Why can mathematics and physics simulate natural phenomena so closely in thought experiments, calculations and computer programs?

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Why can mathematics and physics simulate natural phenomena so closely in thought experiments, calculations and computer programs?

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Because mathematical and physics models that do not correspond closely to reality are deemed incorrect and not used, then the scientists get to work trying to fix what is wrong or looking for a new approach. Your question is quite like asking why a car is so good at travelling on roads: because the attempts at building a car that resulted in something that really sucked at travelling on roads are simply not selected for mass production.

Because science is used to explain what we see. If someone comes up with a mathematical formula to explain something, but the math doesn’t actually work out, then we…don’t use it, because it was wrong.

The math that we use to model natural phenomena is used *because* it works out correctly. If someone in the future discovers a problem with our current formulae and fixes it, then we’d start using their corrected formulae instead.

You have two options:

– The universe has no rules, or
– The universe has rules.

Which do you choose? Option 2, good. Now if it has rules you need a system of rule that is predictable and logical. You have only one option for this: math. You measure a “before”, you add a measurable “event”, and then you get a measurable “result. “before + event = result”. Logic.

Logic says if reality has rules, you can model it with math.

Because that’s the point.

The main reason for much of the work in physics and math is to describe the natural world. And we can come up with lots of models that are just wrong, but we can “easily” verify that they are wrong because the baseline is the observable world.

I always bring this back to the frustrating opening to the Bee Movie:

> According to all known laws of aviation, there is no way that a bee should be able to fly. Its wings are too small to get its fat little body off the ground. The bee, of course, flies anyway because bees don’t care what humans think is impossible.

Think about this for a moment. We have come up with some understanding of how aerodynamics work, how planes work, etc. Maybe this model works really really well for an airplane, or a bird, or whatever. Now we try to apply the model to a bee, and it falls apart. Does that mean that a bee “defies the laws of physics” or does it mean that we just don’t understand how a bee works?

Obviously there’s a way to describe how a bee works. It flies, so if our math says it can’t, our math is wrong. If I’m not mistaken (this isn’t really my area of expertise) the “laws of aviation” that don’t work for a bee are the models for how airplanes or gliding birds fly. If you think of a bee more like a helicopter, then it all works out. As it should.

Math and physics are ways of reasoning about abstractions.

For example, 1+1=2 will be true for anything you can consider to be ‘one of a thing’.

One drop of water plus another makes two.

This kind of logic only works as well as the abstraction involved in calling something ‘one of a thing’, though. Does one wind plus one wind equal two winds? Where does a wind begin and end?

Even the water droplets are constantly losing and gaining molecules as they evaporate or condense from the surrounding air.

The logic of math applies unfailingly to abstractions, but abstractions only sometimes apply well to parts of reality.

When we shape our abstractions carefully to fit reality, the logic of math can carry us to accurate conclusions about the world. The accuracy of these conclusions will still be limited by the lossy compression of reality into the abstractions being considered, though.

For example, talking about the ‘pressure’ and ‘temperature’ of a group of ‘molecules’ can help you make a lot of predictions at that same level of abstraction, but it won’t give you any insight into the individual movements of atoms or the quantum level phenomena happening within/between them.

Physics is literally about figuring out how nature works and what rules govern the processes in our universe. Once you discover the rules, you can predict the outcome of that process in real life. When you learn physics equations in school, thats not just some random equations somebody threw together to torture you. Its a description of how things in nature act.

There isn’t a widely accepted answer to this question. Why should the universe follow consistent rules at all? How can we even judge whether maths and science work well to describe the universe, without assuming that they do work? What even are maths and science? These are not straightforward issues.

The physicist Eugene Wigner wrote a paper called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, in which he argued that it’s actually pretty spooky how neatly and successfully the universe can be described by mathematical theories. This paper was very influential, so that phrase “unreasonable effectiveness” comes up a lot. Some people agree with him, some people agree that maths is very good at describing the universe but think there are reasonable explanations for it, and some people disagree that it’s especially effective.

Well… “closely” and “accurately” are difficult things. The physics modelling done for engineering for example isn’t perfectly accurate, which is why we we have tolerances, coefficient factors, margins.

