The question everyone else is afraid to ask!
It is plausible that very many different types of aliens exist (the number of stars in the universe is mind-bogglingly large, thus the number of planets is very large, thus the number of planets hospitable to life is very large, therefore, since we know that it's possible for life to appear on planets hospitable to it, we can say it's pretty darn likely there's lots of other kinds of life out there).
At least some sufficiently intelligent, curious, and advanced cultures/lifeforms will have things akin to what we humans read in books and study in school ("science", "mathematics", "philosophy", etc.).
Now, assuming they have a means of reasoning and storing knowledge (we have "brains"), a system of communicating with each other (we have things like "French" and "Venn diagrams"), and a system of creating continuity of knowledge between individuals and generations (we have things like "books", "folklore", and "social norms"), what is the overlap between what they "know" and what we "know"? That is, what is taught in their "Universities" that is also taught in ours (and vice versa)? In what ways can we be reasonably certain that what is taught in their "Universities" is similar to what is taught in ours?
(Of course, "University" here need not have anything to do with educational institutions -- we could just as well ask "in what ways are the statements Motherbeing transmits via ultraviolet light to the cryptobreathers similar to what is in our textbooks" -- the use of the term "University" to speak of the hypothetical aliens is to be interpreted completely metaphorically.)
Now then, which branches of our book-learnin will they also probably have?
Stuff taught in our Universities' "Humanities" departments: Eh, mostly conjecture. We assume, recall, that the lifeforms have a sufficiently advanced means of communication; we also assume that they're curious and smart, so they probably have something like our Linguistics (they will care to examine this means of communication). One would think the smart and curious creatures would also care to document past happenings and reason about them, I guess. I don't care to conjecture about much more.
The "hard" sciences: It is almost certain our hypothetical aliens will study physics and chemistry. Further, theirs and ours will converge (molecules over there will do the same thing as the molecules over here; they have the photoelectric effect just like we do). They will also have something like biology ( if they're smart and curious, they'll ask questions about their physical bodies and the creatures around them). As for the interesting question of how their formulation of, say, quantum mechanics could differ from ours -- I'm not remotely qualified to even begin to guess.
Music and art: Not really an interesting question.
Engineering: It is almost certain that they'll have some sort of engineering. Assumedly, the smart and curious alien is a physical being who interacts with a physical environment, and the being will therefore manipulate things to build more complex things, and will want to study how best to do this. Any conjecture beyond that isn't really interesting.
Philosophy: I'd think, in fact, that at least some of our alien friends think about a lot of the things our philosophers think about. I'd posit the following sticky questions are probably present in some form in at least some of the alien cultures:
- identity (when are two things identical? Am I the same thing I was thirty seconds ago?)
- platonism vs. antiplatonism in various domains (does "red" exist? Does the number two exist? Does a collection containing zero objects exist?)
- free will (perhaps -- I'm actually not so sure about this one)
- ontological commitments (do I exist? Do others exist? Does the thing in front of me exist?)
- knowledge (assumedly our alien friends "know" things, and at least some are aware of knowing things. Determining what constitutes "knowing" something turns out to be a huge mess)
It is perhaps out of smallmindedness that I assert that these are universal questions -- a cogitating agent can arrive at each of these questions relatively straightforwardly when it starts really attempting to analyze its condition, regardless of whether this is a human being on Earth or a Znox'x'ow on planet Urth.
Mathematics: This is the interesting one. Suppose the Znox'x'ow have mathematicians. Suppose one of them was taught English and could communicate with a human mathematician. What would happen?
Here are a few conceivable outcomes from the Intergalactic Congress of the two mathematicians:
(a) The human mathematician and the Znox'x'ow mathematician will understand each other perfectly after they get matters of terminology out of the way (what you call "zero" we signify with a pop of our tentacle; what you call "König's Lemma" we call "the dream-consumption hypothesis", etc.). This is certainly plausible -- mathematics is, after all, a description of immutable truths about the universe, and so if the Znox'x'ow are smart and curious, they will discover all of the same math we have and will have (just as they will surely discover the photoelectric effect).
(b) The human mathematician and the Znox'x'ow mathematician sit down and discover, after some period of time, what appear to be irresolvable conflicts in their respective mathematics.
(c) The human mathematician and the Znox'x'ow mathematician will lecture each other about new and exciting things, each of which is genuinely unknown in the other's system of mathematics (but certainly doesn't contradict any currently existing math).
(d) The human mathematician and the Znox'x'ow mathematician will lecture each other about new and exciting things; each will sit down for a while and prove the equivalence of these new and exciting things to things their species is already aware of.
Now, (b) won't happen. That's not how mathematics works. If there is a tension between two statements, something will give. Perhaps one is not well-formed, perhaps one is actually false (and the mathematician was mistaken), perhaps the mathematical system must be amended to resolve this tension (as naive set theory with unrestricted comprehension had to be amended to resolve Russell's paradox, for example). In any case, mathematics does not admit contradictions.
Option (a) could happen. If we had an arbitrarily large number of mathematicians working for an arbitrarily long period of time, and the Znox'x'ow had an arbitrarily large number of mathematicians working for an arbitrarily long period of time, this is probably mostly what would happen. We would know lots and lots of statements about lots and lots of types of math, and we would know, for each of these statements, lots and lots of equivalent statements. They would have the same situation.
More likely, however, we'd have some combination of (c) and (d). Obviously, there's lots of facts about math we don't know, which is why we still pay people to think about math, so option (c) should seem quite plausible.
Now, we humans have many branches of mathematics. There are many different kinds of mathematical constructions and many different types of statements about properties that mathematical objects can have. We often have many different equivalent ways of expressing identical mathematical facts (the Axiom of Choice, for example, has a bunch of equivalent formulations). We'll see seemingly distinct mathematical constructions which end up being equivalent (Turing machines, register machines, and the untyped lambda calculus, for example, all end up doing the same thing). Math is a mess. This is just how it works.
Option (d), then, is plausible. In fact, (d) happens between human beings working in different branches of mathematics all of the time. There is no reason to think it wouldn't happen at our intergalactic math congress.
What, then, can we expect our Znox'x'ow mathematician to know? Will he know what addition is? Multiplication? Will he be familiar with the definition of a Sylow p-subgroup? Will he recognize the Heine-Borel theorem? Will he have finite model theory? Questions abound! I certainly don't have any answers (for what should be extremely obvious reasons), but it's a fun thing to think about.