This is a linkpost for https://charlesr-w.github.io/crw-blog/Every-Measurement-Has-a-Scale/
meters – longer than football fields!
Some people might overlook the hilarity of this remark, so I'd like to draw specific attention to it.
Also, relevant xkcd: 2283.
I thought this was a cool post! I'd find it especially interesting to hear your flesh out how it applies to concepts like "local minimum" and the like.
Nice post!
When a property is discussed in any real analysis-shaped context, its relevance is usually implicitly understood by mathematicians to be in the limit (i.e. for very large numbers or at values very close to some fixed point). For example, the statement "f(x) is a polynomial of degree 26" means that there exists some large enough R>0 such that for every scale L larger than R, the L-degree of f (in the sense of the definition you gave) is arbitrarily close to 26.
This type of "asymptotic" analysis has its limits in all kinds of applied fields (galactic algorithms are an example of this), because we do tend to use math at specific, finite scales : )
Yup! A math-ier version of the insight I was trying to convey is to say that, if you imagine dealing with math where you have a handful of these limit-y properties floating around, it may well matter what order you take those limits in! And at the same time that the limit-y-ness of these properties is usually quite “hidden”, so that this becomes a very useful mental motion.
A worked example of an idea from physics that I think is underappreciated as a general thinking tool: no measurement is meaningful unless it's stable under perturbations you can't observe. The fix is to replace binary questions ("is this a degree-3 polynomial?", "is this a minimum?") with quantitative ones at a stated scale. Applications to loss landscapes and modularity at the end.