Well, we made the claim that dimension counts the number of numbers one needs to describe a point. We might as well just define 4 dimensional space to be the set of all possible combinations of any 4 given numbers.

That is to say, 4 space is just the set of things (points) that look like this (a,b,c,d), where a,b,c,d are numbers.

Similarly we can list n numbers (a_1,a_2,...,a_n) and call any such list of numbers a point in n space. Does this definition fit our understanding of 3 space?