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Noam Chomsky, following in the footsteps of logicians/mathematicians like Emil Post and Yehoshua Bar-Hillel, conceived of language as a formal object: a language is simply a set of structures, namely the set of all (and only) the grammatical structures of that language. On this view, English can be equated with a set which contains Alice runs, Alice sees Bob, etc., but which does not contain Runs Alice, Sees Bob, etc.

Indeed, it's easy to construct a "language" (under this definition). For instance, L = {a} is a language. It has just one grammatical structure, namely a. Not a very interesting language, but a "language" (under this definition) nonetheless.

As Chomsky pointed out, and as did others before him, like Wilhelm von Humboldt and Gottlob Frege, human languages are infinite: there is no limit to the number of grammatical sentences you can construct in a human language. So, English (Spanish, Arabic, ...) is not just a set, but an infinite set of grammatical structures. This means we cannot simply list all (and only) the grammatical structures of English (the list would never end). Instead, Chomsky argues, we need to specify a generative grammar, i.e. a device that generates exactly the (infinite) set of grammatical English sentences. (Chomsky believes that linguistic competence involves generative grammar at the cognitive level, i.e. that we all have generative grammars in our heads.)

Indeed, it's even easy to construct generative grammar that generates an infinite "language". Here's one (using recursion):

1. a is in L.
   
2. For any x, if x is in L, then xb is in L.
   
3. Nothing else is in L.
   
   So, L contains things like a, ab, abb, abbb, and so on (infinitely), but does not contain b, aa, aba, and so on.
   
   Now you can start to see that mathematics is clearly a language under this conception of language, for exactly the reasons you mention: it has a fixed syntax, etc. So 3 + 2 = 5 is grammatical, while = 3 2 5 + is not. And indeed, one can specify a generative grammar for whatever fragment of math one is interested in.
   
   Of course, (human) language is more than just grammar. It's more than just a set of structures. Language has meaning, too. But so do mathematical and logical expressions: a > b has a very different meaning than a < b. The logician Alfred Tarski was the first to give a real semantic analysis of logical language, but he doubted that his analysis of logical language could be successfully applied to human language. His student, Richard Montague, showed that in fact, a formal, logical analysis of human language is possible. In doing so, Montague effectively created the field of formal semantics as we know it today.
   
   As the linguist Emmon Bach succinctly put it, in his book Informal Lectures on Formal Semantics:
   
   > Chomsky's thesis was that natural languages can be described as formal systems. Montague added to this the idea that natural languages can be described as interpreted formal systems.
   
   tl;dr Yes, depending on your definition of "language", mathematics (and logic, etc.) is absolutely a language — one with meaning, no less. Both artificial (mathematical, etc.) and natural languages can be described as interpreted formal systems — a revolutionary idea put forth by Chomsky and Montague (and others) in the 1950s-60s that spawned the entire field of formal linguistics as we know it today. (So, I clearly (and strongly) disagree with /u/raendrop, who instead prefers a circular definition of language, on the basis of which nothing counts as language except, by definition, human language.)

Regardless of what your program's focus is, my best advice is to bone up on syntax and morphology. Of course, if you're studying more practical/clinical stuff like language acquisition, phonology will also be indispensable.

Either way, you're definitely going to need an understanding of the basic principles of syntax. My 400-level class used Grammar as Science by Richard Larson. I liked the diagrams. Another class I TA'd used the O'Grady textbook, which someone else mentioned here, and I'd definitely recommend that, with the caveat that it's designed for 100-level stuff. So start with O'Grady, then move on to other things.
A suggestion: are you into conlanging at all? Because that's a great way to get your feet wet. It's also super fun.

I really enjoyed Latin Alive. It's very readable and entertaining, with lots of interesting etymological connections. It follows the evolution of Latin up through early forms of French, Spanish, and Italian, but not all the way up to the modern languages.

You could start there, and if you wanted to go into more detail, you could try Romance Languages: A Historical Introduction, which is more of a textbook, so less storytelling, but it's still pretty engaging and accessible (at least the parts I've read).

Paul Elbourne's Meaning: A Slim Guide to Semantics is fun, simply-written, and a great way to start thinking carefully about language!

To my knowledge there isn't an online translation tool for Old English. UToronto has a dictionary/corpus that may interest you.

If you can get it, the Mitchell and Robinson Guide to Old English is a great book for learning translation (though they do standardize the OE texts substantially).

This was taken from my textbook

latintutorial is also handy (especially if you're learning Latin) he has two videos on (Classical) Latin pronunciation, one for consonants and another for vowels

Nativlang made a video on the subject and also makes some points which would refer to Vulgar Latin pronunciation (Vulgar Latin being the form that the Romance languages descended from rather than Classical)

Hope this helps

A Book of Middle English by J.A. Burrow and Thorlac Turville-Petre is a great resource if you're looking for a book.

You're basically looking for something like Linguistic Field Methods.

Some universities allow you to focus on language documentation, which is what I did, actually.

>Also, is there any set of sentences for getting it's grammar completely by translating them?

It's way more complicated than that.

That is pretty much the same translation that The Everyman Edition of Sir Gaiwan gave (1962)

I don't have an answer for all three of your questions, but here's a partial answer to one.

> What factors made the various American accents sound the way they do

So in North American English, a lot of the differences between accents are in the vowel sounds. Think of how a stereotypical white New Yorker might say the o in "coffee", how a Canadian might say about, or whether people have the cot-caught merger. There are some differences in how consonants are pronounced as well, such as how often you might say walking or walkin' or saying this as dis. There are some grammatical differences, such as using might could in the South, needs washed in the Midwest, or invariable be in African American English. And there are word choice differences, as in pop vs. soda, put up vs. put away, or roly-poly vs. potato bug.

If you have access to a university library, you might want to look up the Atlas of North American English by Labov, Ash, and Boberg. As a more coffee-table book that more intended for a general audience, try Josh Katz's book Speaking American

A simple answer.

I am positive I read something very similar to that quote in "Though the language glass", a book I read just last year. Wether that is the origin of that quote or if the author was, in turn, quoting somebody else I don't know.

Indo-European Cognate Dictionary

If you need the super basics, there is Linguistics for dummies. Here.

I have the PDF if you need it and can’t find it, hit my inbox.

The Horse, the Wheel, and Language by David W. Anthony

The Unfolding of Language by Guy Deutscher

This is the book you want.

One notable thing aside from just vocabulary differences is that Tangier English takes part in the was/weren't pattern. In standard American English one says "You were home" or "You weren't home." In the majority of southern AmE dialects that use was, one says "You was home" or "You wasn't home." Tangier and Smith Islands distinguish affirmative and negative cases: "You was home" or "You weren't home" but never *"You were home" or *"You wasn't home."

In How To Kill A Dragon, Calvert Watkins managed to reconstruct the poetic formula "imperishable fame" (that exact sequence of words which appears in ancient fables and myths across the globe) back to Proto-Indo-European. He also reconstructs the dragon-killing idiom "[he] slew the serpent" in that book.