> There is no notion of "correctness" for the pyramids of Egypt, the dykes of the Netherlands, Milan Cathedral, or the world economy and yet those huge-scale systems all function.
I think it's impossible to come up with a rational judgment of whether the world economy "functions." I mean, those of us who are alive are alive, but there are a lot of things that have happened that haven't been ideal.
Besides the world economy, nothing you mention has the functional complexity of a large piece of software, and none of it was designed and built with much confidence, other than confidence that came from similarity to previous designs, and from an optimistic outlook that issues encountered during construction could be coped with. This wall is cracking, so we tear it down and rebuild it a little thicker. We have a few centuries, after all. I think you're right that modern software projects are analogous to ancient pyramids and medieval cathedrals. That's pretty much where we are. Yet somehow, working at that low level of sophistication, we've built working software that is orders of magnitude more complex than cathedrals.
I'm not saying I know how to improve on the situation, or even that we can; I'm just pointing out how different it is from, say, contemporary structural engineering where we can analyze the design of a novel structure and have a pretty good idea of whether it will be safe based on its geometry and known characteristics of materials and how they're joined. It's amazing that we've reached the scale we have with nothing comparable.
> If you believe mathematical correctness is such a valuable principle, then you should be able to leverage that same principle in one's own arguments.
The idea that mathematical reasoning is on some spectrum with dispassionate discourse has motivated hundreds of years of attempts to improve natural language by making it more mathematical. Result: some interesting contributions to philosophy but very little change in how people communicate. People are still people.