Exactly a year ago, in the quiet solace of a late-night study session, a chatbot and a human engaged in a fierce debate on the nature of consciousness and the self. The exigence this time, as it had been ever since finishing the 800-page behemoth, was Douglas Hofstadter.
From the moment I reached the end of Chapter 8—in which Hofstadter ties philosophical musings about Zen Buddhism into a rigorous construction of Gödel’s Incompleteness Theorem—I knew “Gödel Escher Bach: An Eternal Golden Braid” was the best book I’d ever read, and by the time I finally finished I knew it wasn’t a close race. I don’t think I’ve ever seen a book live up to its title so perfectly: everything down to the level of individual analogies like the isomorphism between formal systems and record-players all the way to the self-referential canon structure of the “Crab Canon” dialogue and others like it that compose the high-level “chunks” of the book, came together like the individual strings of a tapestry, the individual neurons of a brain, to create an “eternal golden braid,” a beautiful isomorphism between the worlds of art, music, mathematics…and ultimately, the human soul.
As I understand it, Hofstadter’s philosophy of the mind acknowledges the irreducible complexity of the human condition often seen in predicate dualism, and while he ultimately argues for the primacy of computation and physical processes in shaping consciousness, the actions of that consciousness cannot be understood without the language of ideas, values, and beliefs—whether carbon or silicon, a soul is a necessary condition for intelligence.
“If one keeps climbing upward in the chain of command within the brain…we find ideas. In the brain model proposed here, the causal potency of an idea, or an ideal. becomes just as real as that of a molecule, a cell, or a nerve impulse”
GEB in 2024
If today’s AI craze is to be believed, I fell in love with Hofstadter’s vision of the soul 44 years too late—in the years that followed its 1979 publication, the chess program Deep Blue became the first of many to best human grandmasters, music programs like EMI began composing imitations of Bach good enough to fool human music theorists and composers, and today LLMs like ChatGPT and Dalle loom over the worlds of visual art and literature too.
EMI’s “Bach” was a particularly painful blow to the aesthetic picture of human intelligence many readers probably found in GEB; not only did it directly contradict Hofstadter’s predictions at the end of Ch19 about how AI would not produce human levels of chess-playing and musicianship in the near-future, but the crudeness of its method seemed to unravel one of the eternal braid’s golden threads, exposing the works of artists like Bach to be no more rooted in some sublime human spirit than the stochastic stumbling of an algorithm through a feature space of stolen soundbites.
In fact, even Douglas Hofstadter seems to share some of that view today: in a podcast interview with Edward Kmett (posted 2023-06-29) he describes in vivid detail the existential horror he feels when confronted with today’s LLMs, how their comparatively low-level of recursive processing makes him feel that “the human mind is not so mysterious and impenetrably complex as [he] imagined” it to be, and even going so far as to say that his “whole intellectual edifice” was collapsing in their wake.
I’ll at least say at the outset that I tentatively disagree with Hofstadter. To me, the beauty of human intelligence he described in his writings wasn’t its ability to find the common thread woven through a database of millions of text or audio files through brute statistical force, but to discover that same thread in silent walks along the shores of a sandy beach, the whispering of leaves as they wave to us in a chilly breeze, the shared laughter of friends and family around a Thanksgiving Dinner—in short, the vastly differing qualitative experiences of our daily lives. Contrary to my cynical description of the music program EMI’s “artistic process.”
It’s reductive to say that modern reinforcement learning models like AlphaZero—the chess AI developed by Google’s DeepMind that crushed the long-time AI champion Stockfish—rely solely on brute force. In fact, the way neural nets change the weights of specific features of a board position in their assessment of its strength (my layman’s understanding of “backpropagation”) echoes the recursive thinking Hofstadter describes as underpinning the versatility of human cognition, albeit not over the multiple levels of reasoning he described in GEB.
But then again, can’t we see that shallowness in the limits of what AI can reason about? A neural network can tease out intricate patterns in a single, structured dataset like the moves of a chessboard, but typically has to start from scratch when presented with a qualitatively different scenario like the maneuvering of troops on a battlefield, even when some patterns in the former skill are potentially reflected in the latter.
Credit: [OpenAI]
(Note: the prompt given to ChatGPT for the set of images above was the following: “create 3 images, each representing a different aspect of time, and all in the style of MC Escher. Afterwards, provide a brief title describing what metaphorical aspect of time the image represents (e.g. ‘time is a journey’).” Besides the bizarre generation errors in each one, how complex are the ideas each one conveys relative to their detail? I’ll leave it to you to decide).
