This is the entire crux of the article's critique of first-principles (axiomatic) thinking being a full-proof way to guide decision-making.

The inability to logically prove that first principles comport with, or not, (some unknown) aspect of reality due to missing information.

"The map is not the territory", "unknown unknowns" come to mind...

Of course they matter. We're discussing arguments that apply in the real world, not in math theory.

In math, if you have this axiom:

  f(x) > 5 for all x >= 20

you are not then allowed to change it with a later axiom

  except when x is divisible by 240

However, in real life this happens a lot. I have a company that is taxed a fixed amount per year... except for the years where I make over 100k euros, in which case things become quite complicated. If I omit the second part (which is not impossible, given that I never made over 100k euros a year with that company), and suddenly get a big payout from someone, the result will be very different from my initial estimation - as sound as it was WITHOUT that additional axiom / assumption / rule.

A "missing" axiom, in my experience, is not truly a missing axiom that otherwise has no impact on other axioms. A "missing" axiom is one that exposes a bad assumption in another axiom currently being relied upon.

For instance. Socrates is a man, all men are mortal, therefore Socrates is mortal.

But then you discover that a couple of eons have passed and Socrates is still alive. Clearly there must be a "missing" axiom. And after some investigation you realize that Socrates is a Venusian man, and Venusians are immortal.

"Socrates is Venusian" is a missing axiom, but really the problem is that "All men are mortal" is actually false, since it had implicit assumptions that "All men are human" (false) and "All humans are mortal" (true).