How is information inferred from mathematical formulas?

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Science continuously try to refine our understanding about the world, so that we can predict what’s going to happen in certain contexts. Newtonian mechanics for example work really well under every day circumstances, but relativty refine this for when things move real fast.

Math doesn’t really concern itself with the truth about our universe. In scientific contexts, it sometimes gets used to infer how if someting is true, another thing must also be true. Not all things that can be written down in maths even make sense in our universe. Like speed that is faster than light. But maths will still tell you what other things mightbe true, had the faster than light travel been true.

Let’s use gravity in space as an example.

We know (based on experiments on Earth) how gravity works. You release a ball, it falls towards the Earth. We also know that the larger the planet is, the more force gravity pulls with. This is why you can jump higher on the Moon.

These forces determine how planets orbit. It’s the reason we orbit around the Sun; the Sun is massive, so it pulls on the Earth. We’ve seen this sort of behavior with a lot of planets, so we know exactly how orbits are supposed to look.

Now, let’s say we see an orbit that makes no sense. Based on our understanding of gravity and all of the planets in the proximity of the new planet, we know what the orbit should look like. However, it looks entirely different!

There are a few possible explanations. One is that our equations for gravity are wrong. We may have misunderstood how things worked, and need to adjust the equations. A second explanation, however, is that there is an unknown object out there. This could be introducing different forces of gravity that would affect the orbit of our new object. Based on our equations, we could try to figure out where that new object should be based on how the orbit looks.

I’ll use an example.

With Newton’s law’s we can predict the mass and acceleration of objects and tie it back to force. F = ma.

Simple right. Basically if you want something to move faster, you apply a force. That is information inferred from that formula.

Newton also derived the formula for objects falling. F=mg where g is the gravitational constant for earth. On earth this works because the Earth is so big that any other mass value falls off.

Now gravity is not unique to Earth so there’s a more generic formula **F = G m1xm2/r^2**. This formula accounts for the fact that ALL objects with mass have gravitational pull. If you jump up, the earth is attracted to you as well.

This formula tells us that if know the mass of two objects, we can compute the F of gravity between them.

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Ok why am I bringing this up? Well you see, with that last equation, we can model how planets move and orbit. It’s super useful. However there are situations where this model seemingly breaks.

People observed that in certain situations there needed to be way more mass than what we could see in order for certain clusters of stuff (stars planets…) to stay together based on the gravitational forces required.

That would mean that if that equation were to hold true, you need have a lot more mass. Like 80% of all mass that is not observable. This is called dark matter.

Mathematical formulas are just logic, dressed up in some really fancy clothes. By themselves, they don’t tell you anything – but they can tell you that one thing implies another.

For example, suppose you’re modeling the spread of a virus. You know that the rate of new infections will be roughly proportional to the number of infections active today – in other words, (rate of change of infections) = (some constant) * (number of infections). A little differential equations can tell you that this implies that the number of infections over time will follow an exponential curve e^(kt), at least as long as your assumptions are true. (Only observations can tell you whether your assumptions *are* true.)

You mention dark energy in a comment. It turns out that the equations of relativity start from a few assumptions (the speed of light is constant to all observers, acceleration and gravity are equivalent and indistinguishable), and then go from there through a series of complicated mathematical steps. At the end of those steps, you get an equation that describes the relationship between how mass is distributed and how space-time is curved, but oddly, that equation contains something extra: an extra constant that didn’t correspond to any known object at the time those equations were discovered.

This was a suggestion – not a proof – that there may be something undiscovered out there. Since no observations suggested that there was, it was long assumed that that extra +something in the equation would just turn out to be zero, but – decades later – *observations* told us that the Universe behaved in ways that were surprising. And the ways it behaves line up with that extra +something not quite being zero.

The existence of the +something suggested, but didn’t prove, that there may be something out there. Observations confirmed that there was. This is very often how theoretical physics works – the equations you derive don’t quite line up exactly with how the world works, or have extra terms you didn’t expect, and in practice it often turns out that extra stuff is the gateway to the next big innovation in physics.

I’m not knowledgeable enough about this to even ask a question that makes sense. I’m gonna use an example; Dark energy is known to exist because it’s inferred by General Relativity, and by observations of the Universe. What I don’t understand is how information like that is given by numbers…..