So what are the implications of Gödel's theorem? To understand these implications consider a related example: attempts to build a perpetual motion machine. Schemes to accomplish this were worked on for centuries, but they have now completely stopped. Why? The reason is that once the laws of thermodynamics were understood it was realized that the entire enterprise of trying to build a perpetual motion machine was doomed to failure. It could be demonstrated, based on first principles, that the goal can never be achieved, so it is a waste of time to continue working on it.
Next consider an intellectual enterprise that had been embarked upon by human thinkers since before the time of Euclid's geometry -- the enterprise of building, from the ground up, a formal mathematical/logical system within which all mathematical/logical propositions could be demonstrated to be either true or false. The pinnacle of this enterprise was Principia Mathematica, published in 3 volumes by Russell and Whitehead in 1910, 1912, 1913. This entire enterprise has now been shutdown. Why? Because, Gödel, in his paper published in 1931, demonstrated that this goal can never be accomplished so it is a waste of time to continue working on it. Gödel demonstrated that within any formal system, it is possible to construct perfectly valid mathematical/logical propositions that cannot be proven to be either true or false. Attempts to try to evaluate these propositions within the formal system can result in what Hofstadter (D. R. Hofstadter, Gödel, Escher, Bach, Basic Books, 1979) calls strange loops. Constructs that lead to strange loops involve "self-reference". I do not know whether or not Charlie Kaufman has embarked on a consciously explicit attempt to translate the implications of Gödel's theorem to film, but in his previous screenplays (Being John Malkovich and Adaptation), he is obviously using the concept of "self-reference" to create cinematic strange loops.
In his current film, Eternal Sunshine of the Spotless Mind, Kaufman goes a step further in exploring the implications of Gödelís theorem. There is another puzzling property of Gödels theorem that I did not mention earlier. I stated that it is always possible to construct perfectly valid mathematical/logical statements that a formal system cannot evaluate in terms of whether the statements are true or false. But what is in some ways even more surprising is that humans/mathematicians can often evaluate these same statements and come to agreement about whether they are true or false. Some have argued that this proves humans possess cognitive skills that are not based on following formal systems of rules. Instead, humans can make use of mysterious psychological faculties such as intuition. If correct, this argument demonstrates that our brains do not operate by the same kinds of rules as formal systems. Another implication is that it will never be possible to build a computerized robot that can think the way we humans think. The mathematician Roger Penrose constructed an elaborate argument (R. Penrose, The Emperorís New Mind, Oxford University Press, 1989) that the mysterious human intuitive properties are enabled by some characteristics of quantum physics operating in brain tissue. (Not everyone one agrees with the assessment that humans possess mysterious mental faculties impossible to duplicate in any formal system but the counter-arguments would take way too much space to lay out here. Some of these same issues are discussed, in a different context, in my own book (R Boothe, Perception of the Visual Environment, Springer-Verlag, 2002). This brings us to the current movie.
Sunshine of the Spotless Mind dramatizes a battle between a
formal system that is trying to wipe out memories located in brain tissue and psychological faculties
operating within this same brain tissue. The psychological faculties appear to be doomed, but wait! By employing
clever (human/mysterious ?) intuitive reasoning, these psychological faculties may be able to outwit
the formal system. To find out, you will have to watch the movie. I found watching this movie to be a fun ride
into and out of a strange loop.
I gave this movie a rating of 4 stars.