How does math explain the universe and physics so “conveniently”?


Equations like E=mc^2 and stuff like how the force og gravity becomes 1/4 as strong when the distance between the objects dobles. Similarly with braking distance with cars, double the speed and the braking distance quadruples. These all seem to fit so well.
Have we made math to fit so nicely with physics? Am I thinking of all this wrong? Since I feel it like it would be to big of a coincidense that we can so easily use equations to predic physics. What is actually the reason for this?

In: Mathematics

Youre thinking backwards.

For the equations to make sence, theyre taught in the path they were discovered.

We didn’t study vehicles and time their speed and breaking distance/time, and make an equation to fit them conveniently.

The math already existed. Its conveniently convenient because it fits the way it makes sense.

Math is simply an incredibly complex and internally-consistent arrangement of patterns. In fact, any pattern that can exist can seemingly be somehow replicated via math.

This means that we can recreate almost any pattern we find in our universe using math. Once we have recreated a pattern in math, we can use that to make other predictions about the universe.

Someone thought long and hard to come up with equations or inequalities which are mathematically true, and the applications of those equations range from simple to incredibly complex.

Tl:dr; math makes things easier by being hard the first time.

Assuming something can be quantified and is related to another thing in a non random then it should be describable by math. As long the number of things you are trying to describe remains small it make sense to me that the relations aren’t terribly complicated.

Complexity shows up when you try to add more variable. For instance with your breaking example the relationship is simple as long as you are only dealing with breaking force and speed. But in reality as the car slows down the air resistance will also drop, slightly increasing the breaking distance. A lot of commonly given physics examples are chosen because the effects of the other variables is small, but there are lots of places where they aren’t.

It’s probably also worth noting that some relationships between units are simple because they are defined that way. And then we assign a meaning to the bigness of that number. For instance F=ma requires an ugly constant to be made to work if you are working in non SI units.

Many (most?) famous equations a*ren’t* perfectly convenient.

The foundation of these equations are just simple relationships between things we find useful to define. F=ma just means that double the mass requires double the force to accelerate, or the same mass requires double the force to accelerate twice as fast. each side increases proportionally to each other. The fact that m & a and being multiplied together on that side signifies that increasing either of them increases force by a proportional amount. F=ma is nice because each side is proportional 1:1.

**The bigger part of this** is that many (most?) famous equations a*ren’t* perfectly convenient. the ‘c’ in E=mc^(2) is the only hard number in there, and it’s about 299,792 km/s, and it is the number that tells you exactly what the ratio between the two sides of the = is. 1 ‘unit’ Energy is equivalent to 89,875,243,264 ‘units’ mass. Hardly a convenient ratio.

Similarly, the equation we use to calculate the exact force of gravity between two masses is F = G([mass1*mass2]/distance^(2)). The d^(2) is tells us gravity falls off at the square of the increase in distance (2x distance = 1/4 force, 3x distance = 1/9 force, which is because distance is one dimension but gravity spreads out over 3 dimensions; this is true of ectromagnetic fields too). G, however, is the gravitational constant, the only hard number in the equation, and is about 6.67430(15)×10^(−11) m^(3) * km^(-1) * s^(-2). This is, obviously, a weird inconvenient number, but it shows the exact ratio between each side of the equation for the units we created.

TL;DR most famous physics equations record proportional relationships between things, and it’s very common for one thing to double if another thing doubles (like, mass and force, or acceleration and force). “2x = 1/2y” and “2x = y^(2)” are also common in nature. There are relationships in nature that are mathematically more complicated than this, but the most common and most fundamental relationships are often the simplest. That just seems to be the way the universe works.

Math, as I believe you intend it, is a descriptive tool. The reason it explains the mechanisms behind our life is because it’s the tool best suited to do that.

Think of it as a universal language for quantities and measurement (Very reductive, but bear with me). The same way we can describe a tree, a person or a feeling, we can describe a ratio, predict the outcome of a situation and trace the orbit of planets, or of electrons.

In a way, yes, we have made specific equations to fit our observations, much like we make up new words to describe new objects or aspects of reality. We began counting – the elements were there, we just needed a way to say how many – then adding and subtracting to keep track. You must consider that every bit of math has been formulated, analyzed, refined and perfected, or discarded and reformulated, every time a challenge to it ensued. New phenomena or observations may require revisiting existent laws.

What makes it through is what we use as math and as a base for any future theory. As for why things follow certain patterns, that I do not know. But math specifically looks at those patterns and seeks to describe them, it doesn’t “happen to match” right away. If anything, any theory is studied on a sample or an abstraction, and any new fitting result just means that we observed and formulated our hypotheses correctly.

The universe is logically consistent and math is made to model that consistency. More specifically, I think the answer to what you are asking is calculus.

If you are in a vehicle with constant speed, your position will increase at a constant rate. However, if you have a constant acceleration then your velocity will increase at a constant rate and your position must increase at an exponential rate.

The squared term in both e=mc2 and kinetic energy (re braking) exist because of this kind of relationship.

Math is more than the study of numbers. It’s the study of patterns. We can do all of that math without looking at the universe or doing science. You make some marks on a piece of paper, then set up some rules for how to move those marks around, then move them around and see what interesting patterns happen. That’s math.

So when we look at nature, we also see patterns. We can go back to our math games and say that pattern of paper scribbles looks really similar to this pattern of how a block slides down a ramp, or a planet orbits a star. If you’re clever about it, you can then start to come up with mathematical “games” that fit patterns within science from the get-go.

The universe has patterns. Math is the study of patterns.

My interpretation is this, that Math is a human invention, and we invented it in order to describe the rules of this universe.

For the simplest example, why does 1+1=2? Because we observe that whenever we add 1 of something to another 1 of something, we get 2 of such things. And this rule appears to be universal, in that it does not matter what that “something” is. Thus, we invented the mathematical expression of Addition to describe this universal rule.

The same idea applies to more complex mathematical expressions, which are used to describe more complex rules in the universe. In fact, there are very advanced physics questions that are waiting for new mathematics to be invented to help solve.