A Tree of Theorems (and its Shadow)
In the images I see of Douglas Hofstadter, he is so unassuming. Meek looking almost. Frail and human. It’s easy to forget that he wrote a book that sits like a Titan on my shelf next to all the smaller-souled books that surround it.
The book is Godel, Escher, Bach: an Eternal Golden Braid, and the figure above is a drawing from the book by the author. Just one page among the seven hundred. The white tree on the left is a Theorem Tree, growing into a black field of truths. On the right it’s shadow grows into a field of Falsehoods.
The early part of Godel, Escher, Bach is spent teaching the reader from scratch about Formal Systems. These are mathematical, typographical ways of thinking. Hofstadter designs some little game-systems, but most of them are meant to live in the world of numbers. They include numbers, mathematical symbols, logical connectors like “And/Or” and “If/Then”. And the theorems of the system can be built up using logical rules of inference.
Some of the outer boxes of this diagram represent the set of all strings (including the nonsensical), and the set of all well-formed sentences. This area is broken up into the truths on the left (in black) and the falsehoods on the right (in white). Into the truths grows a white tree. These are the theorems: the sentences one gets by starting with some axioms and applying the rules of inference. The shadow tree of negated theorems grows on the right.
Here is the point: The tree of theorems fails to cover the whole field of truths. Many areas are left black, meaning that many things are true, but cannot be proved by the rules of inference.
This is Godel’s incompleteness which Hofstadter leads us up to and explores ever-so-gradually.
One other point worth noting: the fuzziness of the boundary between truth and falsehood. Hofstadter did this on purpose, he writes, to suggest a fractal boundary like the Mandelbrot Set, so that no matter how close one zooms in, a fuzzy confusion exists between true and false so that they cannot be teased apart precisely.