> What is “a language” to you? What is an “isn’t a language” to you?

A language is an abstract concept that describes a method of communication. It need not be spoken (such as English), written (such as what we're doing now). We frequently use body language to communicate, and so do many animals. We have braille, smoke signals, maritime flags, we communicate with knots on a string, and so many more things. You're right that language is quite a broad and vague thing. But recognize that all these things are also not of the universe, but of us humans (or similar of other animals). Something like English is something we may better refer to as a social construct, as it is a collective agreement, though body language may be a bit more ingrained but I still do not think you would refer to it as something other than language or something of the universe (distinct from us being of the universe in the trivial sense).

> What is the “language” here? The word/symbol 2? The subjective boundary that separates something more continuous into discrete forms?

(This is HN, so I'm going to assume you're familiar with programming languages.) If I give you these 14 characters (p, t, k, s, m, n, l, j, w, a, e, i, o, u) are we able to communicate? Maybe after some trial and error, but certainly not something we could throw into a translation machine. It'd be hard to call these even tokens since we have not distinguished consonants from vowels or if that even is a thing here, so we can't really lex. We need words, phrases, and context before you can even from syntax. Then we need to build our syntax, which is equally non trivial despite looking so (build a PL, it is a great exercise for any computer scientist or mathematician. For the latter, build your own group, ring, field, ideal, and algebra. You'd do this in an abstract algebra course). We need all this to really start making a real means to communicate. These are things we take for granted but are far more complex when we actually have to do them from scratch, forcing our hands.

Do you have a problem calling a programming language a language? I'd assume not because we collectively do so tautologically. Great, you agree that math is a language. Thank you lambda calculus. We can have an isomorphic relationship between programming languages and various mathematical systems. I'll point out here that there are different algebras and calculus with different rules and forms, though many that are not deep in mathematics may not be exactly familiar with these. I think this is often where the confusion arises, since we most often are using our descriptions that are most useful, just like how no one programs in brainfuck and just how most drawings are communicative visualizations rather than abstract art. I again remind you of Poincare who says that mathematics is not the study of numbers, but the study of relationships. He does not specify numbers in the latter part, on purpose. Category theory may be something you wish to take up in this case, as it takes the abstraction to the extreme. Speaking of which

> but how do you dismiss the essential and seemingly unrealness of its abstraction from our perceived reality?

I could ask you the same about English. Why is this any different? Is that because you are aware of other languages that people speak? Or is it because you recognize that these languages are a schema of encoding and decoding mechanisms which result in a lossy communication of information between different entities?

You discuss counting, but are not recognizing that you can not place an apple into text, nor atoms, galaxies, or stereoisomers. It is because mathematics is the map, the language, not the thing itself. We can duplicate these at will or modify them in any way. Math is not bound to physical laws like an apple is. Its bound is the same of the apple that exists in my mind, not in my hand. (If you want to make this argument in the future, a stronger one might revolve around discussion of primes and their invariances)

> What language can be used to defend an infinitesimal equating to a finite value?

If this is the essential part, I think this is probably the best point to focus on. Specifically because infinities are not real. Nor are they even numbers. If you disagree then you disagree with physics. Rather infinities are a conceptual tool that is extremely useful. But if we were able to count and use infinities then we'd have the capacity for magic via the Banach-Tarski Pardaox, and completely violate the no cloning theorem. But our universe does not appear to actually have arbitrary precision, rather our tool does due to its semantics. Maybe finite ZF is a better choice than ZF or ZFC set theory. Why not NBG which has a finite number of axioms or why not MK which isn't?

Infinities, singularities, and such are not things in our universe. You may point to a black hole but this would represent a misunderstanding of our understandings of them. We cannot peer in beyond the event horizon, which certainly is not a singularity and has real measurable and finite volume. It is what is inside that is the singularity. But can you say that this is not in fact just an error in the math? It wouldn't be the first time such a thing has happened. Maybe it is at the limits of our math and so thus is a result of the inconsistency of axiomatic systems? There are many people working on this problem, and I do not want to undermine their hard work, and neither should you.

You're biased because you're looking at how we use the tool rather than what the tool is itself. We use mathematics as the main descriptive language for science because of its precision. But we've also had to do a lot of work to ensure its consistency and make it more precise along the way. But I think you may have not been exposed to the levels of abstraction that math has, as this is not seen by most people until well beyond a calculus class.

> I am unsure how you can understand godel but argue against the essentialism of the sur-real abstractions he brings attention to.

And I cannot see the reverse. Are you saying that the universe is incomplete? Are you saying that the universe is not consistent? This sounds like a better argument for the idea that the universe is a simulation (as in we are being simulated, not as in you can represent and draw parallels between the universe and simulation. The former begs the question "on what" and we get turtles all the way down). Rather, as the old saying goes, I do not believe that the map is the territory. Just like how our brain creates an incomplete model of the world we live in, mathematics too is used to create an incomplete model to help describe not only what we can see but what we don't. But do not trivialize or diminish the notion of a model, as I certainly would not claim our brains and senses are useless. Models are quite powerful things, there is a reason we use them. But a model is not the thing itself.