What fields (and operators acting on those fields) might form the basis of alien mathematics? Addition and multiplication, according to most histories, arose in human civilizations out of a need to count a finite number of objects, and then later on especially, to measure land.
What might be the catalyst for the solidification of the notions like "addition" and "multiplication" in alien civilizations (what if they don't really have any "land" to measure?)? Would their system of mathematics perhaps be based upon entirely different common fields and operators than human mathematics because of that? An answer should explore a possible alternative field and operators acting on that field as a viable seed for the development of some system of mathematics.
While this question might seem light-hearted (and undoubtedly is, to some extent), it is primarily meant to be catalyst to try and explore alternative ways of conceptualizing our most basic and cherished intuitions. 
An alternative way to think about this question (but not to answer it!) could be: given the fields and operators that are known and understood by us today, can we imagine how else they might have come about in some history of mathematics? 
 A: I'm studying History of Mathematics at university right now, and so I am inclined to give a negative answer (which isn't interesting, but hey, it is what it is). This is because, regardless of what you are measuring, there is a fundamental need to describe quantitatively how much of something there is. This doesn't have to just be land. A few examples:


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*The Babylonians used cuneiform text to keep track of records for ziggurats - how much of various holy sacraments would be needed to perform x amount of rituals and so on. They also did early work in geometry so that they could figure out the area needed for their buildings, as well as the ratios of their sides and so on, which seems to be of importance to them.

*Similarly the Egyptians did the same for their pyramids, but also developed simple geometry for the use of land allotments.

*The Incas used knot tying to denote the amount of livestock they had, with various types of knots denoting different amounts put in different positions in an early positional system, comparable to a primitive decimal system.

*The Greeks had two mathematical booms. Both were axiomatic, which was a great leap forward. They had an axiomatic geometric system, which people are very familiar with in The Elements. However, there was another note worthy push in terms of elementary number theory. With spectacular progress determining which numbers were rational and irrational, as well as making distinctions between fractions and whole numbers, (which actually was done nowhere else in the world except Egypt. Fractions were denoted the same way as almost every other number, and it was purely contextual which power of 10 a number referred to, including negative exponents!).
This being said, I feel that counting some discreet collection of livestock and making measurements are the only way to give rise to numbers in terms of practicality. So bearing this in mind, if an alien civilization were to not start with either of these two, then it would be most comparable to the Greeks - axiomatic, abstract and deductively certain.
So where could we start? Well, if we were to be truly logical, it would not make sense to discuss any kind of algebraic system using any operators more complex than these, because fundamentally it doesn't make sense to discuss any of these systems without first a development of the real numbers. 
Even if the reals were developed without any of the traditional operations we think about, try to imagine an operator on the real numbers that does not involve the basic ideas of addition and subtraction, multiplication and division, underneath them. They might make less assumptions about these structures and begin the development from the perspective of abstract algebra and group theory with an arbitrary group operation, but it would be meaningless to do so from an application perspective, which would not be very useful to a budding civilization. After all, it has been said that mathematics has only flourished in civilization whose populace have leisure time. 

From a philosophical perspective, I feel the same negative answer creeping at my mind, but for perhaps a different reason. I think this because it is my belief that mathematics is not invented by humans, but rather, discovered by them. The basic notion of a quantifier, and in particular, a first quantifier (namely, 1), cannot possibly be a human invention in my mind, and this is surely the attitude the OP must have, since if quantification of some sort or another were uniquely human, then no alien civilization could even dream up mathematics at all. 
So quantification is the only rational starting place then in my mind. From here, they would have to build mathematics more or less the same way that we do currently - a development from simple quantification to a development of the real numbers complete with the basic operations that we think of.
Now, from there, the places they go can't be said. They would probably have to develop some kind of algebra, calculus and the like, since these seem to be the spring boards into higher mathematics that we are familiar with. From there, they could go almost anywhere with it - the interests they developed and focused on rooted entirely in the paradigm of the thinkers. As an example, consider that calculus and early group theory were worked on by Newton and Gauss respectively. Even though Newton's work didn't predate Gauss's by a very substantial amount of time (only about 40 years), calculus took off in its applications almost immediately, while abstract algebra's sudden growth begins with Galois another 100 years later, 150 years after the publication of principia mathematica.
Further considering that these two fields of math were worked on by two separate groups of people in separate parts of the world, its clear that the direction mathematics takes is in the paradigm of the thinkers - so if a different direction at all is conceivable, it would be more of a question of cognition and circumstance than of mathematics or philosophy.
At least in my opinion.
Hope this helps.
A: I would argue that the key feature that distinguishes maths from other disciplines is logical proof. I think the concept of a proof was first invented in ancient Greece, in the field of geometry, not arithmetic. I could imagine an alien civilisation inventing proof for the first time in almost any area of mathematics.
