The Einstein black hole equation got me thinking. In a universe where so many things seem random, unexplained, and misunderstood, how can numbers on a paper explain and predict how the universe works?
It blows my mind that this concept (math) is so young relative to the universe and can be used to explain how and why things happen. Where’s the connection?
In: Mathematics
Mathematics merely provides a model for the physical world based on logical foundations and calibrated to physical observations.
There’s a deep question about whether mathematics actually explains it, or merely models it.
Think of it like when you look at a picture of a star or a mountain or a cloud or a tree.
You’re not looking at the actual object but a 2 dimensional representation of it that to your eyes looks similar to what it would look like if you were exactly where the lens was.
The image provides a good representation of that object, in the case of the star, much much older than the invention of the camera. You can go deeper and believe the image captured some “essence” of the object but that’s a philosophical question now.
That’s like the use of mathematics to describe the physical world.
Our concepts, or our understanding of math is relatively young. But the actual theorems of math are eternal. 2 + 2 = 4 is a truth of the natural numbers that has always been true. The Pythagorean theorem is a true consequence of the axioms of Euclidean geometry, but we’re able to understand non-Euclidean geometries as well.
The universe, or at least many aspects of the universe, behave in consistent ways, making them amenable to mathematical modeling. If the speed or acceleration at which a given rock falls from a given height were to change every time we tried dropping the rock from the same height, our model of gravity would be wrong (and basketball and diving would be different, if we could even have such sports under such conditions).
The universe hardly ever does random or unpredictable things, and the things that are random appear to obey the law of averages, making them predictable at macro scale. If they didnt, complex structures like life could not exist, so we would never have experienced that universe.
While human understanding of math is relatively new, the inexorable laws of mathematics supersede the universe and exists independently of it (therefore, also before it).
Because the universe is predictable, there is math that models it, its not that math controls the universe, its that there IS math to model everything, and that includes modeling how the universe happens to work. What exactly that math is could vary by universe, and so science uses trial and error to figure out which of the infinite possible equations are the correct one for our universe.
If something can me measured, you can take those numbers and figure out the math to model various properties of the universe.
The physical world doesn’t operate on maths. The operating system of reality isn’t math-based. We don’t know how reality works. We have absolutely no fucking idea whatsoever, and we never will.
But we can use our own language of logic and numbers to try and describe what see so accurately that in some instances, the descriptive (what have I seen) becomes predictive (what do I expect to see next). This we call mathematics and physics. It’s an attempt at making a language that is purely based on logic and reason, in order to most accurately describe what happens, so that we humans can talk about what kind of causes might have what kind of effects.
Maths and physics are languages. They have no direct connection to reality. Their fundamental nature is not that they are *true*, but that they are *precise*.
To be more direct than some of the answers.
Math was invented to explain the physical world. That is why, fundamentally, it is so good at it. We defined it in terms of the things we saw. When we had difficulty understanding something with the math we had, we explored different ways to think about and calculate what we saw.
The basic principles, or axioms, of math are self-true or are simply a definition we set. There is non-eli5 conversation to be had down this rabbit hole if you choose.
Example Axiom:
Things which are equal to the same thing are also equal to one another. (7 is 7, so 7 equals 7, 7=7)
This axiom seems unneeded on the surface, but it begins to allow algebra. As you can substitute anything in for 7.
Most of mathematics is built up from these somewhat simple definitions.
Imagine that you know a box weighs exactly 1 pound. But then you weigh the box and it weighs 6 pounds. There is something in the box. Until you open it, you won’t know exactly what that something is, but using math you know that there is something that weighs 5 pounds in the box.
Similarly, physicists have encountered situations where observable results are not quite matching the predictions they’ve made. For example, light may not follow the trajectory that our math predicted, indicating that some other force has acted upon the light, such as the gravity of a black hole. The math tells us that something with that gravity exists, even before we are able to observe it with our own senses like sight.
The answers so far mostly talk about mathematics being made for and by physically modelling the universe. That is mostly true, but it misses the question:
It is actually quite extraordinary that simple equations describe and model our reality so very well. Life, consciousness, all those things could theoretically also exist in a much less “reasonable” universe with weird magics, time travel, and the sky turning green for an hour each day. More akin to the worlds of [Welcome to Nightvale](https://www.welcometonightvale.com/) and [UNSONG](https://unsongbook.com/), a surrealist horror-fantasy impossible to truly grasp by anybody.
Yet the universe is not, it is unreasonably simple. This is actually a very deep philosophical problem yet to be resolved, if ever possible.
There are some attempts at explanation, none final or perfect, but some at least not completely bonkers. A relatively common one springs from the _simulation hypothesis_, that the universe essentially runs on a computer. If so then simpler code that still spawns life is more likely than complex ones. This especially happens if one has a kind of evolution of programs resulting from simulations again running simulations within them, like we do already at increasing accuracy.
From that point of view, a simpler program-organism is much more likely to appear, and thus much more common, than a needlessly complicated one. Even more, one that is easier to understand by the sapient life within might be more prone to lead to computers and simulations.
But that is just one out of many options.
You may be interested in the essay [The Unreasonable Effectiveness of Mathematics in the Natural Sciences](https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf) by quantum physicist Eugene Wigner. There have been a number of replies to this essay. My favorite was by a statistician; let me see if I can find it!
At it’s root, mathematics is a language we use to describe the physical world. The equations and algorithms derived therefrom are only “true” insofar as they are able to accurately describe, and predict, the behavior of the physical world. The accuracy of these may be further described in terms of probability; the underlying assumption of a “law of nature” is that its accuracy (probability of matching reality) based upon the math is assumed to be 100%
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