GEB Introduction

Gödel, Escher, Bach is a very large book by Douglas Hofstadter. Some of the ideas covered in the book are:

  • self-reference
  • acquired meaning
  • logical systems
  • recursion
  • infinity
  • layers of reality
  • paradoxes
  • strange loops (an idea that ties together many of these concepts)

It relates these ideas to subjects such as music, computer science, mathematics, and art.

Introduction: A Musico-Logical Offering 1

Bach

GEB begins with a story about Bach’s composition of a six-voice fugue Ricercar a 6.

A fugue is like a more complicated canon. A canon uses a single theme played against transformations of itself:

  • Translation over time
  • Scaling of pitch, speed
  • Inversion

These are all isomorphisms - reversible (information-preserving) transformations. Music has at least these dimensions, and probably more:

  • Scale
  • Frequency
  • Pitch
  • Volume

The last canon he describes is an endlessly rising canon, where 3 voices create a cycle that sounds as though it rises without end. It’s not simply a spiral, but maybe like a toroidal spiral? In taiji, it’s like the path power takes in the Positive and Negative Circles#. He calls this a Strange Loop#: where an upwards traversal of the levels in a hierarchical system takes one back to one’s starting point.

Escher

Print Gallery

M.C. Escher visualizes strange loops in his drawings, as the viewer is led in a climb through imagined dimensions that loop without reversing. Across all his drawings, there arises the notion of the tightness of the paradoxical loop: how many discrete steps taken before the loop completes. Although, there is some ambiguity in how to count steps. Interestingly, the viewer is also a dimension in the paradox.

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