Let's say that the aliens exist, do you think that they are smart enough to create their own Mathematics? Or even their own Sciences?
This is a rather philosophical question. Math is assumed to be an universal language of nature, with the ratio between a circle and its diameter being always $\pi$ allover the universe. As Galileo said,
the book of nature is written in mathematics
What can change is how the concepts are expressed: what we call $\pi$ might be called anything else by an alien culture, but its concept will stay the same.
This is not true for all the other languages of humans: we cannot even agree on a set of names for colors or seasons, just to tell you some example.
So, your aliens, given the right conditions, may develop a peculiar mathematical language, but its underlying theory will be the same as the one we know.
The rest of Science (and I am using Popper's definition of science as a theory which can be falsified) are instead less strongly universal. Just see at how the theory of motion evolved from the Aristotle to Einstein.
Hard sciences can be defined as the development of theories about the real world, based on the method of testing falsifiable theories.
- Think about the real world and existing scientific theories.
- Make a theory which seems to explain the real world.
- Use the theory to make predictions about how the real world functions if the theory is true. These predictions should be testable.
- Test the predictions. If the test fails, discard the theory. If it does not fail, keep the theory.
There are no "true" scientific theories, only ones which have not failed yet. But if a theory is tested in many different ways and does not fail, one can have some faith that it won't fail anytime soon. (Consider Newtonian physics. Einstein and others proved that it is wrong, but it works on a human scale. You need to go to space or use extremely fine instruments to see the failures. So it is still taught in school, and used day-by-day.)
- I believe that alien mathematics would be the same when it comes to basics like the natural numbers.
- By adding more axioms, one gets more advanced models.
- Some models appear to explain observed reality very well.
- Other models are useful because their study teaches general truths about other models (which explain observed reality).
- Yet other models are not particularly useful to explain anything.
- Alien mathematics can and probably will differ in what advanced concepts they decide to explore, and what they will largely ignore.
To give you an example, consider Euclidean and Non-Euclidean geometry. Euclidean geometry is a better fit for plane geometry as we know it, so it is taught in school. Non-Euclidean geometry is useful to understand geometry in general, so it is studied at university.
For another example, aliens might largely ignore Bayesian probability and focus on the classical models.
Summarized: Science and mathematics are a way of analyzing the real world. As long as the real world is the same, the results of science and mathematics will be similar.
One could imagine an alien species without knowledge of DNA, even if these aliens have DNA similar to ours.
There might be an alien species without knowledge of quantum physics. That's more of a stretch because it complicates electronics.
A spacegoing alien species without calculus is hard to imagine, because they would have trouble to navigate.
* Calling mathematics a science is somewhat inaccurate because it does not apply the scientific method of falsifiable experiments. But that is close enough for most purposes.
I agree to most other answers that math should be universal. But, some fields and subfields develop mostly because they are needed. Arithmetic follows naturally from the need to count and keep track of things. Architecture would lead to the development of geometry. Advanced mathematics would help the development of advanced science and technology. Everything in math would be centered around the way these aliens are.
To give some examples, if the alien lives underground and most of its life consists in moving through tunnels, they would probably have a pretty rudimentary knowledge of geometry, but they would develop topology quite early.
If the alien was a highly cooperative species and dissent and individualism would be unknown to them, would the concept of mathematical proof ever develop and if it did, would it play the role it plays in our mathematics?
Or if the alien was a being made of fluid would low dimensional geometry make any sense to them? I think they might see inventing 2D geometry as big a leap as us inventing space time.
Mathematics would also look a lot different for a highly intelligent alien. Our best mathematicians are able to make seemingly unexpected connections between various areas of mathematics that seem too far apart. Would such aliens be able to invent and prove problems that are simply impossible for us? If one of us would look through their textbooks, even translated in our language and with a simple and friendly notation, they would not be able to comprehend anything.
Let's say that the aliens exist, do you think that they are smart enough to create their own Mathematics?
Terran animals have been shown to be able to count and do addition. Chimpanzees can count, can tell when they're right, and they use similar parts of their brains (compared to humans) to do so. Crows can count although they lack the layer of the brain that we (humans and chimps) use. It is totally reasonable to conclude that there will exist non-terrestrial beings able to do the same.