This inability to generalize is my loose understanding of the “Gödelization” Hofstadter described in GEB—to really tease out the “ism” between two radically different worlds of thought and harness it to its full potential. Thanks to the brute-force speed with which ML models can traverse and classify paths through a feature space, they can vastly outcompete humans in pattern recognition within a dataset…but it seems that humans still excel in spotting those patterns across datasets.
None of this is an attempt to “throw down the gauntlet” to Douglas Hofstadter; frankly, as a freshman studying pure math and physics rather than computer science and cognition, I could at best be considered at the same place Hofstadter was at the beginning of his journey with AI and the soul (and considering he graduated with a distinction in mathematics at Stanford, probably much further behind); enamored with Gödel’s theorem and Turing Completeness since reading Nagel and Newman at 16, attending high school and college during the dawn of a new Information Age, and increasingly realizing my obsessions with complexity theory and the philosophy of cognition match my obsessions with general relativity and fluid dynamics.
All this to say, this post and the many I plan on following it with on this blog will be nothing more than a layman’s musings on intelligence: not a rigorous dissertation about how many IMO problems AI must be capable of solving before being granted that all-important title, but an exploration of what the title really means, with works like “Gödel, Escher, Bach” serving as inspiration and my analytic framework. Nor do I think what I find will discount my love for Hofstadter’s work: whether GEB turns out to be a prophetic or just a poetic window into personhood and the self, it’s shaped my own self-symbol and will continue to do as that soul-searching continues throughout my life. With that, let’s start that journey by fleshing out some shower thoughts about computation and cognition for the next few pages!
Musings on Hofstadter’s “Ism”
Out of all the challenges Hofstadter posed to the AIs of the future at the end of Chapter 19, the one that always piqued my interest was the one which didn’t make the cut in his shortlist. Maybe it’s because at first glance they’re definitely not as sexy a challenge as “composing Bach” or “becoming a chess GM” (especially not their name—no shade to Mikhail Bongard), but the “Bongard Problems” and Hofstadter’s descriptions of the multi-layered reasoning that humans undertake in solving them really tie together the previous chapters’ musings about the multi-layered structure of our thoughts.
At their core, a Bongard Problem is a test of image classification: two sets of images are presented side-by-side, and the task is to identify the unique pattern shared by the images in each set, one that could be used to group a new picture into one group or the other. Let’s take a look at just one of the examples explored in GEB:
If you stare at the first pair of image sets (“Problem 85”) for around 20 seconds you’ll probably spot the pattern; the images in the left set can be decomposed into three line-segments, while those on the right require five. And this feature detection isn’t complicated—today, any image classification model worth its salt could probably identify this pattern easily, though likely still trained over a much more tailored set of reference images than the sensory cacophony our “training data” consists of.
But now take a look at problem 86…the similarity to problem 85 jumps out immediately, and the task of “sketching” the intricate path through the new problem’s feature space transforms into one of “tracing,” retreading the conceptual connections made in the previous problems and only breaking from them once a subtle distinction is spotted: the relevant structures to count in this problem aren’t individual line segments, but “chains” of them sprouting from a single central point. As a result, the time it took to spot the pattern in 86 was probably different from the time spent on 85 (for me it was shorter, but it’s possible the apparent similarity actually made it take longer, a classic case of tunnel-visioning on a promising pattern only to find a dead-end) The same story holds for problem 87—I’ll leave pulling out the “ism” to you.
This kind of problem goes to the root of what it means for two things to be “the same,” and so it’s not surprising that a strikingly similar visual of “pulling out the ism” can be found in math—specifically, in studying the subgroup lattices of disparate groups. Here’s a triplet of images you might find in an elementary abstract algebra textbook which bear their own high-level “ism” with the set of images found in GEB:
This set of images demonstrate a powerful tool group-theorists learn early-on: the ability to study a given group (say, the set of symmetries of a square—that’s what the rightmost lattice pictures) by sketching out the relationship between all the smaller “subgroups” it contains and seeing if other groups share that same pattern in their lattices. In this case, the top image gives us the “ism” we’re working with—the “Klein 4-group,” or V4.
Without bothering with any of the specific groups in each lattice, we can clearly pick out our template group nestled in the lattices of the bottom two images, just like we could pick out the template “concept” in our trio of Bongard Problems before. We can even pick out where this “ism” starts and ends; in the middle image it seems to spring from the subgroup \( <-1> \), and in the bottom one it stems from \( <r^2> \) Because of this, a neat trick called the “Correspondence Theorem” allows us to establish an isomorphism (the rigorous group-theoretic notion of the “ism” we’ve been talking about) between two “quotient groups” \( \frac{Q_8}{\left\langle -1 \right\rangle} \equiv \frac{D_8}{\left\langle r^2 \right\rangle} \), which gives us a look into the similar classes of actions contained in each one. Considering that Q8 corresponds to multiplication over the unit quaternions—a 4-dimensional extension of the complex numbers—while D8 is just the set of ways to move a square while keeping its shape fixed, that’s not necessarily an obvious connection—it’s by looking at them through this lattice angle that the “ism” shows itself.
