How is information inferred from mathematical formulas?
In: 43
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…..
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.
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 don’t know if this is the kind of formula and information you are interested in … but in the world of finance and business, mathematical formulas are way-pointers. It is less that the formula results provide a definitive answer to the question being studied, but rather that they numerically highlight things that show the analyst the most productive follow-up questions.
The most basic formula results are rankings best to worst, which can aid in setting priorities. It is not necessarily the ‘final answer’ as there may be other factors to consider beyond the numerical rank order of the results.
In an operational system, formulas can help determine which points are weakest. Formulas can be used in any business process but are especially common in manufacturing. Such as: -The most common point of failure in the manufacturing process, or in the product. -The process point where things slow down. -Is it less costly to re-work a problem in an item mid-process, or just discard it.
The formula isn’t the whole answer in business. It usually needs observation and talking to people to really understand why things aren’t working the best. The formula just helps to be sure that we are putting the time and resources into the problem that is the most productive to solve.
Ultimately, all any equation is is a statement.
It tells you how one thing changes when you change another.
So take: Force = mass x acceleration. As a simply thing, or F=ma
It tells you, in essence, that the hard you push something faster it will get faster. And that heavy things require more force to get moving than slow thing.
Whilst equations can get vastly more complicated, you’re always in essence dealing with something like this. I.e, can you find y(x) =ax.
If we observe something changing, in response to something else, we can infer a relation. And then you can do a measurement to find or prove the exact relation.
What you can do in Physics, is further relate statements to each other by units, and conservation laws. Units let us transform different properties into others I. E we can convert Energy Temperature by manipulating the units with some constant. This force instance helps us figure out temperatures things start occurring at, cos it shows a temperature will input some energy into a system.
It’s just manipulation of statements of things we’ve observed to be mostly ‘true’ . We can extract and infer from it by transforming the maths logically.
Math is used to describe reality. F=ma as a simple example. The force an object applies is equal to its mass times the acceleration it is undergoing. What the specific numbers are is essentially arbitrary, but once we’ve all agreed on a standard they become meaningful as we can compare things against each other from the same frame of reference.
Imagine a large complicated equation as a jigsaw puzzle. Each variable is one of the pieces. If you understand the math sufficiently you can look at that equation and more or less “see” that jigsaw puzzle.
If you’re working in a theoretical field, like for instance particle physics, there’s a decent chance that your math is going to be incomplete. This variable doesn’t jive with that variable, there must be something else in between that’s causing them to behave a bit differently. If you’ve done your research properly you can predict the likely nature of that thing based on how it interacts with what’s around it. Like looking at an assembled jigsaw puzzle and seeing there’s a piece missing. Even if you’ve never seen that piece, if you can see where it has to fit in to the larger picture then you can probably describe it fairly accurately as long as you understand what’s depicted by the overall jigsaw puzzle.
Once you know the likely nature of the thing you may then be able to design an experiment to specifically look for it. This is essentially what happened with the Highs Boson a few years ago. Most scientists agreed that it was probably a particle that had to exist in order to fill a gap in their equations. A Higgs Boson shaped gap in the jigsaw puzzle so to speak. So based on its expected characteristics they then designed a particle accelerator experiment specifically intended to detect the expected shape of that puzzle piece. When they ran the experiment lo and behold they found the missing piece just as they expected it to be. If they had been wrong, then the data would have looked different and they would have had to try again, but in this case they were right.
The ability of a scientific theory to predict something before it is found is usually considered strong evidence of its validity. One good prediction doesn’t mean you’re done, in a lot of ways no science is ever truly “done”, but it does tell you you’re probably on the right track.
Latest Answers