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