I wonder how analysis made it into mathematics. Now there is just a schism in the doctrine in a lot of places.
I think it is a "does it 'solve' problems I care about?” kind of thing. I think it is the foundation for validating inferences in math. The choice axiom went from, being considered problematic, to being mostly accepted because new problems from it did not hurt old problems.
Can be done rigorously now, wasn't done so for decades. It was tolerated, because such problematic symbolism was used anyway in physics.
I think anyway, such an area had to be pushed for a long time to make it into convention. Philosophy, not being a moneymaker for math, has had less luck.
@cirnog heard about that, I think ( https://m.youtube.com/watch?v=MiGx8xv6xjE at 2:10). they are aligning the mirror though, so what does it mean that there's no construction? just that they wouldn't necessarily find the solution in some cases, but still know there is one? @jmw150
To guarantee that the phase retrieval can be accomplished, NASA wasn't going to build a telescope with multiple mirrors if it wasn't certain that they could be alligned to form a single image, see archive.ph/20120805213402/http://www.stsci.edu/jwst/ote/wavefront-sensing-and-control for an old explanation from NASA, and opg.optica.org/ao/fulltext.cfm?uri=ao-21-15-2758&id=26002 for a more technical explanation
All of this is apparently based off the work of kilbanov who proved the original uniqueness theorems, which I assume is what that NASA employers were referring to.
They have extra information that allows them to approximate towards the solution. If you only know that there's a misalignment then you're guaranteed a way to fix it, but if you know there's a misalignment AND you have a photo of the mirrors then you can actuate the mirrors to create a smaller misalignment (and repeat this process until minimal error)