GEB is a work of art. Extraordinarily playful, with countless hidden puzzles, GEB is a fugue on the idea of self-reference, braiding together three strands: mathematics (Gödel), art (Escher), and music (Bach). But rather than being ends in themselves, the strands provide a language to explore the deeper concepts of strange loops and heterarchies, which Hofstadter believes can shed light on the truly difficult questions of consciousness and AI.

Ordinarily, I like to try and reduce a book to its core ideas and insights. But it feels profoundly wrong to try and do that for GEB, given that one of the recurring motifs is anti-reductionism, the idea that some systems simply cannot be understood on a lower level of abstraction. Gödel incompleteness is one of the prominent examples of this: there are true statements about number theory that cannot be expressed within number theory.

Doug Hofstadter has a remarkable command of many different fields, which he synthesises together beautifully in GEB. On my first read several years ago, I was most intrigued by the Typographical Number Theory, a minimal number theory (isomorphic to Peano arithmetic) that he uses to clearly explain Gödel incompleteness. But on this read, I was more drawn to his explanation of molecular genetics: the remarkable way that DNA codes for proteins that process the DNA. I think the crowning achievement of the book is the “Central Dogmap”(a pun on Crick’s Central Dogma of Molecular biology – “DNA makes RNA, RNA makes protein”), which provides an astounding mapping between molecular biology and mathematical logic and by extension, computer programming and AI.

Each section of the book starts with an Alice-in-Wonderland-esque discussion between Achilles and Tortoise (and some other minor recurring characters like Crab). These little discussions initially seem innocuous but they are actually astounding intellectual achievements, each acting as a representation of the next chapters of the book on a different level of abstraction: one of them, titled Crab Canon, is a Crab Canon – it reads almost the same in reverse. The final Six-Part-Ricercar mimics Bach’s six-part fugue in the Musical Offering with a high level of fidelity. Also, do you notice anything special in the names Achilles, Tortoise, Crab, and Gödel…

Like the Madeleine biscuits did for Proust, this re-read has evoked wandering (and mostly joyful) memories. I first read GEB shortly after finishing high school and was utterly enthralled by the playful intellectual nature of the book. Not much has changed on this re-read. The book is full of hidden acrostics, self-reference, and subtle humour that I missed the first time, leaving me with a childish grin. I realised just how many things went over my head, and I expect many things will have gone over my head this time round. But that’s a good thing, because I know that the next time I re-read it, perhaps jaded with more years of adult life, I will find that childlike joy once more.

P.S. Forgive me for the weird diction at the start of each paragraph. It is not without purpose

Key ideas

• Music:
  • A canon is a single theme played against itself
  • e.g. shifted by time, pitch, speed (possibly inverted or reversed as well).
  • Fugues are more flexible
• Godel Incompleteness:
  • Essentially, Godel found a way to express statements about number theory within number theory, using Godel numbering
  • He then constructed the sentence G: “This statement of number theory does not have a proof within Principia Mathematica”.
  • If G is true, we have a true statement that has no proof in Number Theory. So Number theory is incomplete.
  • If G is false, then number theory yields a False statement. So it is inconsistent.
• Strange loop - when ascending or descending through a hierarchy, we sometimes end up back where we started (e.g Escher’s staircase, or Bach’s Endlessly Rising Canon).
• MIU mode – machine mode, intelligence mode, un-mode.
• When you write the map of an isomorphism, you've written an interpretation. This can either be meaningless or meaningful, depending on what you map it to.
• Consistency vs completeness:
  • Consistency is not a property of a formal system. Rather, it depends on the interpretation which is proposed for it
  • Omega inconsistency is the opposition between a family of theorems asserting that all natural numbers have a property, and a single theorem which says the opposite. To reconcile this, we have to imagine that there are 'supernatural' numbers.
• Typogenetics – a simple model for genetics
  • DNA is strands of ATCG.
  • Enzymes operate on strands to create new strands. They are made of a series of simpler commands called amino acids (e.g. cut, del, move).
  • The trick: enzymes themselves are coded for by a strand!
  • Enzymes play the role of both data and program – there is a strange loop. Just like in Gödel Incompleteness where statements about number theory are represented in number theory.
• Hierarchy vs heterarchy:
  • Science is hierarchical: we can understand biology without understanding particle physics. There is little leakage between levels.
  • A heterarchy is when we have multiple procedures calling one another cyclically: there is no highest level. The mind may or may not be able to be understood without reference to the underlying structure.
• Quining: "yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.
• In the parallels between the TNT and typographical genetics, the equivalent of Godel's Incompleteness theorem is a virus: a strand of DNA which causes proteins to be manufactured that destroy the cell, resulting in the non-production of that DNA.
• Hofstadter’s Law (see here for an explanation) states that "It always takes longer than you expect, even when you take into account Hofstadter's Law."