A Meeting of Minds on Mars: Charles Babbage • John von Neumann • Geoffrey Hinton

This discussion is about the implementation of computing in matter. That was the topic of John von Neumann’s last book, The Computer and the Brain. He died before he finished it, so it was published posthumously in 1958. I don’t know when I first learned about it, perhaps sometime in the mid-1970s. Though I knew about von Neumann and his role in early computing, it dismissed the book itself, figuring that we’d learned so much about the brain since then, and the nature of computers had changed so much, that it must be obsolete.

I was wrong. When I finally read the book, probably in the early to mid-1980s I was stunned. This was a profound book and free of mathematics beyond some simple back-of-the-envelope calculations. For one thing von Neumann talked of both analog and digital computation; that contrast was central. When I first started reading about computers in the mid-to-late 1960s that contrast was at the beginning of every article or book. But once personal computers appeared and proliferated in the late 1970s and 1980s, analog computing was all but forgotten.

It was good to see it back. For one thing, I had been strongly influenced by the work of William Powers, whose 1973 book, Behavior: The Control of Perception, offered an elegant analog model of the brain. That was central to the work of my teacher and mentor, David Hays, who had been one of the founders of computational linguistics. In 1974 and 1975 he worked out a scheme in which cognitive networks were grounded in Powers’s analog model. This was years before anyone else was working on the problem, before Steven Harnad coined the term, “symbol grounding” in his 1990 paper on the problem in Physica D.

Now, with the success of artificial neural networks, there’s another aspect of physical implementation we’ve got to deal with, distributed representation. You’re looking at an apple. Where is that apple represented in your brain? There might be some one neuron or a tightly clustered group of neurons that represent that apple. That’s a local representation. But the representation might be distributed across a loosely linked population of neurons. That’s a distributed representation. We now know that that’s how the brain works, though I assume there’s some laggards still stuck in the previous century where local representation was widely favored. And that’s how artificial neural nets work. The concept of apple isn’t localized in one or a small group of weights, it’s smeared over a population of weights. The same for the concepts of truth, beauty, and justice, or, for that matter, neuron.

Consequently von Neumann’s 1958 discussion must now be extended to distributed representation. It is for that purpose that I asked Claude to convene a conversation between Charles Babbage, who conceived of an elaborate mechanical computer in the 19th century, von Neumann himself, and Geoffrey Hinton, one of the so-called godfathers of artificial intelligence – a bit of journalistic and pop culture slight-of-hand that obscures a long and complex history. Moreover, since these three gentleman did not co-exist in time, though there’s a bit of overlap between von Neumann and Hinton, I decided that I might also have Claude transport them to Mars for the occasion. Anything is possible in virtual reality. 

Or is it? 

Image by ChatGPT. 

A sun-facing room on Mars. Three chairs arranged loosely, angled toward a broad window. Through it, the Martian plain stretches red and absolutely still, and the sun — recognizable but wrong, too small, too distant — hangs in a pale pink sky. Three men stir and look about them, bewildered but unhurt.

I. Awakening

Charles Babbage

I am not dead — I can confirm that much. My pulse is steady and my faculties appear intact. But this room is not any room I have entered by ordinary means. That light through the window — is that the sun? The colour is right but the disc is too small by half. And that plain outside... brick red, entirely without vegetation or track or any sign of habitation. Gentlemen, I should very much like an explanation.

John von Neumann

I was in Princeton not five minutes ago. I notice the gravity is reduced — perhaps sixty percent of normal. The sky has the ferric oxide colouration one would predict. The solar disc subtends roughly two-thirds the angle it does from Earth. I believe I know where we are, and I find the hypothesis almost too extraordinary to voice.

Geoffrey Hinton

Mars. We're on Mars. I was at a conference in Toronto. But that sky, the dust, the sun at that distance — I'd stake my reputation on it. He exhales slowly. I've recently left a position at Google partly so I could speak freely about alarming possibilities. I did not anticipate that the alarming possibilities would include this. May I ask what year it is, from each of your perspectives?