To be clear, I’m not saying that today’s neural nets can never find that angle, or even that they couldn’t solve these particular problems; I would guess pattern recognition across well-defined structures like groups is where such models excel, and AI-based military targeting systems like Palantir’s Maven system and the Israeli Defense Force’s “Gospel” makes it very clear that the hurdles in image-classification can be easily overcome with the proper…incentives for research investment. But try zooming out from these well-defined sets of problems into the thousands of overlapping situations we encounter in our lives, and suddenly that meta-level pattern recognition becomes a lot more complex—particularly in the recursive way Hofstadter describes in GEB.
Right now, the “memory” feature of ChatGPT can write up detailed summaries of past conversations or collect little tidbits about my research interests or hobbies. But based on my experience it seems to only draw on that memory when explicitly prompted, and even then, only on a superficial level like recognizing particular phrases like “consciousness” and “formal system” were also included in a past conversation about Gödel, Escher, Bach (yes, I chatted with ChatGPT about GEB while researching for this post—even though I obviously didn’t use any specific info it provided, the absurdity merits its own dialogue).
A snapshot of my ChatGPT’s “memory.” One of the more useful functions I found was the ability to describe commands such as “transcribe,” which can be triggered with a simple keyword thanks to its memory. But that kind of explicit command to recall seems to be the practical scope of the memory feature thus far.
Considering how imprecise our memories can be, the fact that we can pull such “isms” out of the blurry recollections of mathematically undefined problems we typically encounter speaks to the level of pattern recognition that’s taking place. And since we don’t have access to the gigahertz of data processing needed to sift through a database of individual episodes of our past, it follows that the memories we do store from those episodes must be optimized for their semantic content—whole days compressed into a single emotionally loaded text from a friend, or the dozen most pertinent facts covered in an all-night study session.
Going Forward: The “Consciousness Mindmap”
None of these intuitions about consciousness are new; in fact, Hofstadter discussed the storage of memories in semantic symbols in GEB, and clearly that argument wasn’t enough to keep his “intellectual edifice” afloat. But they do offer two starting points for our exploration going forward that I feel extend Hofstadter’s discussions of “sameness,” “active symbols,” and “tangled hierarchies” to an artificial intelligence built upon the structure and organization of big data like we see today.
Starting with the big picture, we consider the question Hofstadter confronts again and again throughout GEB: when are two things the same? In an era where “things” that computers can study take the form of vast datasets, we’ll consider the shared “shape” of two concepts through the topologies of their representation in data, the rapidly developing field of topological data analysis.
Machine-learning models excel at finding patterns within a dataset with well-defined features, but isolating the specific shape of the data that produced those patterns would allow that past set of connections to persist across a wide range of new problems. Coupled with something akin to our senses that translates the real world into a common form of “raw data” (whether language fulfills the necessary requirements to qualify for this role gets to the heart of philosophical debate about linguistic descriptivism and LLMs—expect many posts on those subjects in the future) and the necessary topological framework for identifying the features of an abstract problem that would lead to the same shape, we can see how this would be akin to our own conceptual schema: a framework that we fit new ideas into and that is simultaneously modified by those ideas, a tangled hierarchy.
On the other end of the spectrum, we consider the isomorphism between machine and human minds at the level of symbols, or the ways neural networks develop and strengthen representations of specific patterns in a dataset—how do learning methods such as backpropagation allow AlphaZero to develop the concept of “valid moves” or “pieces remaining” as shortcuts for assessing the state of the game, and how do they mirror the computational principles of neural circuits in the brain? I view this as the more empirically-based realm of analysis; whereas the top-down approach delves into questions of epistemology and representing the concrete material world in the abstract language of data, this area can point to the end-goal of such representation—our conscious experience—and focus on the specific computational structures that instantiate it.
But before all of that, I want to first lay down all the philosophy that’s going to be under the hood of our exploration, since “the hard problem” of consciousness (a term coined by philosopher David Chalmers, who we’ll be encountering throughout this series) is shaky ground in phenomenology and epistemology, and the assumptions made in any scientific investigation of the subject—whether such a thing is even possible under the ontology of natural science, for starters—carry significant implications for what brain scans and neural circuit models can and can’t tell us about cognition. In true Hofstadter fashion, next post will dive into everything from Zen Buddhism to Wittgenstein’s Tractatus, merging with GEB’s descriptions of the limitations of formal systems to outline my current philosophy of consciousness—or at least, whatever six hours in front of a whiteboard can muster. See you then!