Mathematics really is building a logical model based on assumptions that you consider true regardless, called axioms. When your axioms match what you can observe in nature, the logical system that follows is pretty accurate at explaning nature. An exemple of an axiom that we use is that a + b = b + a, and you can indeed place two objects on you left and one on your right or vice et versa and you will still have three objects in total. You could totally build another model with different axioms where a + b =/= b + a, but such a model wouldn’t predict well the natural world.
Edit: Typo
Experimental and modeler in physics here.
Fundamentally the reason we can’t do it is because it is too complex.
We can model the motion of TWO particles essentially perfectly. Once you get to three particles or more very small initial conditions can result in massively different final results. This is known as chaos.
When we model systems we run multiple ‘samples’ and look at the average to predict a final results. This takes significant computer ‘power’. We can run millions of samples… but not billions. This because the runs of millions might take a few days. Billions would be tens of years.
Now consider that one cubic centimeter has 10^19 particles….or ten billion billion particles. Such a simulation would take millions of years.
We base them on the assumptions made about what we are modeling so if we’re modeling cellular division we base it off the math of one becoming two and two becoming 4 and so on. It gets more complicated but that’s the gist of it is that we just write the assumptions science is wanting to test in math form and they can extrapolate into models of the physical reality.
**”All models are wrong, some are useful”**
This is a famous saying in mathematical modeling, the field of science that uses math to mimic natural processes and predict outcomes. Often, we play around with models enough that they are actually quite reliable at predicting real-world outcomes. They are almost always oversimplified, missing components that exist in the real world, etc, but they are *good enough*.
It’s worth noting that the more stochastic/random the system is, the more complicated the models often need to become in order to be reliable. This happens in weather/climate models. It can take days to run simulations on super computers because there is so much data and so many calculations, and so many interactions between variables to account for. But in more simplified/controlled systems, it might be possible to use quite minimal math to model what is happening.
Mathematics and Physics were built based on what we observed in nature.
If I have 🍎, then put another 🍎 next to it, I now have 🍎🍎. We needed a name for this, so we called the number of apples “2” and called the process of adding more apples “addition.” Nature came first, and then we wrote rules to fit our observations.
Physics was the same way. We threw a rock and realized that if we put the same amount of “oomph,” into the throw and aimed at the same spot, the rock always landed in the same place. We then realized that by measuring the speed and angle, we could use math to predict where it would land and again wrote rules to match these observations.
Both of these situations involve writing rules to match our observations about nature. We know we’re correct (or at least correct enough) when we can take those rules and use them to predict things outside of our testing scenarios.
A great example of this are black holes. We saw stars moving in strange ways, and since we could estimate the mass of the stars, we could make an educated guess that there was a very large “invisible,” object exerting force on the stars. By doing more math, we realized an object might exist that was so massive that not even light could escape it’s gravity, and that might be our “invisible,” object. We built a massive telescope the size of the planet, pointed it where we thought one of these things might be, and bingo, [a black hole](https://www.nasa.gov/sites/default/files/styles/full_width/public/thumbnails/image/20190410-78m-4000×2330.jpg?itok=SGK55kJs), confirming our guess was correct. If it wasn’t, we’d need to go back and re-adjust until we could reliably figure out what was causing those stars to move.
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