From Self-Reference to Transcendence
From "Gödel, Escher, Bach" and Lewis Carroll to the Philosophy of Consciousness
It was in the year before I finished high school that I discovered a world I had not known about: AI - artificial intelligence - the field in computer science that studies how a software program can learn beyond what it has been programmed to learn, so that its performance improves by learning from itself as it runs. Yes, this last sentence is very self-referential, intentionally, if somewhat awkwardly. Since I was a teenager with a mind full of books and a room of full notebooks setting about to explain the world, the idea of AI, of autonomous learning by machine, was too irresistible in my impressionable youth. Instead of engaging with the world on its own material and monetary terms, I found I could conceptualize more easily than others (perhaps this quality was already nagging me — and others), and I lost interest in dealing with the world on primarily concrete terms. I took up jazz guitar, computer programming, and philosophy at a time when (perhaps like now), a teen kid following any one (much less) of these three strands was bound to become an introvert. But if ideas, especially philosophical ones, seemed the Promised Land for my innate way of thinking, then ideas that create ideas (which is my definition of “interesting”) were to be the true Holy Grail of insight. And so I dove hard into that rabbit hole.
At exactly this time, and by serendipitous coincidence, a friend lent me an incredibly thick book of over 800 pages which had just been published, and it delved deeply into how ideas create wholly new ideas by using aspects of their own structure (what is known in computer science and math not as self-reference but as recursion). By means of this kind of conceptual growth, a system could learn more and more from its own self-analysis, and thus grow more intelligent.
The book was “Gödel, Escher, Bach: An Eternal Golden Braid” (often abbreviated as GEB), written by Douglas Hofstadter, and published in 1979. It explored perhaps the most profound concepts in mathematics, art, and music through the works of mathematician Kurt Gödel, artist M.C. Escher, and composer Johann Sebastian Bach — these geniuses made important contributions to mathematics, visual art, and music by utilizing recursion to produce insights that had never been encountered before.
For example, in the imponderably complex world of mathematics, the book discussed Gödel’s work, whose Incompleteness Theorem states that in any sufficiently complex mathematical system, there are true statements that cannot be proven within the system itself. In other words, we could conclude that all systems are formally incomplete because no system can ever rely on mathematical proofs to produce evidence that it is complete. Gödel gets to this proof by using recursive logic, something that Hofstadter explains with gusto and humor. This is not easy, given that self-reference and paradoxes play a crucial role in both mathematics and human consciousness, but are not intuitively evident to everyday thinking.
Turning similarly to visual art, Hofstadter drew parallels between M. C. Escher's visually paradoxical artwork and Gödel's theorems, showing how both challenge our perceptions of reality. This is because almost all of Escher’s work is entirely recursive, which is to say, its visual references appear at first to portray the world as we know it, but in fact, each of his woodcuts, etchings, paintings, and drawings is rife with points and “exits” that lead back to the other regions of the work itself.
For a glimpse into M.C. Escher’s way of working see here:
and for a deeper foray into M.C. Escher’s life and work, this feature-length documentary will do nicely:
Going further still by finding self-reference in the realm of music, “Gödel, Escher, Bach” explored Johann Sebastian Bach’s frequent use of recursion. Hofstadter highlighted four examples of Bach’s work to illustrate this concept. The first form is the fugue, a musical composition where a theme or subject is introduced by one voice and then successively taken up by others, creating a layered, recursive structure. That is, the melodic theme recurs in different voices, often with variations, reflecting the idea of recursion in music. This is perhaps best illustrated when we can see the notes on a sheet of music, as in this case of Bach’s “Little” fugue in G minor, where multiple voices harmonize with each other, although that harmony and thematic exposition derive from essentially the same methodic theme. It's like hearing a melody that has cloned itself several times, yet the fact that each copy begins at a separate moment in time over the others makes the resulting composition sound different than wear we would expect from hearing the same line three times. A new creation is thus born from self reference:
Next, Hofstadter explored the musical form known as canons, which are another example where a single melody is played and then imitated after a certain delay by another voice, creating a complex, interwoven pattern. Bach's “Canon per Tonos” (Endlessly Rising Canon) from “The Musical Offering” is particularly noted for its recursive, self-referential nature as it modulates and returns to its starting point:
No canon more beautifully captures self-reference than Bach’s own Crab Canon. This composition can be played forward and backward, much like a palindrome, which we can see cleverly depicted here, first in one, then in two voices:
One might think that this is already more than enough for a single book to weave together — self-reference being obvious because, in the book, three geniuses in different fields drink from the essence of recursive thought to produce new ideas, which in turn refer to each other and produce new ideas, which in turn refer to each other and produce new ideas, which in turn refer to each other and produce new ideas. But no - there is more.