Charles Babbage

The year of Our Lord 1862. I am Charles Babbage, formerly Lucasian Professor of Mathematics at Cambridge, Fellow of the Royal Society. I have spent the better part of forty years attempting to build mechanical calculating engines, with results that have been — a pause — mixed, as regards the support of Her Majesty's government.

John von Neumann

John von Neumann. The year is 1955. I work at the Institute for Advanced Study in Princeton, and consult for various government and military bodies. Mr. Babbage — I know your name extremely well. Better, I suspect, than you know mine.

Geoffrey Hinton

Geoffrey Hinton. For me it is 2024. And I know both of you — Mr. Babbage, you designed the Difference Engine and the Analytical Engine. Dr. von Neumann, you are among the most formidable mathematicians of the twentieth century and you gave your name to the architecture that every conventional computer on Earth is built upon. You are, in a real sense, my ancestors. The field I work in — machine learning, artificial intelligence — descends directly from the problems both of you were grappling with.

Charles Babbage

A long pause, during which he stares at Hinton with an expression mixing hunger, pride, and something close to grief. The Analytical Engine. Did anyone build it?

Geoffrey Hinton

Not in your lifetime. The government never provided the funds. But the ideas were entirely right — and they were eventually built, first in relay and vacuum tube and then in silicon, by people who in some cases had read your work and in other cases had arrived at the same conclusions independently. You were approximately a century early.

Charles Babbage

Very quietly. A century. I had hoped twenty years would suffice. I petitioned the Chancellor. Three times.

II. The mill and the store

John von Neumann

Mr. Babbage, allow me to tell you what your Analytical Engine set in motion — because it bears directly on the work that has occupied all three of us. You made a distinction, in your design, between what you called the Mill and the Store. The Mill performs the operations — addition, subtraction, multiplication. The Store holds the numbers awaiting operation and the results of operations completed. That separation of active calculation from passive memory was the foundational insight.

Charles Babbage

It seemed to me the only sensible arrangement. The columns of number-wheels in the Store are passive — they merely hold values. The Mill acts upon them. To mix the two functions in the same mechanism would create hopeless confusion.

John von Neumann

And yet it is precisely that separation which I have spent the last years of my life questioning — not as an engineering choice, which was entirely sound, but as a principle of intelligence itself. My colleagues and I formalized your Mill-and-Store distinction into what is now called the stored-program architecture. The processor executes instructions sequentially; the memory holds both data and those instructions passively until called upon. It is a serial machine, one operation following another, orchestrated by a central clock. The computers being built in my era all follow this pattern.

Geoffrey Hinton

And in my era they still do, at bottom. But you've just described the tension at the heart of everything, Dr. von Neumann. Serial, precise, with a strict wall between computation and memory — that is the von Neumann architecture. And it is, in a sense, the architecture that human intelligence refuses to use.

Charles Babbage

You are saying the brain does not separate Mill from Store?

Geoffrey Hinton

Exactly. In the brain, every neuron is simultaneously a memory element and a processing element. It holds information in the strength of its connections to other neurons, and it also fires — it computes — based on what it receives. There is no central Mill. There is no passive Store. Computation and memory are fused at every node in the network, and the whole thing operates in parallel, millions of neurons active at once.

John von Neumann

This is precisely what troubles me — and I am glad to find the trouble is still alive in your era, Mr. Hinton, because it means I was not merely chasing a phantom. I have been writing a manuscript, unfinished I'm afraid, called "The Computer and the Brain." My central puzzle is this: where, in a neuron, is the Mill? A neuron receives signals, sums them, and if the sum exceeds a threshold, it fires. That threshold operation is a computation. But the synaptic weights — the strengths of the incoming connections — those are also the memory. The neuron is its own Mill and its own Store simultaneously. I have been calling such things active elements, to distinguish them from the passive memory elements of our conventional machines. But I confess I have not yet worked out the full implications.

