Why can mathematics and physics simulate natural phenomena so closely in thought experiments, calculations and computer programs?
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The simple answer is, we don’t know. Check out the well-known essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”: http://www.hep.upenn.edu/~johnda/Papers/wignerUnreasonableEffectiveness.pdf
Physicist look at what they see around them, create rules about what they see, and use math to find the consequences of those rules.
In thought experiments they just look for a contradiction. Aristotle’s original principle of all things naturally being in a state of rest contradicted the movement of comets in the sky.
In calculations and computer programs they will often know what will happen a very small moment in the future. It is easy to predict the weather tomorrow but hard to predict the weather next year. They can then keep predicating a small step in the future until they finish quite far from where they started.
Basically because math is the language of the universe! We do math to figure out how things are in the world around us and physicists apply it to things that we can see repeatedly. When we do that to one thing that’s called a specific case. When we do it to a lot of things that all behave similarly that’s called a general case and we can basically assume that anything that belongs to a group of behaviors will behave according to the general case. Of course we do as much testing and grouping as possible to make sure that the right math is being applied to the right thing.
Computers are basically just really complicated calculators that we’ve made to do a lot of stuff and be easier to use so it’s really very good at doing math. You feed in a bunch of calculations that you think might apply as a simulation and then see which one is closest to the way the world actually works! We’ve been doing this for a few thousand years now (minus the computer part) and have gotten pretty good at it.
It’s important to note that it’s definitely not perfect. A large portion of the time it *works* and that’s important, but each time it works for the first time there’s a whole human history of thought and math and a bunch of people trying and failing and partially succeeding that gets us to where we are today.
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.
To me it is because we know how things react with eachother. Computers and calculations are kind of linear. But the fact that there are so many combinations of this reaction happening in nature, we can’t really define what is happening on a let’s say quantum level.
**”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.
The language of the universe as it exists is spoken through math. Think of it as the foundation and structural integrity of your house. Now how we decorate the house is up to all of us that would be emotions and such.
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.
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.
Mathematics abd physics exist to allow us to understand the world we live in. It’s a framework/model. Physic try to explain how things are work. Mathematics is the framework used to define it.
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