Comment by StopDisinfo910

> does this paragraph mean something in the real world?

It's actually both surprisingly meaningful and quite precise in its meaning which also makes it completely unintelligible if you don't know the words it uses.

Ordered field: satisfying the properties of an algebraic field - so a set, an addition and a multiplication with the proper properties for these operations - with a total order, a binary relation with the proper properties.

Usual topology: we will use the most common metric (a function with a set of properties) on R so the absolute value of the difference

Finite-dimentional: can be generated using only a finite number of elements

Commutative: the operation will give the same result for (a x b) and (b x a)

Unital: as an element which acts like 1 and return the same element when applied so (1 x a) = a

R-algebra: a formally defined algebraic object involving a set and three operations following multiple rules

Algebraically closed: a property on the polynomial of this algebra to be respected. They must always have a root. Untrue in R because of negative. That's basically introducing i as a structural necessity.

Admits a notion of differentiation with reasonable spectral behaviour: This is the most fuzzy part. Differentiation means we can build a notion of derivatives for functions on it which is essential for calculus to work. The part about spectral behavior is probably to disqualify weird algebra isomorphic to C but where differentiation behaves differently. It seems redondant to me if you already have a finite-dimentional algebra.

It's not really complicated. It's more about being familiar with what the expression means. It's basically a fancy way to say that if you ask for something looking like R with a calculus acting like the one of functions on R but in higher dimensions, you get C.