From a response to an email:
Keeping the equations in balance is a good example: we’re taught we “must” keep them equal. I prefer to think of it like this:
If we have two things that are the same size (a = b), and we want to keep them the same size, then we have to do the same thing to both sides.
For example, if a = b, then
a + 3 = b + 3
If we only make a change on one side:
a + 3
then we know we’ve tipped the scales in favor of a, so we have
a + 3 > b
That’s a perfectly valid equation, by the way, but it’s not as helpful or precise as saying what something is equal to. (“This shoe will cost you $14.73” vs “This shoe will cost you something over $10.”).
I think that’s the unstated assumption, we want to keep things as specific as we can, because knowing what something is equal to is better than knowing what it’s greater than.