[G]eometry and number[s]…are unified by the concept of a coordinate system, which allows one to convert geometric objects to numeric ones or vice versa. …
[O]ne can view the length ❘AB❘ of a line segment AB not as a number (which requires one to select a unit of length), but more abstractly as the equivalence class of all line segments that are congruent to AB.
With this perspective, ❘AB❘ no longer lies in the standard semigroup ℝ⁺, but in a more abstract semigroup ℒ (the space of line segments quotiented by congruence), with addition now defined geometrically (by concatenation of intervals) rather than numerically.
A unit of length can now be viewed as just one of many different isomorphisms Φ: ℒ → ℝ⁺ between ℒ and ℝ⁺, but one can abandon … units and just work with ℒ directly. Many statements in Euclidean geometry … can be phrased in this manner.
(Indeed, this is basically how the ancient Greeks…viewed geometry, though of course without the assistance of such modern terminology as “semigroup” or “bilinear”.)
[O]ne can view the length ❘AB❘ of a line segment AB not as a number (which requires one to select a unit of length), but more abstractly as the equivalence class of all line segments that are congruent to AB.
With this perspective, ❘AB❘ no longer lies in the standard semigroup ℝ⁺, but in a more abstract semigroup ℒ (the space of line segments quotiented by congruence), with addition now defined geometrically (by concatenation of intervals) rather than numerically.
A unit of length can now be viewed as just one of many different isomorphisms Φ: ℒ → ℝ⁺ between ℒ and ℝ⁺, but one can abandon … units and just work with ℒ directly. Many statements in Euclidean geometry … can be phrased in this manner.
(Indeed, this is basically how the ancient Greeks…viewed geometry, though of course without the assistance of such modern terminology as “semigroup” or “bilinear”.)
