Godel Escher Bach


“Every system will contain a problem that cannot be solved from within the system itself. We have to go outside the system, and soon this new larger system will become redundant.” – Kurt Godel

Happy Birthday Kurt Godel!

Abelard made a more poetic appeal to doubt, but my philosopher hero, Kurl Godel, is famed for mathematically proving no system is fundamentally infallible.

One of my favourite all time writers (see left-hand bar) is former Scientific American columnist and University of Illinois Professor Douglas Hofstadter and his crowning work is ‘Godel, Escher, Bach’.  A playful exploration of the most convoluted and yet most elegant of concepts reflected across a range of daily life.  He explores self-referential paradoxes in music, art and most significantly mathematics/philosophy in a sort of Alice in Wonderland allegory for modern mathematical concepts.

In many ways, Hofstadter set me on my course in life as his whimsical examination of recursion led me to study LISP, which led to my first computer engineering job for an AI/expert system company which led to another across the street of ‘AI Alley’ which sent me to England which led to me joining Microsoft UK where I have spent 15 years.

A number of fine descriptions of Godel’s Theorem are listed at Miskatonic and my favourite is Jones and Wilson’s taken from their ‘An Incomplete Education’

  • “In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms … of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules. Gödel’s Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths … It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you’ll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.”

In many respects, Godel’s Theorem is the philosophical codification of failure.  A methodical proof of the failure of any ‘system’ to be both ‘complete’ (no gap failures) and ‘consistent’ (no logical failures).  Of course, it’s all pretty esoteric stuff.  But curious nonetheless that this structural failure lies at the very heart of our theoretical constructs of the universe.