If we were to find life on another planet at least advanced as ourselves or more so, do you believe that they will have created a system of mathematics in any way similiar to ours. I tend to think they would and here is my reasoning behind that: It seems the first step in mathematics would be simple addition and subtraction which, at least in our history, has been an important ability even in our most primitive stage. Also considering that animals can, to a degree, also understand simple addition it seems likely that this would be the base for almost any life form capable of understanding it. The next step would likely be multiplication and division which, much like addition and subtraction, are fairly simplistic and helpful. Our entire system of mathematics seems to follow a logical path and so it seems reasonable to assume that if somebody started from the exact same idea of simple mathematics, that their more complex ideas would progress at least similiar to ours. The only problem I can see with my logic is assuming that their brain would function similiar to ours. For example, we use various different methods to find convergence of sumations. If an alien mind were capable of performing thousands of sumations in a very short amount of time I can see how their take on math might be entirely different from ours.
Discuss. 
Not only would said aliens use mathematics similar to our own, but it's logical to assume that they would look a lot like us, too.
The SAME laws of physics that produced us would also produce them. It is reasonable to assume that evolutionary processes there would produce superadapted animals (Like cats) that don't need to further evolve beyond tooth and claw... and only the most generic animal would develop reason. Likewise, the foundation of mathematics is counting units, a concept that is universal. And PLEASE! Noone pull out any crap about 'alternate universes' where these constants might not apply unless you've actually BEEN there and seen it. 
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We could still be stuck trying to multiply Roman numerals (what a chore that was) and, because of it, not progressed any farther. Whole branches of mathematics might not have been discovered, like topology, nonEuclidian geometry, calculus. Your alien might, earlyon, have contemplated nature and decided to define a line as a contiguous series of points in 3space (translated to English from the Asdfjkah dialect). They might have concentrated on curves and how to manipulate them, treating straight lines as very special and not very useful cases. I am afraid that it is arrogant to think that aliens who know how to form sustained toroidal bubbles of air in their water environment (Porpoises) will recognize the universality of Maxwell's equations. Apologies to Sagan and "Contact". 
While xenomathematics may have been conceived differently, that does not change that fact that their system would be totally compatible with ours and an almost (if not totally) 100% accurate translation could be made between the two. This would be more accurately translatable than any terrestrial human languages have ever been. Viewing a line as a contiguous series of points might allow for dramatically different methods of calculation, but that does not change the nature of the line itself, and calculations made about that line would be compatible between systems. The calculus may not have ever been developed in the same way, but given a continued interest in the natural world and mathematics, to assume that they would develop equally effective ways of determining differentials, integrals, and c would not be unreasonable.

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The METHOD of expressing the basics may seem alien or incomprehensible to us at first, but it WILL be there. Just as hydrogen and oxygen on their planet can form water under the right conditions... there will be things they need to count. The basics will always be there. When teaching a child math, do you start with trig? No! You start with "If I have 2 apples..." There is no reason to assume aliens won't do this too. 
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It is true that, once a rich mathematics is known, it can be derived, after the fact, from first principles starting with the distinction between zero and nothing. It is not at all certain that an intelligent baby placed alone in the middle of a forest with no other human contact would or could develop any part of math during its entire lifetime. I expect that when we meet your aliens, they will not be listening to ipods so we will label them unintelligent, retarded savages and will do our best to work them to death. 
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stupid whores. They just need a good shag instead of munching the furry pi. 
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The basics will ALWAYS be there. 
Of course, if they were living in an alternate universe where the laws of physics were different, and mathematical equations even to us would be unrecognizable. 2+2 might = 7, cat may be spelled D O G and even... :o
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antix, you must masturbate three times in the moonlight top make amends......

Well... THAT wasn't easy. Amazing the looks people will give you when your sitting, leaning against the sde of the house masturbating furiously at night. But hey. It had to be done. I believe this discussion can get back on course now.

I imagine the biggest difference would be the base of their number system  ours is base ten, but how many fingers do these aliens have?

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How do we know they're not tentacles, or tractomorphic patches of luminescent white marshmallow skin? 
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I personally think that a lot of mathematics is cast into us as we evolve. Bear with me.... The pressure a primitive being feels when water pushes against him, for example, has a describable amount of force. While he doesn't have the brain power (yet) to quantify it, he could at least understand large vs small. If you think about it, birds are a good example of that. How much is a bird likely to know about what we consider as mathematics, even something as basic as addition? Likely extremely little. Yet, watching a turkey vulture hover in a thermal against a headwind, remaining in place, you have to realize that it is naturally balancing a large array of forces to remain stationary. The math is then "innate" I guess you could say. Of course, how many millenia is it going to be before the birds are to the point where they can quantify and communicate that fact to another bird? I surely have no answer. Still, the math is there, whether or not it is described by the animal. In the same way, a spacefaring alien would be likely to have a mathematics that is quite similar to ours we live in the same physical universe. Their notation would be different, of course, hell the Russians use a different notation in their vector field theory, but at least it only looks like a convention thing, and you can get used to reading it after awhile. I would say that a really good way for two alien races to communicate with each other would start with mathematics, since that is relatively unambiguous, compared to the rest of the things we can describe. Pick a simple example like the name for an elevator in England (I'm American, but know this): "lift". In America, that would only mean we were going up, right? And "drop" means going down. I'm not knocking England, just pointing out how easy it is to miscommunicate. But there is a large amount of information contained in Mathematics (and Physics, for that matter) that is unambiguous enough to give a good start on reliable communication. 
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