Two agreements all maths uses
Two rules work on rice, on rupees, on lengths. Two things balance when they weigh the same. A link is a middle thing that matches both. Switch the link and see.
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- Sack A balances crate M.
- Crate M balances sack B.
- So sack A balances sack B.
- Add 5 kg to both pans. Still level.
An axiom, in full
An axiom is agreed at the start, with no proof. Euclid listed more of them. These two are enough here.
Things equal to the same thing are equal to one another.
If equals are added to equals, the wholes are equal.
Neither sentence names rice, numbers or lines. That is why every part of maths can use them.
What an axiom cannot do
The first axiom passes equality across one shared middle. Sack A balances crate M. Crate M balances sack B. So A and B are equal.
Now cut the link between M and B. Nothing connects B to anything. We know nothing about B.
A length example: take three segments AB, PQ and XY. If AB = PQ and PQ = XY, then AB = XY.
Rule: Equality passes only through a link. The link must be given or already shown. Without one, nothing follows.