Gödel
Godel's incompleteness theorems are fundamental to understanding the limits of scientific and mathematical knowledge and the power of self-reference.
"It was something to be expected that sooner or later my proof will be made useful for religion, since that is doubtless also justified in a certain sense." -Kurt Gödel
It proves the Taoist saying by demonstrating that we can never universally determine what is true using the language of mathematics.
The tao that can be described is not the eternal Tao.
The name that can be spoken is not the eternal Name
Gödel, Escher, Bach should be considered required reading for a full understanding of metaculture.
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Gödel and Fractals
The implications of Gödel are metaphorically and visually represented by the fractal boundary between the black and the color in the Mandelbrot set. Consider the black areas of the Mandelbrot to be the Truth--theories about the universe that can be proven, and are consistently true when tested. All of the colored areas are incorrect ideas--imprecise estimations that are often true in simple cases but break down when you consider more complex variables.
Visualizing it this way, it becomes clear that why truth is such a slippery concept. On the macro scale, obvious truths and falsehoods can be clearly identified. But as soon as you start looking at the edge cases, and considering more and more complex variables, simple theories and models break down, and we need to revise them to get a more accurate, higher resolution picture of reality.
Consider Newtonian physics versus Relativity. Newton's laws of motion predicted the behavior of matter at macro scales and low velocities, but began to give incorrect predictions at higher velocities and subatomic scales. Relativity allowed us to zoom in further on our fractal picture of reality, revealing new, complex patterns that were not visible with Newton. Quantum mechanics further enhanced the picture, allowing us to model the behavior of subatomic particles. However, the more we learn about the complexity of the universe, the more questions it raises. We realize that despite the accuracy of our models, there is an infinite amount of detail yet to be discovered. Even if it isn't literally infinite, it's big enough to be effectively so.
Gödel proved that universal objective truths are impossible. The truth is infinitely complex, and science can only continue to refine our models to create closer and closer approximations of the true laws of physics.
The same logic also applies to ethics, which is why every "moral absolute" has a thousand exceptions when you consider all the possible external factors. Simple rules always break down, and we must consider all of the specifics in each edge case to determine the best course of action. No matter how many precedents are set, the Supreme Court will always have new cases to decide.
There is also the fact that all fractals are generated with recursive, self-referential equations, which is the same mathematical tool that Gödel used. It works on multiple levels!
Those who are nit-picky about the application of theories about formal logic systems to the domain of empiricism should note that this is an allegory to show how the necessary incompleteness of our math is like the incompleteness of our models of reality. It's not to imply that Gödel's theory directly addresses questions of empirical knowledge. The idea is to draw parallels and connections in order to demonstrate that the principles of self-reference and infinite nuance are universal.
Incomplete Videos
Here's an incomplete explanation of Gödel's incompleteness theorem.
A free online course from MIT is available to help get the most out of this seminal work.
Technically this is based on another book by Hofstadter but it does a good job explaining a lot of the concepts of recursion and consciousness that are in GEB.
Many science fiction stories have used Gödel's technique to short-circuit machines that operate on logic.