Incompleteness Theorem Doesn't Mean "Stop Trying"
May 3, 2015
Incompleteness Theorem Doesn't Mean "Stop Trying" It just means, "Revise Your Objective" Kurt Gödel's incompleteness theorem is a lot like John McCarthy's Frame Problem, which essentially postulates, "the world changes behind your back". A more Gödelian explanation might assert, a system cannot account for (be resilient to) Frames it hasn't seen. One such relevant demonstration was the insufficiency of euclidean postulates (the axioms of euclidean geometry) to satisfy alternate (e.g. spherical) geometries, which are relevant because of the specific application that spherical geometry better models our earth (i.e. an advancement in our understanding) Rather than appealing to a specific anecdote, I'll generalize: the reasons we cannot derive comprehensive systems (for logic, math, etc) which are absent of inconsistency and achieve verifiable "correctness" are (a) Occam's Razor, Search Space & Expense; there's only 1 way to be right, an infinite number of ways to be wrong, and a potentially combinatorially explosive search space of possibilities and inter-dependencies to verify (b) this rightness can only be measured by the currently known use cases (known/visible Frames), and (c) there will always be views or functions (as we can contrive infinite permutations of abstract cases) which are not considered by out current models. The Frame problem & Gödel's incompleteness theorems can be somewhat mitigated in one of two ways. First, by addressing or eliminating each inconsistency as it arises (derive a new model completely, monkey-patch or ignore the "edge-cases") or, by listening to the wisdom left by John McCarthy and championed by Monica Anderson and others, who I believe have a productive mindset: that the very nature of Artificial Intelligence and systems (like math) which try to model the intricacies of the real, evolving world *need not*, and in fact as proof by Gödel' and McCarthy, cannot be both verifiable and comprehensive. That we should instead, (where necessary) limit ourselves to scoped, safe mostly-deterministic environments wherein uncertainty, stochasticism, and inconsistencies cannot harm us, or alternatively, build systems which attempt to continuously improve as inconsistencies (Frame changes) are discovered with the best accuracy we can achieve, and not rely on unrealistic promises of verifiability. For further insight on this topic, I recommend watching Monica's video from 330 seconds in: https://vimeo.com/monicaanderson/dualprocesstheory#t=330s The important TL;DR? Because the world (even the "assumed", limited world we model in our math) has not been made fully visible/known/exhaustively explored or proven deterministic (other than for basic, highly restrictive / self contained universes of discourse / domains dealing with discrete and finite components under specific functions, constraints and use cases) we both shouldn't expect fully verifiable systems, *nor should we let this arbitrary reality deter progress*. Disclaimer: This is my first time really thinking about the Frame problem as a parallel to the Incompleteness Theorem and I have not read such works as Godel Escher Bach (which is why Mark is cc'd for his thoughts), only Godel's Proof (http://www.amazon.com/G%C3%B6dels-Pro…/…/ref=pd_bxgy_b_img_y). cc: Mark P Xu Neyer, Jessy Exum, Kartik Agaram, Anthony Di Franco