Consider $f$, some mapping between two ordered subsets of the natural numbers, $A$ and $B$. I see two ways $f$ can be expressed. The first way is an equation that defines $f$ (i.e. $f(A) = 2A$). Alternatively, $f$ could be a mapping that can only be expressed explicitly. In other words, $f$ can only be defined by an infinite list of explicit mappings for the elements in $A$. By "explicit" mappings, I mean something like this:

$$a_1 \to b_7$$ $$a_2 \to b_{323}$$ $$a_3 \to b_{42}$$

and so on.

Intuitively, I would guess that more functions can be expressed with an infinite list of mappings between the elements than with an "equation", since equations can only be permutations of mathematical concepts. The "infinite list" can also be a mapping between two subsets of real numbers, if we don't require it to be countably infinite.

Even if the set of mappings that can be expressed by an equation isn't a proper subset of the mappings that can be expressed via an infinite list, it has to at least be a subset.

All the physics I've encountered so far has basically been a mapping between variables. And literally every mapping I've come across so far has been been expressed with an equation.

I'm curious as to why so many mappings in physics can expressed as "equations". I don't know of any mappings in physics that have been shown to only be able to be described by "infinite lists", and therefore impossible to completely know. Even if by their very nature of them, these relationships are impossible to completely know, I would assume that the phenomena should at least be able to be observed in some way.


closed as primarily opinion-based by Kyle Kanos, ACuriousMind, user36790, Norbert Schuch, John Rennie Dec 11 '15 at 7:27

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    $\begingroup$ Most physical "mappings" must obey some kind of constraint, like $f(ax) = a^n f(x)$ for some exponent $n$, and even if not that, they typically must be smooth, so you can always Taylor expand them to get an "equation". I don't really see a physics question here - a description in terms of a mapping we can't write down would be pretty useless, so we don't use it. $\endgroup$ – ACuriousMind Dec 10 '15 at 20:27
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    $\begingroup$ @ACuriousMind I'm not asking why we don't use descriptions that we can't write down, I'm asking why we haven't encountered any phenomena that can only be described with a description we can't write down. I believe that's a question about the nature of the physical universe, so I thought it would be appropriate here. $\endgroup$ – Phidias Dec 10 '15 at 20:41
  • $\begingroup$ unreasonable effectiveness of math in physical sciences, wigner / wikipedia. on the contrary side, significant areas of physics are known to be undecidable & new areas are found all the time. $\endgroup$ – vzn Dec 10 '15 at 21:02
  • $\begingroup$ The world just doesn't seem to be that way. Physics is about describing and predicting the behaviour of the physical world, not about explaining "why" it can be described and predicted in a particular way. That's a question for philosophy. $\endgroup$ – ACuriousMind Dec 10 '15 at 21:41
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    $\begingroup$ @ACuriousMind. I'm almost inclined to disagree with "The world just doesn't seem to be that way" because I'm more inclined to see that "Humans don't seem to work that way". A mathematical description of a phenomenon that ends up being like the OP's example wouldn't be useful, so we don't use it. In some sense, things we describe as random processes do end up being like the OP's example function, but that's why use probability and statistics to tease out the "mathematically nice" relationships among average quantities. $\endgroup$ – march Dec 10 '15 at 22:51