To clarify and be fair, the people who actually invented geometry, or at least worked on the majority, were never interested in building things. Or at least that wasn't their primary concern.
It's more accurate to say that mathematicians were just interested in intellectual exploration, everyone else was interested in building things, and there's an uneasy alliance between the two that is about to be shattered.
Use a wire to get a straight line? Used in construction
Use a determined length of wire to compare lengths ? Used in construction
Use a determined length of wire to draw circles? Used in construction
Use 2 lengths of wire to construct parallel lines? Used in construction
Use lengths of wire to construct a perfect 90 degree angle? Used in tool-making (ie. to make a jig that you then use to measure)
The list goes on and on and on. If you've never used CAD packages, you should. The amount of geometry that goes into part design in everything from construction to "tools and construction engineering" as it's called in my country is incredible. Incredible as in they reserve 8 months in university to make sure people who already would have had pretty difficult math in high school are up to date on advanced geometry (plus teaching them how to properly extend what they know into 3d)
There's been updates since Greek times, such as using lasers for drawing straight lines instead of wires, but not as many as you'd think.
This is great because as anyone will point out, doing things this way (with correct equipment) results in accuracy across large distances (up to dozens of kilometers) that you just cannot match with analytical geometry + measurements.
Geometry was obviously invented and fine-tuned for thousands of years, for construction. That's what it was used for 4000 years ago ... and that's close to the only reason to still use (non-analytical) geometry today.
> I learned mathematics because others gave their time and knowledge freely, and I wanted to do the same for the next generation. The possibility that a student would go on to do something I could not do was part of the point. But I had imagined this as a continuation of a human activity. [...] I could not imagine treating this knowledge as something to be guarded.
But soon after, AlphaZero was trained purposefully without human games in the training data. The "extractive" training was sufficient to show that a certain machine learning model worked, but unnecessary to produce a machine that performs beyond the level of humans.
For coding agents I imagine it is similar: humans introduce so many subtle bugs, and they're only subtle relative to human cognitive capacity. I'm sure subtle mistakes pass peer review in mathematics too, even if it's generally more rigorous.
What I'm getting at is that labs have demonstrated that a certain model architecture is capable of producing mathematics at a (super)human level, and their extractive methodology was sufficient, but possibly not necessary. You can fight in court whether their methodology is against the law, but it would be naive not to look beyond that, which is my main gripe with the AHM statement the author refers to [1]
Fortunately the author of this article has a more nuanced view:
> I can understand the anger, but I am not sure that withdrawing from AI is the answer
[1] https://www.ahmath.org/statements