This is a philosophical post about ideas, communicating them, and math.

Formalizing is the act of taking an intuition, a vague idea, and putting it into words — in mathematics, those words are usually in the language of logic, and usually in a specific flavor of logic, that which underlies the Zermelo-Fraenkel theory (by far the most common kind of set theory, which is a very handy framework for formalizing all sorts of mathematical ideas). It is heavily helpful in modern math, as it enables the thinker to rely, even when they have nothing else to use, on the power of definitions and the heavily refined toolkit of formal logic. We will now focus on cases where the idea to be formalized has been had, but its rigorous formalization has not been successful yet.

In mathematics, there is the concept of a field. A field is a kind of mathematical object, usually seen as a set, containing some kind of elements, which obey some precise rules that formally define what a field is. Among these rules is the fact that a field must contain at least two distinct elements. Therefore, the one-element field is, properly speaking, an impossible object. However, some results that apply to fields do find a natural extension to the case where said field has one element — more strikingly, some other results tend to find a clear interpretation only if one supposes that there exists a one-element field, which as we said, is a logical impossibility.

David Madore calls this a unicorn. That is, an idea that under the current rules we believe it should fall under, makes them fail and does not fit neatly with them at all. It then requires an ingenious thinker to find other rules, which to some extent should extend the previous ones in order to also be fit for the "normal" cases that did not break anything, but under which the unicorn does work as intended. This is hard work, and earned Laurent Schwartz a Fields medal for being able to formalize the theory of distributions, the "extending rules" underlying the Dirac spike, previously a unicorn.

I believe, if we allow ourselves to broaden the definition a little bit, that infinity is also a unicorn. Many results in very classic math do allow quite elegant extensions to the cases where one or several of the relevant variables "go to infinity", but infinity obviously is not a number. There are many different ways to formalize infinity, but I'm going to focus on only one of them: ω (omega), the first infinite ordinal. In order not to get too formal in what should essentially be a philosophical post and not a mathematical one, I'll summarize the idea of an ordinal as "something you can count up to", for a sufficiently broad definition of "counting". That is, ω is not a natural number, but it is defined as something which comes after every natural number. You cannot count: 1, 2, 3,... hoping to some day get to ω, but there is nothing simultaneously lower than ω and greater than every natural number. That is, ω is the limit ordinal of natural numbers. You can already see that ω is a pretty good formalization for the vague idea of "infinity", as a child attempting to count "up to infinity" can only hope to one day reach it, but already has the intuition that it is "something greater than every natural number".

If you were to invent ω without already knowing its formalization (which is pretty rigorous and based on standardized set theory), you'd probably struggle quite a bit — just as we said before, formalizing such a vague idea as the one of infinity is hard work. However, the same goes for formalizing natural numbers, which are way more concrete and easier to think about. I believe that the main difference is, once you've formalized them, the formal natural numbers are concrete proxies for the "actual" natural numbers, that is, the intuitive idea of them. (I do realize that by using the word "actual" I'm supposing a Platonicist view of the world, but you can replace it with "intuitive" if you will. I just find it a bit less clear for the reader who's not used to think about these things.)

What I mean by the word proxy is, any property you could think of regarding the "actual" (or "intuitive") natural numbers, you should (if it's a good proxy) be able to find an analogous property for regarding the formal natural numbers. For instance, one basic intuition about "two" is that you can obtain it from "one" and another "one". This is proxied through the equation 1+1=2, which is about the formal natural numbers, not the intuitive ones: as soon as you write representations such as the written digits 1 and 2, and you're manipulating symbols which apply following the rules of logic, such as + and =, you're actually working on the proxies. One could define a proxy (in the way I personally think about it) as an idea (here, the formal natural numbers) that enables clarifying the properties of another idea (here, the intuitive natural numbers) through analogy (here, the logic-abiding symbols of addition and equality are useful for representing, respectively, the intuition of getting two items together, and that of "having the same result").

The formal natural numbers are concrete proxies in that you can always develop them on to simpler forms: that is, you can write 2 as 1+1, 3 as 1+1+1, and so on, in order to only have to handle the "simple" proxy represented by the digit 1, and the addition symbol. You can reach every single natural number that way (except 0, if you want to be nitpicky), at least every one that it is physically reasonable to think about (you cannot count up to 10^80 in a practical way — one could also argue that 0 is non-physical). You can handle those proxies in a way analogous to that in which you'd manipulate concrete objects. Now, an abstract proxy is another kind of beast entirely — I argue that ω is one (among others) abstract proxy for the idea of infinity. That is, you cannot handle it in a way analogous to the physical manipulation of an object in the physical world, because no physical object has the properties of infinity. You have to think about infinity, as represented by ω, in an entirely different way from how you'd think about the intuitive natural numbers. And I say "as represented by ω", because as soon as you represent infinity through other proxies, you realize that they differ in various properties, between whom there is no perfect analogy (which is logically obvious — otherwise they would be the same proxy; mathematically-literate readers may relate that to the notion of isomorphism, isomorphic objects being essentially "the same one, just seen under a different light").

This underlines a critical difference between concrete and abstract proxies: an idea that can be represented by a concrete proxy, is represented "in the same way" by any two proxies (that is, it really only has one proxy up to isomorphisms). An idea than can only be represented by an abstract proxy can, however, be represented in entirely different ways by different proxies. Physical objects can obviously be represented by concrete proxies — but as seen with the example of the intuitive natural numbers, the class of "concretely representable objects" is larger than that of the physical objects. All proxies are therefore useful for communicating "refined" (for instance, by rigorous formalization) versions of ideas, but abstract proxies seem to tie into a strange class of ideas, that are only well-defined by said refining, and cannot exist in a non-nebulous way before they are proxied. Communicating ideas of this kind is therefore an even more destructive endeavor (by which I mean that it unavoidably changes the shape of the ideas which are being communicated) than communicating "concrete ideas" (that is, ideas that can be concretely proxied).

This probably has some implications regarding ways to think about mathematical objects, which I may or may not explore in some other post(s). In the meantime, enjoy this maybe unclear wandering, I guess.