Why is $\mathbb{R}^1$ different than Euclidean space $\mathbb{E}^1$? Roger Penrose road to reality In Roger's book, the following is stated: (I'm paraphrasing because my book is in spanish)
"We consider time as part of a space, namely $\mathbb{E}^1$, instead of it just being a copy of the line $\mathbb{R}$. This is because $\mathbb{R}^1$ has the privileged element 0 which would represent the origin of time"
I don't get why this would be true, because, according to what I know, $\mathbb{E}^1$ also would have its zero. 
He goes on and talks about how theres also no origin for space in $\mathbb{E}$. But I'm also confused about this.
I understand the idea, of course, what I don't get is how you can mathematically model a space with no origin?
 A: My guess would be that $\mathbb E^n$ denotes Euclidean space. In addition to having geometric structure (angles and distances) and motions (rotations, translations, reflections) - not all of it terribly useful in the 1-dimensional case - it is an affine space.
Affine spaces have no notion of distinguished origin or zero point. We can use a vector space like $\mathbb R^n$ to represent an affine space, but we have to be aware that not all operations defined on a vector space are allowed in the affine case. In particular, addition and scalar multiplication make no sense in affine spaces, whereas substraction still does, but will result in a displacement vector that is not part of the affine space itself, but an associated vector space.
As an example, consider dates: What's the result of adding March 13th to December 4th? That's nonsense. However, you can ask how many days lie between two dates, and you can add such a duration to a date and get a well-defined result. Displacing a point by a vector is fine, but adding two points is not.
Note that this goes even further in general relativity (and analytical mechanics): In curved spaces or spaces without the necessary structure, there might not be a coordinate-independent notion of finite displacement. The best we can do are infinitesimal displacements (tangent vectors) tied to specific points of the base space. It's probably a good idea not to think of these in terms of displacements at all, but velocities, even if we sometimes do have things like the exponential map that blur that distinction.
