“A barber (who is a man) shaves all and only those men who do not shave themselves. Does he shave himself?”

If you can’t solve the above riddle, don’t worry — nobody can.  The above riddle is an example of a self-referential paradox.  Self-reference is problematic in logic because it allows for many such paradoxes.  Perhaps the oldest such paradox is the liar paradox,”Everything I say is false,” which has been known since the time of ancient Greece.  Over two millennia later, the Austrian logician Kurt Gödel proved that such inconsistent statements may occur in any axiomatic system of logic — including mathematics — which allows for self-reference.

Why is self-reference important?  Seeing as how we live our daily lives largely unimpeded by such paradoxes, how can they have relevance to human thought, much less the brain?  While Gödel’s theorem is highly abstract, it has important implications for such questions as, “Is the brain a computer?” and “Can machines think?”  It is also central to two classics of cognitive science: The Emperor’s New Mind by Roger Penrose and Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter.

The Emperor's New MindIn The Emperor’s New Mind, mathematician and theoretical physicist Roger Penrose argues that the human mind transcends the limits of formal logic imposed by Godel’s theorem.  Penrose emphasizes insight and understanding as unique properties of the human mind that cannot be realized by computers.  As a mathematician, his arguments largely consider the ways that mathematical insight, which he and other mathematicians have experienced, could not have occurred were the human brain a strictly computational machine.  What, then, is the alternative?  Penrose points to the strange world of quantum physics as a source of the brain’s allegedly non-computational workings.  Quantum particles, such as photons and electrons, behave in a random manner that cannot be predetermined.  Furthermore, such particles are often said to exist in many seemingly exclusive states simultaneously (a phenomenon called quantum superposition) until an event (called wave function collapse) gives the particles a definite, material existence.  Imagine a billiard ball that exists in many places simultaneously until it or another ball is pocketed…  Such is the strange world of quantum physics.

The Emperor’s New Mind is a long and winding journey through formal logic, computing, quantum physics, and relativity, only reaching a firm climax in the territory of the brain and cognitive science in the final two chapters of the book.  The book’s lack of focus is often described as both its greatest strength and weakness: like a particle in quantum superposition, The Emperor’s New Mind is simultaneously a tour de force of physics and a 500-page tangent.  Once the book finally returns to the questions it set out to answer — how does consciousness arise and is human thought computable? — many of Penrose’s arguments feel vague and unsatisfying.  How the workings of the brain are rooted in quantum physics remains largely unanswered, though Penrose later collaborated with anesthesiologist Stuart Hammeroff to write a sequel of sorts, Shadows of the Mind, in which it is proposed that tubulin dimers in the neuron’s cytoskeleton are really qubits (quantum bits or switches that can take two states simultaneously) in the quantum computer of the brain.

Gödel, Escher, BachWhile Penrose’s book is often vague and unfocused, Hofstadter’s Gödel, Escher, Bach is a similar grand tour which remains largely coherent and focused.  Like Penrose, Hofstadter uses Gödel’s theorem to understand the nature of human consciousness and the limits of artificial intelligence.  Central to the book are the concepts of recursion and strange loops.  Recursion is a phenomenon where something (e.g. a statement, law, function, program, painting) contains self-reference or a copy of itself.  Similarly, a strange loop is a paradox in which one reaches the top of a hierarchy only to return to its beginning!  Examples of strange loops in art include M.C. Escher’s depiction of two hands drawing each other and J.S. Bach’s endlessly rising canon, which returns to the same key it started in after rising many times in tone.  Hofstadter implies that the human mind is also a strange loop: by containing a representation of itself, the brain is capable of self-awareness and abstract thought.  While reading this book, the act of thinking about thinking seems to prove Hofstadter’s point!

DrawingHandsWhile Penrose’s The Emperor’s New Mind is largely a review of physics and mathematics crowned by a final consideration of consciousness and the physics of the mind, Hofstadter’s Gödel, Escher, Bach is a journey through logic, art, and music with the nature of human thought as a unifying theme.  Like Penrose’s book, Hofstadter’s book is dense in parts, yet made more accessible by playful yet profound dialogues between talking animals, inspired by Lewis Carroll, at the start of each chapter. While exploring seemingly unrelated topics, Hofstadter emphasizes the ability to perceive sameness between almost disparate concepts as an essential quality of human intelligence.  For Hofstadter, metaphors and analogies are the soul of human cognition.  One such analogy is drawn between ant colonies and brains: ants in the colony can be likened to neurons in a brain (the colony), with pruning of individual ants as an essential and healthy process similar to the pruning of extraneous neurons in the developing brain.

Ultimately, Gödel, Escher, Bach is a book that fundamentally changed the way I look not just at brains and computers, but, well … everything!  The implications of recursion and Godel’s theorem are so broad that you will come away from this book never seeing the world the same way again.  For instance, many intellectual problems such as, “Do people have free will?” and “Can God be all powerful?” must be framed in the context of self-reference to be fully understood.

Which book should you read?  If you have a broad interest in physics and don’t mind a lengthy detour through fractals, time dilation, and quantum entanglement, The Emperor’s New Mind is a great place to start.  My personal recommendation, however, is Gödel, Escher, Bach, a book that will change the way you view everything from art to social interactions to the human brain, all the while teaching you to think more abstractly.  Whichever book you choose, you will come away from either with huge ideas … and even huger questions.



 Images adapted from Wikipedia and made by Jooyeun Lee.

One Comment

  1. “If you can’t solve the above riddle, don’t worry — nobody can. The above riddle is an example of a self-referential paradox. Self-reference is problematic in logic because it allows for many such paradoxes. Perhaps the oldest such paradox is the liar paradox,”Everything I say is false,” which has been known since the time of ancient Greece.”

    FYI: Barwise and Etchemendy’s 1987 book ‘The Liar’ might be good read for you if you can concentrate on reading over the sound of how awesome this blog is… Seriously people! I. Need. To. SLEEP!

    Keith Devlin also does a fair tl;dr on their work in ‘Goodbye Decarte’ in Chapter ten which resolves* the paradox through applying ‘Situation Theory’ and ‘Non-well-founded set theory’ and has the advantage (I think) over Penrose in that it eschews ‘logic transcendence’.

    In any case, you guys just made it one above xkcd and one below mindhacks.com (whence I came upon you) in my favourites list.

    *for a given value of ‘resolve’ of course…

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