Each of the "ordered field", "inital R-algebra" etc. are the names of a set of rules and constraints. That's all it is. So you need to know those sets of rules to make sense of it. It has nothing to do with brain size or IQ :)
In other words, you define a new thing by simply enumerating the rules constraining it. As in: A Duck is a thing that Quacks, Flies, Swims and ... Where Quacks etc. is defined somewhere else.
Each of the "ordered field", "inital R-algebra" etc. are the names of a set of rules and constraints. That's all it is. So you need to know those sets of rules to make sense of it. It has nothing to do with brain size or IQ :)
In other words, you define a new thing by simply enumerating the rules constraining it. As in: A Duck is a thing that Quacks, Flies, Swims and ... Where Quacks etc. is defined somewhere else.