Equal to the same are equal to each other - Is it geometry?

Question

Is this geometry or algebra "things equal to the same thing are equal to each other"?

thank you in advance.... Joseph Whelan CPA Bedford, NY

Answer 1

Things which equal the same thing also equal one another.

It's what Euclid (in his book Elements, Book I) considered a "common notion", and the "things" which he was referring to were geometric "magnitudes", but is a generally accepted in every branch of mathematics, and fundamentally, in logic. A "common notion" states propositions which cannot be otherwise defined nor proven, but are propositions no one (sane) would object to (i.e., that which we find to be obviously true that "it goes without saying", so to speak).

Answer 2

In general, this is just a philosophical statement about one thing that an idea of "equality" (or "equivalence") should entail. There are a few illustrations of this in geometry, but it appears many more places than that.

In most systems of synthetic geometry there are axioms asserting this fact for congruence of segments and congruence of angles.

For example in Hilbert's axioms for geometry, in axiom group IV, #2 and #5 make exactly those assertions. As amWhy remarks in his solution, Euclid also adopted this idea when thinking about foundations of geometry.

The same idea appears throughout mathematics. The general property of a relation that goes "If A is related to B and A is related to C, then B is related to C" is called transitivity. All branches of mathematics use transitivity, at least as an essential ingredient for equivalence relations. In a nutshell, equivalence relations establish different ideas of equality between items of a set.

The congruence axioms above simply assert that in geometry we wish segment and angle congruence to be a transitive relationship.

Actually there is one more thing similar to the above hiding in axiom group III about parallels. By asserting that through any point $P$ not on a line $L$, there is exactly one parallel line to $L$ through $P$, we are actually asserting that "being parallel" is a transitive relation. For, if $L_1$ and $L_2$ were distinct parallels to $L$ but were not parallel to each other, they would intersect at a point $P$ (which of course could no be on $L$): but there can only be one line through $P$ parallel to $L$... a contradiction!

As an easy example of a non-transitive relationship, you could consider "friendship." I'm certain somewhere there are two people who don't consider themselves friends who have a friend in common. This basically shows that friendship is not suitable to be an equivalence relation. If friendship were transitive and also symmetric, then friendships in this world would be much simpler!

Answer 3

This is an axiom of equality. As Umberto P. suggests, it is closely related to the definition of equality wherein we say that equality is a predicate which satisfies reflexivity, symmetry and transitivity.

If you say algebra is study of functions and operations and geometry is study of relations, this might be close to, but not in, geometry. It is far more fundamental than geometry.