I’ve heard things like the integers form a ‘ring’ while natural numbers do not, but I don’t quite understand what they *are*.
In: Mathematics
Mathematician specializing in abstract algebra here!
For context you should probably realize that Mathematicians are mostly doing things with arbitrary logical constructions rather than just numbers and geometric shapes.
We figured out that some results are much easier to think about when we look at the set of all things sharing a property rather than just the property alone. For example, people spent a long time knowing self evident results like 2+3= 3+2, and understanding that it doesn’t really matter which numbers you pick because it’s true for all of them, but leaving it as just that, a property of numbers.
But at some point people stopped and thought that maybe it’s not really something true about numbers, but rather something that is true about the way we’ve decided to represent numbers. And so people basically kept repeating the same thought experiment over and over: what is the smallest amount of assumptions that anyone would need to deduce how numbers work? In doing so they started to make discoveries, because if you have too few assumptions you can actually come up with really weird (and interesting!) patterns that operate under those same assumptions, but are clearly very different from how numbers work.
These patterns became so interesting and popular that we felt the need to package the assumptions together so it was easier to talk about the patterns. One of these packages is called a “ring”. There is no reasonable explanation for why exactly it’s called a ring, it’s not ring shaped or anything, it’s just a name. Another comment talked about the other packages we call groups and fields, and with a more detailed explanation. If you relax the assumptions in a field you can find what’s called a ring, and if you relax the assumptions in a ring you can find what’s called a group.
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