🔗 Why Gödel, Escher, Bach is the most influential book in my life.
Gödel proved in 1931 that mathematics is not decidable, an earth-shattering result. He proved that there are statements in mathematics, which are true but not provable within the system. Worse yet, it turns out that you can’t build a more powerful mathematical system. Once a system becomes sufficiently complex, there will always be statements which are undecidable. You’re left with a choice: either have weak system of mathematics or accept that there will always be theorems out of reach. A rough analogy to incompleteness Heisenberg’s Uncertainty Principle, which shows that physics makes it impossible to determine both the position and velocity of a particle with exact precision.
A third major theme of the book is isomorphism, which is unique to Hofstadter’s vernacular. In formal mathematics, “''isomorphism''” takes on a version of “equivalence.” For example, it turns out that many different formalizations of mathematics are provably isomorphic, like Turing Machines, arithmetic, set theory, and formal logic. Hofstadter deliberately uses the term more loosely to describe two systems that are structurally similar. I find this quite useful because it forces one to define the structures of the system, why they are similar, and why other parts of the system are less important. We might describe the way that planets fly around stars as isomorphic to the way that electrons fly around nuclei.