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Should We Expect The Mathematical Constant (Pi) π To Appear In A Alien Mathematical System?
Humans developed Pi (π) as part of our need to survive under certain evolutionary conditions. We had to invent the wheel and from there the way to circles and trigonometry and calculus is short.
While Pi (π) is likely to be known to any advanced civilization, because of the centrality of its role in several fundamental aspects of geometry, it is very unlikely to appear in any alien mathematical system as it appears in ours. The likelihood of it appearing in the same form, designated by the letter Pi (π), is vanishingly thin. So far as I know, the letter Pi (π) arose for reasons having nothing to do with the nature of the mathematical concept. As such, like forces are unlikely to control the symbolic representation in any other civilization.
If it is thought “as it appears in ours” was a reference to the numeric representation (3.1415…) then I am lost. It is unlikely alien mathematicians will use Arabic numerals or even a base10 number system. So the answer still remains “no.” The fact that they might understand pi as being the ratio of circumference to diameter, in the alternative, seems not to involve a meaningful difference in form since it can implicitly be reformulated to a parallel definition without the uncertainty of result or the need for innovation.
As for the idea that this would only be true if they existed in the same universe as us, I am baffled. Is there a universe in which Euclidean circles have circumferences that are not 2*pi around? How could that be?
The caveat that this would only be true of smart aliens seems unnecessary unless we are prepared to admit that in addition to true mathematics, there is something byways of a false body of beliefs about numbers that is still properly called “mathematics.” But this seems like yet another unreasonable aside.
One could image two cases, where there will be some differences:
 Volume instead of length: we use length as a base in our units, but it is possible to start from volume instead of length.
 Complexity instead of simplicity: normally, we consider the formula Circumference=π Diameter, an ideal formula, it cannot be any simpler. However, this has not always been the case and one could actually construct very complicated relationships.
I can imagine too many ways where intelligent creatures (smarter than humans) can develop without the need of circle or even calculus as we know it. If I would restrict the discussion to creatures that are very similar to humans made out of carbon and water that live on other Earthlike planets (in size, gravity, temperature, etc..). Then, I would expect that small changes in the materials making the planet crust would change the entire picture. For example, what if one of the following materials exist in a probability that it is much larger than on our earth: strong magnetic materials, strong piezoelectric crystals, strong pyroelectric materials, room temperature superconductors, or others. I can imagine that these would lead to early development of electronic based energy sources that would be used for essential needs without the need of wheels or Pi (π).
To conclude, I claim that Mathematics is only our language to describe the world. Physics of the universe can still be understood with different languages other than the Mathematics we developed along the years.
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