For example, we have differential equations that model, for example, resonance, vibration, heat transfer, the motion of a spring, etc. How were these equations first calculated and derived, and to such a degree that we can be sure of their accuracy in modeling real life phenomena? What is the process for actually deriving them like?
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They just understand math better.
The best analogy I can think of is language.
It’s fairly easy to learn some phrase in a foreign language. You may even learn that there’s some word in the phrase that you can substitute in to change the meaning a bit.
But you’re almost certainly using the phrase wrong. You’re probably using it in the wrong context, you’re probably mispronouncing words, you may not even be remembering the phrase correctly and you certainly can’t figure out how to create a new phrase in that language.
How do people create new phrases in a language? They learn the language. They learn the patterns of how different ideas are expressed. They learn how to put those components together to express more complex ideas.
It’s the same thing with math. You may learn some formula, plug in values and get an answer but the people who create those formulas think of math as a language.
I’m not a physicist but I used to study economics and I can give you an example of why my thought process was.
The formulas generally describe some relationship. Variables are basically nouns so I’ll need at least two variables, maybe more if I think there are more factors involved in some relationship. Next I start to think about how they’re related. If they move in the same direction I can just put them on opposite sides of the “=” sign. If they move in opposite directions I’ll add a “-” to represent that. I’ll also look at the nature of the relationship; depending on how sensitive one factor is to an other I might add a placeholder for a coefficient or an exponent to represent that. That takes care of a lot the simple models. If I suspect that terms interact with each other I multiply them together.
After all that I would run regressions of my data on that proposed formula. That will get me empirical values for the coefficients and some other metrics that can tell me how likely my formula is to match the data.
I can also provide an example of how I think about physics equations. Eg F=MA. You can just plug values into that but what does it really mean? That formula says that there are 3 things that matter when it comes this relationship; force, mass and acceleration. For one thing that means the formula leaves out a lot of things; we don’t care about color, texture, shape, time of day, or phase of the moon. It also describes the relationship between F,M and A; for any given mass if we want more acceleration we need more force, if we want to accelerate something at a certain rate we’ll need more force as the mass gets bigger, if we’re applying a constant amount of force it less massive objects will accelerate more. That’s all pretty intuitive. It also tells us that the relationship between mass and acceleration is scalable rather than constant. That is, I can’t generally add something to mass and and subtract the same amount from acceleration and hope to get the same force, but if I double mass and halve acceleration I’ll have the same force.
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