If physics involves so many assumptions and simplification in the process of calculating, how does the math actually check out?

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For example, I’ve seen that an electron is assumed to be a point-particle, i.e, occupies negligible space.

But it obviously has mass, so it’s gotta occupy *some* space!

However, the math seems to add up (pun intended), and the behaviour of the electron can be explained.

How?

(This was inspired by the *assume a spherical cow* thing)

In: Physics

>For example, I’ve seen that an electron is assumed to be a point-particle, i.e, occupies negligible space.
>
>But it obviously has mass, so it’s gotta occupy some area!

You’re confusing *mass* with *volume*. If something has volume then it must occupy some area. It is not clear that elementary particles such as electrons and such have any sort of volume which is why it is fine to treat them as point particles.

There are proposals, such as string theory, which give them a volume (of sorts) but they are currently untestable in these regards. Basically, if they do have volume, it is currently below our threshold to detect, which is why ignoring it hasn’t raised any problems.

>However, the math seems to add up (pun intended), and the behaviour of the electron can be explained.
>
>How?
>
>(This was inspired by the assume a spherical cow thing)

Mainly because our math is derived from past observations. So it should be little wonder then why future observations are in line with our math!

But the real answer is: our math *doesn’t* check out. All of the math is approximations based on our finite set of observations made within the degrees of accuracy that we are capable of. As we make more observations and are able to measure things with greater accuracy, we find that our math doesn’t check out in some areas.

It’s why we moved from Isaac Newton’s math to Einstein’s math and why we are trying to find new math to move to from Einstein.

You have it slightly backwards. Reality apparently conforms to the math first, and all those assumptions and visualizations are secondary artifacts that we as humans use to try to understand what the math is telling us.

For instance, the point-particle electron: the universe doesn’t seem to care if it’s a point or not. If anything, an electron seems to be more like a vague haze of “it’s probably here, but it could also be there at the same time.” Yet, as soon as you get far enough away from the electron, you can treat it as a single, geometric point with mass and charge and the rest of the math just works (and by that, I mean experimental evidence completely agrees with math that assumes it is a point).

The problem with quantum mechanics is that it’s just *so weird* that even scientists studying it have no idea what’s going on down there. Weirder still, there are a couple of theories that produce multiple, wildly-different interpretations that are somehow all true at the same time.

In many cases the math doesn’t actually check out but we don’t have refined enough instruments to measure the difference until much later. When we do experiments we can usually tell with what precision we’re going to be getting from our results and that’s often enough to be reasonably sure that the theory is accounting for enough complexity to be accurate. For a long time it’ll be close enough for doing practical engineering. When we do refine instruments enough to make measurements that prove our mathematical understanding is actually wrong, that leads to new theories and new math and our understanding of the nature of the universe deepens.

Theories are not absolute nor do they need to be. They can be close enough that we can still do astounding things like split the atom, land on the moon, and peer into a working brain by placing a person inside an MRI.

Assumptions for physics depend on the problem you’re solving

If you’re working on electronics then its fine to assume an electron is a point particle, the size of the electron doesn’t have an impact on your work

If you’re designing a new type of electron microscope then you can’t make simplified assumptions about the electron because all their quirky properties have significant impacts on what you’re working on.

Most of the math you’ll do in physics or engineering classes makes a boatload of assumptions because it gives you a good enough answer.

If you’re just solving projectile motion for a heavy cannonball fired at 20 degrees at a fairly slow speed then air resistance doesn’t particularly matter and we ignore it to make life easier. If you’re making actual artillery firing tables in advance then you’ll need to factor in air resistance and the shape of the shell. If you’re calculating a real time firing solution and you want to hit *that* pillbox, then you can’t just use 1.225 for air density, you need to factor in altitude(which changes over flight), temperature, and humidity to get an accurate firing solution.

The level of assumptions you can make is entirely dependent upon the level of detail required for the application. For most things you can assume gravity is 9.8 and pi is 3.1 and end up just a couple percent off the answer and its fine because most applications have margins of safety built in to accept the unknowns of the real world.

Margins of error.

I don’t want to talk about electrons as I’m aware there are lots of theories that I’m not fully versed on. However, I can talk about some basic engineering.

Say you’re making a bridge across a river. You measure the river to be 10m wide. Your bridge has a bit of an arch, so you’ll need 12m of material to make it. Great.

But what if your measurements were slightly off and it’s actually 10.1m?
What about when it gets warm, since your bridge is made of metal it will expand. Or contract when it’s cold.

We build in margins of error that are perfectly acceptable to account for these issues. On a bridge if it wasa few mm too big for the holds you built either side you could probably just squeeze it in, compressing it slightly. Same with basic wooden furniture: IKEA don’t have precision measuring, but we know wood is a little bit flexible so whatever it’s fine.
For larger bridges heat expansion can be an issue, so they often have expansion joints or even roller wheels built in to allow for small changes.

Basically we can do maths close enough that our assumptions don’t matter, it’s close enough that noone will notice the difference.

Because we only use assumptions which we can show are valid for the problem we’re working on. For example the spherical cow thing, if you are working on a problem that doesn’t care about the exact geometry of the surface of the cow, only it’s total surface area then it is completely fine to approximate it as a sphere with a radius such that the surface area matches.