Can one rigorously define "meaningful" versus "arbitrary" in math?

Question

Often we regard certain mathematical expressions, or elements thereof, as arbitrary, in the sense that they have no apparent reason or cause, whereas more beautiful or natural seeming expressions feel more meaningful or useful. For example, $ e^{i \pi} + 1 $, $ \ln 2 $, and $ \sqrt{x^2 + y^2 + z^2} $ as opposed to $ e^{1/2}-1 $, $ \log 15 $, or $ x+2y-1016z $. Mathematicians, through their creativity, have endlessly defined, invented or discovered objects, ideas and relationships, but only the most somehow compelling results end up in final theories, while the more apparently superfluous stuff gets disregarded and forgotten. The perception certainly varies from person to person, and with the amount and kind of mathematics they've been exposed to (here I'm thinking of the monstrous moonshine and the value in some modular form expansion, or Hardy's taxi number on the way to Ramanujan), so is the concept necessarily subjective or could a rigorous definition of the idea be crystallized out of this sort of thing? It sounds sort of foggy and intractable like "irreducible complexity" but perhaps it could fare better in the pure abstract.

How might we quantify meaningfulness vs. arbitrariness in mathematics, if only in principle - i.e. even if actually computing such a measure is in general infeasible with our limited resources and knowledge?

Answer 1

You can if you like give some precise mathematical definitions for "meaningful" and "arbitrary", but such definitions would be arbitrary rather than meaningful.

Less preciously: you're not going to be able to give such a definition that agrees with the way actual mathematicians use these terms. Proof: the use of these terms is only somewhat consistent, not perfectly consistent. "Meaningful" and "arbitrary" are more like sociological concepts than mathematical concepts. You might have better luck studying them as a naturalist would: i.e., rather than trying to envelop them in a single formal theory, study them in practice and see what various meaningful mathematical concepts and constructions (usually) have in common.

By the way, you haven't given any motivation for wanting to formalize these concepts. Why would you want to do so? As you seem to realize, it is very unlikely that some kind of formal theory along these lines would be helpful in one's actual study of mathematics.

Answer 2

I think this question can be taken into several different directions. If you substitute "interesting" or simply "nonarbitrary" for "meaningful" (as Nate Eldredge and Pete L. Clark did), at least three criteria come to mind: The first is simplicity; this can be made precise in various formal systems, e.g. the natural numbers are in some sense the simplest infinite set, etc. Second, if some object appears naturally while researching something else, that can suddenly make it significantly more interesting. Actually, I think such cross connections are often a boost for both sides. So for example, if $x+2y-1016z$ happened to be, say, the only polynomial with integer coefficients satisfying some non-obvious property, it wouldn't be arbitrary any more -- perhaps even if the subject where that property came up didn't get a lot of attention before. The third factor is how many nontrivial theorems can be proved about an object. I suppose this only makes objects more interesting if they are already interesting for other reasons, though.

The issue of complexity is especially subtle, even if you can probably find a good metric. For example, the definition of a Turing machine is rather lengthy with quite a few arbitrary decisions. What makes Turing machines interesting anyway is the Church-Turing thesis. In other words, the truly interesting property is Turing-completeness, and the truly interesting object is the class of all Turing-complete machines, while the Turing machine itself is just an arbitrary representative of that class. The special gem here is that there does not seem to be any simple/natural/nonarbitrary/interesting way to formally define that class. This suggests that one should distinguish between mathematical objects, which may be interesting, and their descriptions, which may be arbitrary nevertheless.

This brings me to another possible interpretation of the words "meaningful" and "arbitrary." Axiomatic set theory (e.g. ZFC) can be said to trade "arbitrariness" (is that a word?) for complexity. Take the Kuratowski definition of an ordered pair, for example: It is not just asymmetric, it even implies that (under the usual definition of natural numbers) the pair $(0,1)$ is a proper subset (yes, subset!) of the number $3$. You can replace $3$ with any larger number, but not with $2$...

Here, one might argue that the question whether something is a subset of the number $3$ is not meaningful. This may not be exactly what you had in mind, but I consider this concept of meaningfulness quite important, and it can be made precise in various ways. Type theory provides a very satisfactory formalization, but since it is an alternative to set theory, it leaves open the question of why we consider a particular statement meaningful or meaningless in (set-theoretic) mathematical practice. This question has been researched on philosophical grounds, and one can also try to find a formal system that only permits meaningful statements. (I have been working on this.)

A weaker version of "meaningful" vs. "arbitrary" comes from mathematical incompleteness. The independence of e.g. the continuum hypothesis from ZFC gives objects like $\aleph_1$ a more "vague" feel, thus some people consider such questions or objects less meaningful than others. But there are also very concrete independent statements, most famously (in this case) the question whether ZFC itself is consistent. A key difference is that the continuum hypothesis does not have any arithmetic consequences.

To conclude, if your question is interpreted sufficiently broadly, a lot of well-researched foundational topics come to mind. In any case, there cannot really be a single answer.