Geoffrey Hinton

With genuine feeling. Dr. von Neumann, the full implications are what I have spent my career working out. And you had named the essential thing: active elements. That is exactly what the nodes of a neural network are. Each one holds a weight — that is its memory — and each one applies a non-linear function to its inputs — that is its computation. Fused. Inseparable. Replicated millions or billions of times, connected in layers, and trained by adjusting all those weights simultaneously until the network's outputs match the desired answers.

New Book Project: Language, Memory, and Mind: A Supplement to The Computer and the Brain

As you may know, I’ve been working on a book project, Play: How to Stay Human in the A.I. Revolution. For some reason I’ve been unable to finish the proposal, though I’ve got lots of stuff and a number of the chapters are substantially drafted. But I keep finding myself distracted into thinking about basics, very basic things about computing and A.I.

At the very end of his life, John von Neumann wrote a slim book, The Computer and the Brain (1958). It grapples with the problem of how computation can be implemented in a physical medium and does so in a way that is basic, both simple and straightforward and profound. We’ve learned a great deal about both the brain and the computer since then, but as far as I know, no one has revisited von Neumann’s project and extended it to include what we have since learned. That’s what I propose to do in this book.

Now, I have no intention of trying to summarize what we’ve learned on those two topics since 1958. That’s working at the wrong level. When von Neumann was writing he, and by extension, we, had no conception of distributed representation much less how it could be achieved physically. Now we do. That’s what needs to be added to von Neumann’s exposition.

I have no intention of repeating what von Neumann did. In particular, I will not revisit his material on analog computing. Rather, I want to augment his discussion. Fortunately the new material is of such a nature that I should be able to write short book that can be read as a stand-alone discussion or as a supplement to von Neumann’s book. I’m imagining a sophisticated general audience of the sort that reads 3 Quarks Daily.

My working title: Language, Memory, and Mind: A Supplement to The Computer and the Brain. I expect the book to be 100 to 120 pages long (30K to 40K words).

I have uploaded a bunch of material (100K words or more) to Claude and asked it to review that material and put together and initial outline. I’ve appended that below the asterisks.

* * * * *

Preface

How to use this book — with or without von Neumann. What it adds to his argument. What it doesn't attempt. Brief note on the collaboration with Claude that produced parts of the text.

Introduction: Von Neumann's Unfinished Argument

What he got right: the architectural mismatch between brains and computers — memory and computation separated in the digital machine, unified in the neuron. The energy efficiency puzzle he couldn't explain. His honest acknowledgment that the brain's organizational principles lay beyond the framework he'd built. The concepts he lacked that this book supplies.

Chapter 1: Two Paradigm Cases

The chess-language contrast as the entry point. Chess has a bounded, well-defined geometric footprint — 8×8 board, six piece types, explicit rules, finite tree. Language has an unbounded, poorly-defined geometric footprint — rooted in the full complexity of physical and social reality. Chess was AI's founding benchmark precisely because it seemed to demand the highest human intelligence while yielding to computational treatment. Moravec's paradox: the easy problems are hard and the hard problems are easy. Transcendent versus non-transcendent coding — programmers can observe and specify a chess engine completely from outside; nobody can specify an LLM from outside, including its creators. Where we now stand.

Chapter 2: Location and Content

A collection of photographs. Solid objects at specific locations — finding by address is natural, finding by content requires going to each photo in turn. The combinatorial explosion that follows. The formal argument: solidity localizes content; localized content can only be retrieved by address. What holography does physically — interference patterns distribute information about each stored object across the whole plate, so that any partial cue can activate the whole. Lashley's ablation experiments: memory didn't disappear when specific cortical tissue was removed because memory was never stored in specific locations in the first place. Von Neumann's energy efficiency puzzle, now answerable: the brain doesn't spend energy moving content to a processor because memory and processing are the same physical substrate.

