The following is from Analytical Mechanics, by Valter Moretti:
Definition 1.11. A metric space is a set $M$ equipped with a function$d:M\times M\to R$, called distance, satisfying:
(1) symmetry:$d(P,Q)=d(Q,P)$;
(2) positivity:$d\left(P,Q\right)\geq0$, with equality if and onlyif $P=Q$;
(3) triangle inequality:$d\left(P,Q\right)\leq d\left(P,R\right)+d\left(R,Q\right)$for $P,Q,R\in M$ .
A function $f:M\to M^{\prime}$ between metric spaces $M,M^{\prime}$with respective distances $d$ and $d^{\prime}$ is called an isometryif it preserves the distances:$d\left(P,Q\right)=d^{\prime}\left(f\left(P\right),f\left(Q\right)\right)$for any pair $P,Q\in M$ . $\diamondsuit$
This is from Modern Geometry with Applications, by George Jennings:
The most important measuring tools for the Euclidean geometer are a ruler (for measuring distances), the protractor (for measuring angles), and a sense of orientation or rotational direction for distinguishing clockwise and counterclockwise rotations.
Since Jennings discusses n-dimensional space, he clearly intends his "ruler" to be abstract.
In Tensor Analysis for Physicists, J.A. Schouten calls basis vectors "measuring vectors" even in the context of affine geometry.
My question is: When discussing distance in the context of Euclidean geometry, is one obligated to introduce some notion of a measuring device, such as a ruler?
Stated differently, is geometry distinct from numerical mathematics?
Consider this statement from Albert Einstein's The Meaning of Relativity:
"One is ordinarily accustomed to study geometry divorcedfrom any relation between its concepts and experience. Thereare advantages in isolating that which is purely logical and in-dependent of what is, in principle, incomplete empiricism. Thisis satisfactory to the pure mathematician. He is satisfied if hecan deduce his theorems from axioms correctly, that is, withouterrors of logic. The question as to whether Euclidean geometryis true or not does not concern him. But for our purpose itis necessary to associate the fundamental concepts of geometrywith natural objects; without such an association geometry isworthless for the physicist. The physicist is concerned with thequestion as to whether the theorems of geometry are true ornot. That Euclidean geometry, from this point of view, affirmssomething more than the mere deductions derived logically fromdefinitions may be seen from the following simple consideration."
Does that mean that geometry is devoid of any exercise of spacial reasoning? Or does it mean that the spacial reasoning of pure mathematics has no direct connection to objects of experience? And specific to my question, is a concept of a rigid object which can be used to compare separations of points required in order to do geometry?
As a more dramatic example, if we punctuate the top of the sphere upon which the Euclidean plane with rectangular Cartesian coordinates is projected, the points on the plane map bijectively to the sphere. Using the the coordinates of the plane, we can define "distance" on the sphere using $\left\Vert P-P_{\circ}\right\Vert ^2=(x-x_\circ)^2+(y-y_\circ)^2$ with the $x$ and $y$ being the plane coordinates of the projected point. How can one distinguish between the sphere and the plane without some kind of standard of comparison that "honestly" measures distances?
This image is from https://virtualmathmuseum.org/
What follows is quoted from this resource:
https://sites.math.washington.edu/~king/coursedir/m444a04/notes/02-bbaxioms.html
Birkhoff's Ruler and Protractor Axioms
This is the "informal" version of the axioms as found in a high school text Basic Geometry by Birkhoff and Beatley (abbreviated B&B). In that text these Axioms are called Principles 1-5. There is another more sophisticated version published in a mathematics journal for mathematicians.
Undefined Terms
- Point
- Line
Axiom 1. (Ruler Axiom: Line Measure)
- The points on any straight line can be numbered so that number differences measure distances.
Axiom 2. (Two points determine a line.)
- There is one and only one straight line through two given points.
Definitions: At this point, some definitions are made, including the definition of a half-line and straight angle.
Axiom 3. (Protractor Axiom: Angle Measure)
- All half-lines having the same endpoint can be numbered so that number differences measure angles.
Axiom 4. (Straight Angle Measure)
- All straight angles have the same measure.
More Definitions: At this point, some definitions are made, including the ones used in the final Axiom.
Axiom 5. (SAS -- Congruence version)
- If in two triangles, two sides and the included angle of one are congruent respectively to two sides and the included angle of the other, the two triangles are congruent.