Lakoff and Núñez wrote the book Where Mathematics Comes From to answer the question posed by the title. Their conclusion is that the mathematics used by humans is a result of our minds and mental structures.
Their conclusion is:
Mathematics as we know it is human mathematics, a product of the human mind. Where does mathematics come from? It comes from us! We create it, but it is not arbitrary - not a mere historically contingent social construction. What makes mathematics non-arbitrary is that it uses the basic conceptual mechanisms of the embodied human mind as it has evolved in the real world. Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history.
The authors deny that there is some sort of Platonic "ideal" of math that every sentient race will use. Furthermore, it is not useful to even consider what a non-human mathematics would consist of. This was mentioned in the preface (under the section "The Romance of Mathematics" - the idea that there is some universal, transcendent mathematics) as well.
Why can't our math be only our interpretation of our surroundings? I do not believe in this universal paradigm. Same goes for physics. The space-time fabric that can be bent? ... well... I truly think that time is measurement that we came up with. Think about it; Let's say that Einstein had the means to carry out his theory and put it to practice. By changing his onw time perspective what would happen to everyone and everything else?
In the 60's a golden disk was sent across the galaxy in hope of being intercepted by some other race and had information encoded in binary because it was assumed that some other race at least as developed as we, if not more, would understand that code. Boule wanted to create a system (to me another math per se) in which there were only two possible outcomes regardless the matter. And it fitted well electricity and computers, the old 0 and 1. But now were evolving to the quantic field, so soon it's not going to be the simple on/off, is/isn't, 0/1. The foundation of math is axiomatic. Why one plus one is equal to two? If we had not currency and economy maybe numbers were not so important.
If I'd be the only living human on earth why would I need math?
In the end I think a lot about this and I'm just not convinced about that universal(ness) of math.
Since we don't know anything about your setting or what defines "aliens", I'll offer the most out there example of a science-based science-fiction novel I know of: The Planiverse.
This is a novel by mathematician and computer scientist A K Dewdney. In it, him and his students make contact with a 2-dimensional alien, viewed side on like a side-scrolling video game, and follow him on an adventure across their 2-dimensional realm. They see 2-dimensional life, technology, and society. The book even contains a rather complete appendix of 2-dimensional physics and biology.
For example, in the Planiverse, gravity decreases linearly with distance, so there is no escape velocity. Perfect seals are very easy to make, so there's numerous safeguards against accidentally suffocating. People must walk over each other to pass, so there's an etiquette about who lies down. There can be no tubes for blood vessels, eating, or excrement; instead there are zipper cells.
So while the basic fundamentals of math are the same, and the scientific process still works, the results are extremely different because their reality works rather differently.
You can take this further and decide that in the alien's universe the basic axioms of mathematics are different. For example, in our universe when you add 1 to something you always get a bigger number. What if you didn't? What if there was no such thing as infinity? What if there's a point where adding 1 caused it to wrap around back to 0? Like "clock math". This isn't so far fetched, it's how numbers in computers actually work, they have limits. For example, with 8-bit unsigned integers 255 + 1 is 0. In a computer this is known as integer overflow, but in math this is modular arithmetic. As non-natives, we build up a fiction of arbitrarily large and precise numbers in computers to match how math works for us. But to an alien native to that reality it would be common knowledge that when you add 1 to a big number you get 0. Similarly for floating point numbers, they would accept that if you try to specify something too closely things get fuzzy. And they'd build their mathematical axioms to match this reality.
Maybe your aliens are living in a giant simulation, a serious question even for our own universe. Greg Egan's Permutation City wonders about life inside a computer simulation. The people in that simulation build a simulation within their simulation with its own physics and chemistry which develops its own intelligent alien life. His follow on Diaspora in the far, far, far, far, far, far future explores life forms in increasingly bizarre simulations and situations. For example, aliens which live in a holographic universe entirely within giant mats floating in the atmosphere of a gas giant. Who knows how math and physics work in there?