[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”.)

Terence Tao

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