# How do scientists and mathematicians create mathematical equations from real world phenomena?

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Hi everyone!

Basically what the title asks. I was wondering if there was some sort of one to one pairing between certain phenomena and math concepts. For example, I’ve heard that multiplication is an indication of interaction between two variables.

Another good example of what I mean is Maxwells equations. How was he able to figure out all the details to comprehensively describe electromagnetism? How did he know what math tool to use to depict a real world phenomena?

How can one read these equations and discern what would happen in reality?

In: Physics

the philosophical answer is that everything can be, and will eventually, has a mathematical explination.

we all new gravity existed before Issac Newton, we all knew if you dropped something it fell towards the ground…..what we didnt have was a way to express it in math so it could be used consistently and reliably. The same thing happens in most of math and science.

we observe something, then try to define it better

Lots and lots of experimenting.

Let’s say you wanted to find a formula for how wire length affected voltage and current.

You’d start by… getting a piece of wire and a power source, preferably of constant voltage. You then stick an ammeter and run current through differing lengths of wire, and chart your measurements on a graph.

You get a straight line pointing down! This tells you that if voltage is constant, current is inversely proportional to length, or I α 1/L.

Now you do the same thing for voltage, adjusting the power source until the current read about the same value each time and then taking the voltage across the wire sections. You plot them on another graph and you get another straight line, pointing up this time! This tells you that when current is constant, voltage is proportional to length, or V α L.

Putting these two together you get V α IL, or V = kIL, where k is a constant.

And then to test your formula you run a bunch more tests and see if your measurements line up with your expected values! If they line up fairly well (and k is indeed a constant) you’ve got the beginnings of a mathematical formula.

TL;DR lots and lots and lots of experimenting.

>How can one read these equations and discern what would happen in reality?

With a good deal of training. That’s a problem I’m actually facing now, and it’s not easy.

So the easiest way to explain this is by pointing out a very crucial fact:

Everything described by math is a MODEL of reality, not reality itself. An approximation. And the job of people developing these models is to make sure that they fit reality as close as possible, but everyone knows they will not be perfect. And sometimes they’re imperfect on purpose, because making them perfect would be too complex. Otherwise we would have wrapped up science by now, nothing more to do

As for how it’s done, there are functions and operations in math that represent some specific things, universally. It’s difficult to talk specifics because that would require explaining the math, and that’s beyond ELI5, but there are universal ways to describe amounts of stuff, how they change, how they move, how they can interact.

Because they have specific meanings, you can put them in different combinations to make them represent more complex things. It’s sort of like a language.

As simple example, if something is changing in proportion to how much of it there is (which most things do), you use the e^x function, because that’s what it does mathematically.

A slightly more complex example. If you want to specify that something isn’t created or destroyed, say water flowing through a piece of pipe, there’s a specific operation that universally describes an amount of “stuff” moving, which you can then define to be water by putting in appropriate numbers. You then specify using that operation that amount entering plus amount exiting equals 0. You have just defined a conservation of mass law, water cannot pop into existence or disappear inside the pipe, only enter through one side and exit the other.

And now for something interesting. But what if you say it’s not 0? Does that break physics? You’re saying that water can be literally conjured from vacuum, quantum style?

No, because remember, it’s not reality, it’s a model of reality. What you can claim is that there’s a water source or drain there. In actual reality, that would be another water pipe of some sort, or a hole in the pipe, and the water keeps existing somewhere and you should keep track of it. But because it’s just model of this one piece of pipe, you can just say “yeah, a specific amount of water just appears or disappears”.

Mathematically it’s literally indistinguishable from magic, and by just looking at the equation you wouldn’t be able to tell where the water comes from and goes. You need the context of what you’re reading to actually understand what it represents, physically.

That’s where the training comes in. You learn to recognise these contexts without being explicitly told. You see some specific symbols in a specific combination and recognise “oh, that’s a liquid”. Then you logically assume that it will obey laws of liquids, that you know.

And if something weird and unusual is going on, then the person making the model has to specify for everyone reading.

You play around with things until you get something that seems to work.

A good example of this (if slightly historically inaccurate) is [epicycles](https://en.wikipedia.org/wiki/Deferent_and_epicycle). Observations of the planets showed that they moved in slightly weird paths – sometimes moving backwards, sometimes getting further away. The solution to this was to add in “epicycles” – or cycles within the cycles (as you can see in the top right diagram on that page). This leads to a [wonderfully complicated](https://en.wikipedia.org/wiki/File:Cassini_apparent.jpg) but largely accurate model of the Solar System. It works as a mathematical model, and could predict the motion of the planets.

Of course it is largely “wrong.” The Earth isn’t the centre of the Solar System; the problem they were dealing with was not that there were circles on circles in the motion of the planets, but that the planets *including the Earth* were moving around the Sun. We can still use the epicycle model to track the position of planets relative to the Earth (the maths is fine, mostly), but we’re better off using the heliocentric model, with at least Newtonian gravity, to understand what is really going on.

You asked about [Maxwell’s equations](https://en.wikipedia.org/wiki/History_of_Maxwell%27s_equations), and they are a lovely example of how science works, step by step, to produce useful results. Maxwell didn’t come up with them alone. Their current form was first expressed by Heaviside in the 1880s, alongside Gibbs and Hertz; originally they were known as the Hertz-Heaviside equations, or the Maxwell-Hertz equations. Heaviside’s versions are slightly different to Maxwell’s, but built on them, using new notation and new mathematical tools developed in part to help with these physics problems.

Maxwell’s versions are scattered through a paper published 20 years earlier. But they are also built on earlier work. Of the four modern “Maxwell’s equations” they are Gauss’s law (applied to electric fields), Gauss’s law applied to magnetism, Faraday’s law of induction (also called the Maxwell-Faraday equation), and Ampère’s circuital law (with Maxwell’s addition).

Maxwell’s contribution was adding in an extra term to that last one, which was needed to make them all fit together neatly – which he also did – to create a coherent and consistent system of equations. Maxwell was building on the work of Coulomb, Ampère, Faraday, Lenz and any number of other physicists and mathematicians of the 18th and 19th centuries. There is a quote from Newton, referring back to Descartes, that if he had “seen further it is by standing on the shoulders of giants”, but the reality is all science is collaborative; we do what we do by building on the work of brilliant, but normal scientists and mathematicians who helped us on the path.

> How can one read these equations and discern what would happen in reality?

By understanding what all the maths means, and plugging in values.

Maxwell’s equations – in their modern form – relate the electric field **E** and magnetic field **B** (or **H** if we’re using an H-field), with the local charge density *ρ* and local current density **J**.

Like any mathematical equation we take the things we know, plug them in, and rearrange to get something useful. For example, if we plug in *ρ = 0* and **J** = 0 (i.e. no current or charge), we can combine all four equations to get a thing called the wave equation; we get rippling, self-propelling electric and magnetic fields, that move at a fixed speed. We can use the equations to predict things; given certain values for *ρ* and **J**, we should get certain values for **E** and **B**, or in other combinations.