What had also always captured me about this book was how, in each chapter of “Gödel, Escher, Bach,” Hofstadter had channeled a bit of the spirit and words of Lewis Carroll, through the kind of playful dialogue, logical paradoxes, and recursive structures that also characterized many rich and complex ideas in Carroll’s own literary work. For example, Hofstadter incorporated whimsical dialogues featuring characters inspired by Carroll’s stories, particularly “Alice's Adventures in Wonderland” and “Through the Looking-Glass.” Notably in evidence throughout “Gödel, Escher, Bach” were Carroll-esque conversations between Achilles and the Tortoise (remember them?). Incidentally, for a documentary on the incredible life and work of Lewis Carroll, you may enjoy this:
Achilles and the Tortoise were borrowed from Carroll’s “What the Tortoise Said to Achilles,” an essay that explored logical paradoxes, and Hofstadter used these two affectionately cordial characters in a series of dialogues throughout “Gödel, Escher, Bach” as a way to gently introduce complex concepts like recursion, self-reference, and formal systems. Consequently, Hofstadter’s own dialogues between Achilles and the Tortoise are written using some of the playful, paradoxical elements reminiscent of Carroll's style, including puns, wordplay, and logical puzzles that ignited Carroll’s fascination with the quirks and intricacies of language and logic. Perhaps because Lewis Carroll and Douglas Hofstadter are Aquarians (the mixture’s rich enough here, so we might as well also bring astrology into it), as am I, and because Aquarius is the most conceptual sign in the horoscope, Hofstadter found a harmonious mind in Carroll, as I did in Hofstadter (in fact, my Ph.D. dissertation was inspired by Hofstadter, whom I coincidentally encountered at a book signing on the very day that I completed my thesis — perhaps a memory for another day). In any case, since Hofstadter’s dialogues and the book’s format explore how recursion and self-reference appear in different domains, I thought Achilles and the Tortoise would do well in another kind of deep conversation that I could imagine these characters having.
Summarizing his own intention, Douglas Hofstadter later said that his aim in writing “Gödel, Escher, Bach: An Eternal Golden Braid” was to explore and illustrate how self-reference and recursion are fundamental to human cognition and understanding. And to that end, he sought to show how these concepts manifest in the works, as we mentioned, of logician Kurt Gödel, artist M.C. Escher, and composer Johann Sebastian Bach, as a way to offer insights that demonstrate how complex, beautiful structures can arise from simple, recursive processes.
Hofstadter’s are thus primarily cognitive interests; they explore the degree to which mind can detect structures in the world and use them to enhance creative intuition and problem-solving. Since my own life aim has been devoted not so much to cognitive but rather to spiritual and metaphysical problems, my own variation on all of these ideas today is to spawn a similar yet distinct form of these efforts. Specifically, I wanted to write a dialogue, calling upon Achilles and the Tortoise to engage in a friendly discussion that argues for and against the logic of attempting to accomplish anything in the material world when the real rewards, according to so many cultures and religions tell us that transcendence, overcoming the mortal, material world, are instead the highest rewards to strive for in life. Is there a simple way to explore this conundrum between the physical and the metaphysical? In attempting an answer to that question, the conversation between Achilles and the Tortoise, I imagined, could resemble something like this:
Achilles: Good morning, Tortoise. I have been pondering a rather perplexing question. Why should we strive for material accomplishments when various cultures and religions emphasize transcendence and spiritual rewards as the ultimate goal?
Tortoise: Ah, Achilles, you tread upon a well-worn philosophical path. Indeed, many spiritual traditions assert that transcending the material world is the highest aim. For instance, in Hinduism, the concept of Moksha represents liberation from the cycle of rebirth and union with the divine. Similarly, Buddhism teaches Nirvana as the ultimate release from suffering and samsara.
Achilles: Yes, and in Christianity, Jesus said, “For what will it profit a man if he gains the whole world and forfeits his soul?” (Matthew 16:26). This seems to imply that material gains are trivial compared to spiritual salvation.
Tortoise: Precisely. And yet, these teachings do not necessarily denounce all material pursuits. They often emphasize the importance of living virtuously within the material world to cultivate the qualities necessary for spiritual growth.
Achilles: But doesn't the pursuit of material success often lead to attachment, greed, and distractions from spiritual goals? It seems contradictory to engage in worldly endeavors when the real rewards lie beyond.
Tortoise: It might seem so at first glance, but consider the Bhagavad Gita's advice: “Perform your duty without attachment, for the man who does his duty without attachment attains the Supreme” (Bhagavad Gita 3:19). This suggests that the key is not the renunciation of action but the renunciation of attachment to the fruits of action.
Achilles: So, are you suggesting that material accomplishments can coexist with spiritual aspirations if pursued without attachment?
Tortoise: Indeed, Achilles. The practice of Karma Yoga, or the yoga of action, teaches us to engage in our duties and work in the world while maintaining a focus on the divine. This approach harmonizes the material and spiritual aspects of life.
Achilles: This reminds me of the Zen saying, “Before enlightenment, chop wood, carry water. After enlightenment, chop wood, carry water.” It implies that the material world and our actions within it remain, but our perception and engagement with them transform.
