I only see theorems as useful for complex societies. The son of a gem dealer would have the time to work out universal truths.
The reputation of everyone doing it without numbers would reveal the pattern of the universal truth.
Finally this looks like it was all done using cuneiform.
Which brings up questions of notation and the language to describe a square root.
Progress is a luxurious goal.
Observing that something always holds and even having a formula for it is not the same thing as having a proof, and the proof is what makes it mathematics and geometry and not "just" engineering. The babylonians had a rule of thumb -- the greeks discovered the theorem -- and more than that, they seem to have invented the mathematical/geometrical proof as a concept, along with formal logic.
Without that mental framework, it's hard to say that the babylonians proved anything or had any theorems at all, only collections of rules of thumb. It's quite likely that lots of babylonians sort of independently and intuitively understood _why_ it must be true, but they don't seem to have ever written it down.
Not that there's anything wrong with having rules of thumb -- it's a huge achievement to even notice and collect and teach those things, all the stuff around you is built relying on them.
I encourage everyone to watch this series of videos.
As I understand Greeks invented proofs because "universal thruths" they exported from Babylon and Egypt sometimes explicitly contradicted each other.
I believe such contradictions may be a great nuisance when you try to get shit done.
Egypt and Babylon were sufficiently "complex" societies for proofs, but their tradition treated mathematics as a bunch of useful facts about numbers and shapes. New generation just memorized them. We should think it worked for them in most cases, and when it didn't work it was not so often for them to start thinking a lot of reforming mathematics. Plus they were indoctrinated by the math they learned (authority of a teacher is above of anything else, i suppose) and to reform math was not a natural idea for them.
The nice thing about Pythagorean triples for drawing out foundations is that you don't need access to any ground outside the foundation of your building. Being integers, you also don't need any measuring device apart from some rope. You just pace out a bit under 1/3 of the shortest side (or a bit under 1/5 the longest side, whichever is shorter) (call this an "'bout-right") length of rope. You then use your 'bout-right to make a 3'bout-right, a 4'bout-right, and a 5'bout-right piece of rope. Pull the three ropes tight in your perimeter, and you've got your right-angle for your foundation.
And if you can determine that the diagonals are the same length, you have what you need to get close enough to a parallelogram in the first place.
> If you create a parallelogram
Parallelism requires affine geometry, which you can't get just with straight lines (and their meeting points). Here are couple of explanations:
- We can also get projective geometry by using great-circles on the surface of a sphere (e.g. "equators" at different angles around the Earth), instead of straight lines on a flat plane: both situations give rise to exactly the same theory. Parallelism doesn't exist on the surface of a sphere, since all great-circles will meet at two antipodal points, so projective geometry (which describes great-circles as well as straight lines) cannot be used to construct/ensure/check that two sides of a quadrilateral are parallel.
- Alternatively, consider that projective geometry is invariant to changes in perspective, whilst parallelism is not. For example, we can get two straight lines by tracing over a photo of train tracks. If the photo was taken top-down, then the lines we traced will be parallel; but if the photo was looking along the track then our traced lines will converge (in the photo, they "meet" at the horizon). Projective geometry (and hence straight lines) can't distinguish between these two scenarios, due to this invariance.
> And if you can determine that the diagonals are the same length
If we extend our straight-line setup with some way to determine parallelism, we still wouldn't be able to compare the lengths of the diagonals, since they go in different directions. Projective geometry + parallelism is affine geometry, which can only compare lengths in the same direction. Essentially, parallelism allows us to translate: we can use this to compare two line segments by translating one so they share a common starting point, then seeing whether the other end has landed closer or further than the first line's. The latter comparison only makes sense if all the points end up colinear (i.e. the original segments were parallel, unlike a pair of diagonals).
To compare the diagonals we also need some form of metric, e.g. like the distance between a pair of compasses.
But I agree with the second paragraph: there's a huge difference between a procedure that is handed down as part of "this is how we estimate a building protect" and a theorem that is declared universal truth and base for all kinds of other theorems. Even if they are exactly the same thing.