If aliens were certain to exist, would they have math? It is very tempting to assume that mathematics really exists and that it has been discovered if it was conceived by a very different kind of life, not on this earth, but with the same observations on the universe. However, there is no scientific evidence that aliens exist, yet. Let alone that they would have math….
The same rephrased question: did we create math to understand the universe or is math the native language of the universe itself? Discovery or invention? One could say, for example, that the sun would still exist if man doesn’t, but one could not say cars would exist. That is the difference between a discovery and an invention; something made by man and something that already existed before man.
Just adding numbers
This is definitely not a new question. Actually, this question has been asked since ancient times and has been hotly debated. The Pythagoreans of the fifth century believed that numbers were both universal principles and living entities, with each number its eccentric nature. They called number one, “the monad”, the source of creation and generator of all the numbers, since every number can be made by adding up ones. Another example, Plato asserted that, independent of human existence, numbers were as real as the universe itself. Moreover, Euclid argued that nature was a manifestation of mathematics, similar to what the Pythagoreans thought.
Philosophers of our time have thought about mathematics as being in the exterior of our thinking, already existing before mankind. The foundation of mathematics is the number system, which could be described as an extension of the natural numbers (the natural numbers already existed because counting has always been done). One extension could be made by asking: what do I need to do to go from 100 to 40? Simple! just take another number with another sign, -60. Integers were born. Want to make from 2 a 1? Just multiply by this new symbol, ½. Rational numbers were brought to life. When having a one-by-one square, what is the length of the diagonal? That is the square root of two and that is just a number that fills the gap between the rational numbers whose square root is less than 2 and those rational numbers whose square root is greater than 2. Hey! That is a real number.
But how can these operations of extending the system be justified? With the train of thought that mathematics is discovered, natural numbers are assumed to be already in existence and thus all its successors too, since the only constraint for those successors is that they have to follow the rules of the already existing system. Another way to put this is to say that these numbers are not really added, but were already there. This sounds vague, so here is a helping analogy. A consumer asks at a grocery store if there is any lickerish left. The worker says that there is nothing left, but another consumer comes and tells the two that there is some lickerish (outside this store). This is not a lie, it is rather an extension of the area in which the lickerish can be found, and less strict. The same is with numbers; by taking a less rigorous conception of the area in which, these numbers can be found.
In contrast, others may argue that mathematics is invented and thus a creation of our own thinking. From that it follows we are indirectly thinking about ourselves; we use our thinking ability to set a framework of the universe. Though, it works and applies to the real world: bridges don’t fall (for the most part), we know how to predict the weather (for the most part) and man has visited the moon, to name a few things.
One particular point of view is that of Stephen Wolfram: in an interview he said that mathematics can be seen as an artifact, an axiom system peculiar to this humanity. How is that? His following explanation may clear this up. If you look back to the history of mathematics, it could be noticed that it is just a processive generalization of arithmetic, geometry and one abstract idea of logic. The development of arithmetic was there to count sheep, for example and in the first place, geometry was probably used to find the maximum area of a field. The idea of logic is based on an axiom system that happens to be useful. This originated with the mindset: let’s make up a theory and then find the most general thing it satisfies, based on everyday experiences. Then follows the question: which theory will be just true and which will be provable? Some theorems became axioms (accepted as true) and some became proved theorems (based on those axioms). The remarkable thing is that millions of papers in mathematics are based on these few axioms, but are those axioms the only possible axioms? Is our mathematics the only possible mathematics? Probably not. There likely is a whole universe out there with other possible valid mathematicses. There would be nothing special about our mathematics. So, to answer the introductory question, yes, if aliens exist, they would have math, but a different kind.
Another argument for why math can be seen as an artifact is that the counter argument “Math is a model of the world.” is a circular argument. The development of math has been driven by modelling the natural world, but the things that we can observe about this world, the things that has been successfully addressed in science, are just the things that our mathematics allowed us to successfully address. So, that argument could also be read as: the world is a model of math.
From this it follows that there are several points of view on this question, but let one thing be clear, the most useful thing to do is to look at mathematics as a game and the numbers are the peonies. Speculation about the existence of a number or its eccentric nature is generally disregarded within the field of mathematics. After all, math continues to help us solve problems.
Book: Elementaire Deeltjes 14- Wiskunde by Timothy Gowers