If we were actually able to accurately model stress on a structure and the behaviour of the structure and material, we would be able to make structures that exactly have the correct properties in size and material. However we are not, so we scale those up just to be sure.

But now to the “how”. The rules that govern everything in this world, as far as we know; are actually rather simple for the most parts. Especially since we simplify them to be simply enough. Example: gravity. We just define gravity as a constant acceleration towards the ground. That is not what “gravity is” but for our purposes it is good enough of an definition. If I throw a ball and want to know where it approximately lands, things like spacetime and quantum particles are irrelevant to me. Now consider that humans and animals are instinctively really good and just estimating this, while not having any concept about gravity or how it works.

So we have defined the world to function according to certain rules. We have huge collection of these rules and we keep adding them together. For example something that one might not even consider is that if you put a glass on a table, it weight more than that glass in a vacuum chamber. Ok… the glass doesn’t weigh more, it weighs the same as an object, but if you put in on a scale that is on your table and one that is in a vacuum chamber, you must remember that on the table the glass is filled with air, and this air does have mass, and that this glass also displaces air which means there is buoyancy. This is not significant on the scale of a drinking glass. However when you are at the scale of the *National Institute of Standards and Technology*’s *Million-Pound Deadweight Machine*, to get accurate calibrations you actually have to consider air’s buoyancy (along with getting more accurate measurement of the gravity. Generally the 9.80665 m/s^(2) is good enough, however for the accurate calibration that lab provides it isn’t).

But how can we then simulate the behaviour of a steel beam, accurately enough for most purposes, for example? This once again is shockingly simple. When I sat down on the lectures on this subject and realised what the matrices meant, I was shocked. To simplify it: We take the steel beam and treat it is a collection of point that are connected to each other with springs. We know the properties of these “springs”, we know what the starting state of each of these points, and then we simply start to calculate the interactions of these points until we get to a point where we decide it is accurate enough. We use the same principle for fluid dynamics. We just calculate interactions of imaginary points according to rules of interactions we have decided to be good enough descriptions of the world around us.

Now here is the important point I want reinforce. These simulations we do and formulas we use are “good enough”. Something that physicists being insufferable creatures that they are constantly like to poke at us engineers about… how we aren’t accurate enough or don’t actually simulate reality because we don’t work with quantum-whatevers. Then we proceed to ignore them and go out drinking with all chemists who also have had enough of physicists going on about quantum-whatevers. Granted… They are right… we don’t simulate reality, we simulate close enough approximation of it; however we are not going to give them the satisfaction of admitting that when we can just ignore them and get drunk with people who understand that liquor is a solution.

P.S if you want to kinda understand what we do with the point, in the example of a steel beam for example. Then take a marble/bearing/ball, and connect 6 rubber bands to it. You can tie the other ends of these bands to whatever you want, in our simulations you can get really crazy relations of these “springs” (Since the computer does the calculations we really just fiddle with them if the results are “wrong” based on the estimations we made of them (basically if you do the simple calculations on excel then simulations results should be in the ballpark of those, if not you need to adjust your simulation or consider where you excel goes wrong)). Then if you pull on one of the rubber bands you see that the marble in the centre adjusts accordingly. All we do, is that we calculate the movement of that point according to the properties of the rubber bands influencing it. We do this for solids, gasses and liquids.

If you want more accurate results, you add more points and connections. However at some point you get to a scale so small that those dreaded quantum-whatevers start to poke their ugly heads and the faint smell of day old yogurt fills the room as physicists rush in to act as if they are better than you.

Why do I fit on every door, and into my clothes, and into my car? isn’t it an amazing coincidence?

Of course it isn’t – they were designed for that purpose. Same deal.

Because natural phenomena are incredibly complex and often happen incredibly quickly. For an example how quickly they happen drop a object off of your desk, it falls on the ground in maybe a second. Developing a mathematical model that explains it took till Issac Newton. For complexity just look at 3Blue1Brown [explain the physics of a ball swinging on a pendulum](https://youtu.be/p_di4Zn4wz4?list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6&t=350), a simple introductory physics problem. After about 30 seconds he says the components of the equations are deceptive ‘lies’ and that after about 1 minute I at least get really scared by the horrors that is differential calculus.

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.