[dissidents]

Thanks Ross for the reply. I believe that this is a misunderstanding of how propositional logic works. If the propositions or axioms that you start with are sound, and if you correctly apply all inference rules, then the propositions that you derive will also be sound. "Missing axioms" that you did not use do no matter, regardless of their soundness.

[PKop]

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...

[mdpopescu]

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.

[dissidents]

This has definitely nothing to do with any of what I said.

[tunesmith]

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).

[dissidents]

Again, I am sorry for being direct, but this does not make sense. If a Venusian man is immortal, then the "axiom" (preposition) that all men are mortal is false. In other words, the issue is not that the preposition "Socrates is Venusian" was missing but that the preposition "all men are mortal" is false.

It is possible to develop significant mathematical theory without using some axioms. For example, mathematicians sometimes choose not to use the "axiom of choice" when working with Zermelo–Fraenkel set theory. That does not mean that mathematical theorems proven without using the axiom of choice are invalid, even if you later assume that this axiom is true (or false).

[tunesmith]

The point is that the axiom that all men are mortal was thought to be true, and then was later discovered to be false. My comment was actually in agreement with your previous comment.