I went to an amazing wedding this pi day for my cousin Jonathan and his now wife Kd where the following question was posed on the dance floor by Kd’s friend Jade, “Would math be the same on Jupiter?” I don’t know if it was the occasion, the alcohol, or the dancing, but I was inspired to finally write again after getting some moderate research done over the quarter.
The real gist of the question is whether or not math is something subjective which would be completely different on some alien world. Her answer to the original question was a big “no,” that is, that math would be different on Jupiter and hence is subjective like all other things.
I agreed with Jade that everything is subjective, but said math is the only true exception! This is what makes math the coolest. I’ll try to explain why math is the same everywhere as best as I can.
I first need to mention science because a lot of people mistakenly believe that math and science are heavily intertwined. In the past this was true because scientists often came up with math accidentally while trying to explain things, but in modern times math is so specialized that only mathematicians come up with the abstract nonsense. Sometimes scientists will use modern math but typically old math is sufficient so they rarely come up with new ideas. Their relationship is better described as parasitic now. Science the parasite uses math everywhere, but math can exist on it’s own without science. Scientists observe things, make guesses, and try to show that their guesses are correct in the best way possible. The guesses and theories they make can be heavily influenced by location (but some physical truths are universal).
For example, in X amount of years which I can’t be bothered to Google, there will be no stars in our sky because they will all be too far away to observe. A scientist of that time would have no ability to observe celestial bodies and might never figure out what we’ve figured out because of that. Yes, the truth is the same either way, but a scientist cannot figure out that truth without something to study.
Math is a different beast. Math is not done through experiments first of all, we have no fancy labs to work in, we just sit there and think for a while. Then we jot something down, cross it out because it was the dumbest thing a human has ever written down, and keep thinking. Math is all about thinking and proving your thoughts with logical rules. Proofs are just collections of statements which start at some assumed statement and end at a conclusion. In between those two things you are only allowed to say objectively true facts to get to the end of the proof. You cannot use things that you believe are true, only things that are known to be true.
For example I proved in another post that there were an infinite number of prime numbers. See here for details. To prove the statement you have to assume several things, like that we have a number system, and that there is a definition of a prime number along with some more subtle assumptions that go way too deep to mention. I use these facts to eventually conclude that the statement was true.
This leads me to the next big point, assumptions.
It was long ago realized in math that you have to start somewhere. The late 1800’s saw a surge in Foundational Mathematics. People wanted to redo all of math starting from the bare minimum. The problem is that you cannot define everything because that would require an infinite amount of words. For example, if you go to a dictionary and look up a word, then you look up the words in that definition, then keep doing the same with each of those words and so on, eventually you are going to end up going in circles. Not every word can be defined by other words. Math is similar.
If I tell you that a “Set” in math is defined to be a collection of elements, you might ask, well what does a collection mean? What does an element mean? And I’ll tell you that a collection is just a grouping and an element is a thing. Then you ask what is a grouping or what is a thing? And it never ends so you simply accept the intuitive idea of what it is supposed to mean! How definitions interact with each other is what really matters! In this example, the fact that a Set consists of Elements is what matters. What we do in math is start from some beginning definitions like this, assumptions, or axioms and see what pops out when natural questions are raised.
Here is a link to David Hilbert’s axioms of Geometry. Axiom is just another word for foundational assumption. It is the most basic thing you can possibly assume because it doesn’t follow from a different assumption. All the things about geometric shapes that you know and love can be proven from the starting points in the above link and if I take these assumptions to Jupiter, all my proofs will still be true. Universal truths simply pop out of assumptions and our job is to find them. If the universe ends, all the facts about shapes will still be true starting from these axioms regardless of whether or not someone is alive to say so. The shapes don’t even have to exist for the proofs to be true. There is no such thing as infinity in our universe but we still have tons of theorems about infinity that are true regardless.
Now the cool part is that you and I can start from different axioms and get different but also true statements. With the above axioms and a lot of free time you can prove the Pythagorean Theorem as it is commonly known () for the sides of a right triangle. If you change one axiom, the Parallel axiom, to say that parallel lines are allowed to cross (like on a sphere),
you get a completely different version of geometry with a different Pythagorean Theorem and they don’t contradict each other because they started from different assumptions.
This kind of stuff also allows new math to be created from minimal things. We don’t need a universe or observations to make math. We just assume some arbitrary stuff and see what happens. The assumptions don’t have to go along with physical facts, they can be whatever so long as they don’t contradict each other and are foundational. Obviously assuming some stuff and not others makes for more interesting questions and answers.
You might exclaim here that I was incorrect! An alien could come up with different theorems because they used some other axioms, I even admitted it! Yes that is absolutely true, however it doesn’t make their math different from our math. Our math is still true on their planet and theirs on ours. If an alien starts with the same assumptions, they cannot contradict what we’ve figured out. They will get the exact same results. Getting different results from different axioms is totally fine and doesn’t make our maths any different in the sense that people usually think about. One plus one is not suddenly going to equal three anywhere in the universe.
“It does not matter if we call the things chairs, tables and beer mugs or points, lines and planes.” – David Hilbert when referring to Geometry.