Tortoise: Exactly. The material world can be a training ground for spiritual discipline. The Stoic philosopher Epictetus also echoes this sentiment: “It is not what happens to you, but how you react to it that matters.” Engaging with the world teaches us resilience, patience, and wisdom.
Achilles: But what about those who argue that material accomplishments are illusory and fleeting, like the Buddhist teaching that all conditioned things are impermanent?
Tortoise: They are not wrong. Material accomplishments are indeed transient. However, the process of achieving them can foster personal growth and insight. The Tao Te Ching advises, “He who knows contentment is rich” (Tao Te Ching 33). It’s about finding balance and understanding that while material gains are temporary, the virtues and lessons we develop are enduring.
Achilles: I see. So, the material world is not to be shunned but rather engaged with mindfully and virtuously. This way, it can support rather than hinder our spiritual journey.
Tortoise: Precisely, Achilles. The material and spiritual realms are not mutually exclusive but intertwined. By integrating both, we cultivate a life of purpose and meaning that aligns with the ultimate goal of transcendence.
Achilles: Thank you, Tortoise. This dialogue has illuminated the path ahead. I shall strive to balance my worldly duties with my spiritual aspirations, understanding that each can enrich the other.
Tortoise: You are welcome, Achilles. Remember, the journey itself is as important as the destination. Engage with the world, but do so with a heart anchored in the eternal.
Achilles: Indeed, the journey is as important as the destination. But, Tortoise, how do we deal with the constant temptations and distractions of the material world? It seems easy to get caught up in the pursuit of wealth, fame, and pleasure.
Tortoise: That is a significant challenge, Achilles. Many spiritual teachings offer guidance on this matter. In Buddhism, mindfulness is a key practice. By being fully present and aware of our thoughts, actions, and desires, we can better discern which pursuits align with our higher goals and which lead us astray.
Achilles: Mindfulness sounds beneficial. It helps us stay focused on what truly matters. But what about ambition? Is it wrong to be ambitious in the material world?
Tortoise: Ambition, like any tool, can be used for both constructive and destructive purposes. The Bhagavad Gita advises us to act without selfish desire. Ambition that stems from a desire to contribute positively to the world and to fulfill one’s potential can be virtuous. However, ambition driven by ego, greed, or a need for external validation can lead to suffering.
Achilles: So, the key is to align our ambitions with our values and to remain unattached to the outcomes. This reminds me of the Stoic idea of focusing on what we can control and accepting what we cannot.
Tortoise: Precisely. The Stoics teach us to focus on our own actions and attitudes, rather than external results. Marcus Aurelius wrote, “You have power over your mind—not outside events. Realize this, and you will find strength.” This aligns well with the concept of non-attachment in Eastern philosophies.
Achilles: It seems that wisdom from various traditions converges on similar principles. Engage in the world, but with awareness and virtue. Pursue goals, but without being consumed by them. This balanced approach seems both practical and profound.
Tortoise: Yes, Achilles. The integration of these principles helps us navigate the complexities of life. Remember the Zen concept of the Middle Way, which advocates for a balanced approach, avoiding extremes. This applies to our engagement with the material world as well.
Achilles: The Middle Way… It sounds like a path of moderation, where we neither renounce the world entirely nor become overly attached to it. But how do we find and maintain this balance in our daily lives?
Tortoise: It requires constant self-reflection and adjustment. Practices such as meditation, prayer, or contemplation can help us stay grounded and connected to our deeper values. Engaging with a community of like-minded individuals can also provide support and accountability.
Achilles: That makes sense. By regularly examining our actions and intentions, we can ensure they align with our higher goals. And being part of a supportive community can help us stay true to our path.
Tortoise: Indeed. Additionally, service to others can be a powerful way to transcend our own desires and connect with something greater. Many traditions emphasize the importance of compassion and helping those in need as a means of spiritual growth.
Achilles: Service to others… It reminds me of the Christian teaching, “Love your neighbor as yourself” (Mark 12:31). By focusing on the well-being of others, we shift our perspective from self-centered ambitions to collective well-being.
Tortoise: Exactly, Achilles. Acts of service and kindness not only benefit others but also cultivate humility, empathy, and a sense of interconnectedness. These qualities are essential for both material and spiritual fulfillment.
Achilles: Thank you, Tortoise. This conversation has provided much clarity. I shall strive to balance my material pursuits with mindfulness, non-attachment, and a commitment to serving others.
Tortoise: You are most welcome, Achilles. Remember, the path is ongoing, and each step is an opportunity for growth and learning. May you walk it with wisdom and compassion.
Achilles: I shall endeavor to do so, Tortoise. And I look forward to our future conversations, where we can continue to explore these profound truths together.
Tortoise: As do I, Achilles. Until then, let us each strive to live harmoniously within the world, while keeping our eyes on the ultimate goal of transcendence.
Thinking further, it seems inevitable that a version of this dialogue should repeat in a future, self-referencing post, where these ideas, and our chatty friends will make a thematic return.
Thanks to Douglas Hofstadter.