How is information inferred from mathematical formulas?
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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.
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