Approximation in physics: assuming the best

Most view physics and mathematics in the same light. Both are STEM subjects after all, especially for those who know how many times deferential equations rear their ugly mugs. In reality, however, these aren’t as alike as you would(a) like to think , because there is one general distinction in how these two fields go to work.

I shall elaborate on, but not prove (wow such foreshadowing), this difference during this actually shorter blog (not like last time guys I swear >_<).

The methodologies

So wat is this difference I alluded to? Well this lies in what is defined as an acceptable amount of evidence for new knowledge. For pure mathematics something does only get seen as fact or is generally excepted once a (correct) mathematical proof is provided, which means that it has been shown with only logic, which is the bases of maths, (and not data) that it can only be this certain way.

While, on the other hand, we have our goofier brother physics, where the ‘what is widely accepted as true’ is based on what explanation or model represents our found data the best. Now this doesn’t really mean physics is less “true” than mathematics, but it does mean that we have to view it differently. In this case, a theory is considered true until it doesn’t anymore align with our data, where it either has to be amended or replaced with a better theory.

We could take, for example, the elementary particles (like the quarks, leptons and bosons), where we haven’t “proven” their existence in the mathematical sense, yet the theorem holds up with all of our data and is a mighty solid explanatory tool which has actually helped us discover new thing (which where later verified by gathered data). It’s also momentarily the best explanation we have. However, in string theory, for example, there aren’t any electron but everything consists of strings.

The particle-theory isn’t any less true, because it still holds up as of now, which is why we haven’t tossed it aside, but it isn’t true’er either. Which also shows that there can be multiple “true” explanations in physics for the same problem, while in maths this isn’t really the case. Currently string theory isn’t widely adopted because there is little empirical evidence for why it is better than the standard model, but we’re just gonna let the string theorist play with their yarn for a bit.

This is also why we work with p-values in physics, because we use that to show wat the possibility is of our results being significant or true. Here, it is important to note that this p remains small and never reaches 1 (meaning that it is 100% true).

A concrete difference would be the Riemann hypothesis, for example, which is a complicated (or at least for me) statement in mathematics which still hasn’t been proven. We do know, however, that it does hold up for all the examples we have calculated with our fancy smancy technology (which is a lot a lot). Of course this doesn’t actually describe a physical phenomena but let’s just assume it does for the fun of it. In mathematics, this is still an open question, however, if we were to have a similar problem in physics, it would shows to be holding up for all our data so we would see it as true.

The world’s programming

Ha! I tricked you with the title of this part! Weren’t expecting my mischievous side today where you? This is in the sense that most of the equations are, as far as we can tell, not actually baked in to the world.

Let’s take Newtons theory of gravity. While being a simple formula, it also explains how gravity works very well and can be used to make predictions on the movements of planets, for example. We all know, however, that this had certain inconsistencies with our data, with mercuries orbit, for example, which is the reason why general relativity is now adopted as the more accurate explanation. But this doesn’t mean that we just made an oopsy, and that general relativity is the actual truth (or at least the whole truth and nothing but the truth); it just means we have gotten better at explaining and predicting gravity in our world.

This is also the reason why Newton’s theory of gravity didn’t become obsolete after Einstein showed him up, because it can still be used explain a lot of things and is way easier than applying general relativity. (For someone who wasn’t good at maths, he really did make it hard on everybody).

Physics, like all things in life, is about choosing the right tool for the job. When we are calculating the speed of the metaphorical mike Einstein dropped after releasing his theory, we should just use the simplified form of Newtonian gravity (F = m • g) instead of breaking out our cumbersome, more accurate theory. At the end of the day there are both approximations of reality.

General assumption

This doesn’t only apply to physics as a whole, or just the equations, but also to how you go about solving individual number.

This is actually a very important note for if you’re studying physics, which is tactfully choosing what to and what not to ignore. A good trick for this is to look at everything as being relative to each other (not if you’re from Alabama), and especially look at what magnitude or powers of ten you’re looking at. If you notice that something, while not zero, is really small compared to the rest of your problem, you can probably ignore it (as long as it doesn’t mess up the equation).

This is also, for example, how we get the infamous sin(x) = x, which is never (besides zero) actually true, but because the numbers are so small and insignificant we can assume it is so that our problem becomes easier or sometimes even possible to solve. So, like always, work smarter (and less 100% precise), not harder and make them mathematicians cry.

Final remarks

This was the end of our more meta blog about what we consider as evidence and fact in physics. This does kind of stil leave laws like conservation of energy, but these aren’t approximations, but more closely resemble that of mathematical truths. These are also the most common methods by which we discover that one of our theory must be slightly adapted, but that’s a topic for another time.

So, have an approximately splendid week, and I’ll see you in the next blog.