “make up” math, all we did was create systems of how we understand math. Numbers and operations mean things, regardless of what we call them or how we do them. You can add 1+1 to get 2 in base 10 or you can use binary, but the information that operation conveys is the same. It was the same 5 million years ago before humans existed and will be the same 5 million years from now when humans (probably) don’t exist anymore. I mean, the universe has existed for 13.7 billion years. Are you really suggesting that the laws of physics didn’t exist until humans came around to describe them?
Math, and subsequently, physics, also *don’t* describe the universe with perfect accuracy. We know there are pieces of the puzzle missing. We know that, for example while relativity and quantum mechanics are extremely successful, they aren’t complete. When we discover these missing puzzle pieces, we incorporate them into existing theories or make new theories.
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