For the computer scientist, a vector is a tuple of numbers. For the physicist, a vector is a direction and length. For the mathematician, a vector is an object which you can add with another of its own kind and scale with a real number and leads to an object of the same kind.

The last definition sounds aweful. But is it pure freedom. A moment ago, I was thinking about half-life of caffein and ask myself if I drink two coffees they sum. Yes. Then I realized, that I scale half-lifes by a scalar. Turns out, it is enough to qualify as vector. I can use half-lifes to construct a vector-space and use all my knowledge of linear-algebra on them.

Mathematical definitiions are minimalisic and allow you to reuse existing knowledge in unexpected areas. Math is pure elegance!

Btw, A vector is not a list of numbers - or a tuple.