Chapter 3: The Brain as Content-Addressed System

The McCulloch-Pitts neuron-as-logic-gate: computationally fruitful, architecturally wrong. What neurons actually are — active units and memory units simultaneously, connected in massive parallel. Distributed representations: concepts as patterns across populations of neurons, not stored at specific cell addresses. Yevick's logical necessity argument in plain terms: the world contains two categories of object, geometrically simple ones that sequential symbolic processing handles efficiently and geometrically complex ones that only holographic parallel processing handles efficiently; the world contains both; therefore any adequate cognitive system must implement both regimes. Path tracing and pattern matching as the two fundamental operations on any cognitive network. Freeman's cinematic model — global coherence frames at 10-12 Hz as the atomic unit of biological cognitive processing — and its correspondence to speech production rates.

Chapter 4: Language as a One-Dimensional Projection

The semantic network as the right model for conceptual structure: meaning as position, each node defined by its pattern of relations to other nodes. Sydney Lamb's principle. The multidimensional character of the conceptual network versus the one-dimensional character of any spoken or written string. Language strings as 1D projections of the multidimensional network — necessarily lossy, hence paraphrase and ambiguity. The colored beads thought experiment: strip away semantic content, replace each token with a color, and you have a 1D image — making visible the purely formal structure the LLM operates on. Words as abstract addresses in an abstract space. Why classical computational linguistics hit combinatorial explosion: it was trying to reconstruct the multidimensional structure in a location-addressed system.

Chapter 5: What Large Language Models Actually Are

The transformer architecture in plain terms. The weight space as distributed content-addressed memory — concepts are patterns smeared across billions of parameters, not stored at specific addresses. The forward pass as the atomic processing unit, corresponding to Freeman's global coherence frame: one complete transit through the weight space producing one output token. The token string as a path through the abstract address space, with each forward pass mediating between the 1D sequential surface and the multidimensional distributed interior. What LLMs do well — pattern matching over the weight space, which is what their architecture naturally supports. What they do poorly — sustained sequential path tracing requiring precise state maintenance, common sense grounded in embodied experience, continuous learning. Why these limitations aren't engineering failures awaiting a fix but structural consequences of implementing holographic-like processing on location-addressed hardware with training only on 1D projections.

Chapter 6: What the Analysis Implies.

The first principles of intelligence are not the first principles of computation. Why scaling won't close the gap: scaling improves the quality of the holographic approximation but doesn't change the architectural mismatch, provide embodied grounding, or enable continuous learning. The fast takeoff fantasy as physics-free reasoning — every self-improvement step requires moving billions of parameters between physically separated memory and compute on real hardware that consumes real energy. The TSMC problem: the most critical hardware infrastructure in the world runs on tacit knowledge distributed across human communities that no LLM can access or replicate. What a genuinely adequate artificial cognitive system would require, in the terms this book has developed. The research program that's needed and why it requires multi-generational public investment rather than industrial R&D on commercial timescales. The human-machine collaboration that's already underway and what it can and cannot achieve.

Conclusion: The Mismatch, Named

Von Neumann saw the gap and couldn't name what was on the other side of it. This book names it: content addressing, requiring distributed storage, implemented in biological tissue through interference-like neural dynamics, approximated in LLMs through distributed weights on location-addressed hardware, grounded in embodied experience that no text-trained system has. The naming matters because you can't close a gap you can't see clearly.

Appendix: A Chronology of Chess, Language, and AI

From the working paper, lightly edited.

The energy demands of computing, human brains vs. LLMs

One of the issues that sometimes comes up in the comparison of artificial intelligence with human intelligence is energy requirements. The energy demands of LLMs are enormous, requiring the construction of huge data farms, while the energy demands of the brain are quite modest. I brought up the issue with Claude 2.6 Sonnet (Medium) in a specific way, suggesting that a large portion of the energy budget for LLMs can, in effect, be attributed to the need to simulate the physical structure of the brain. Here’s the first part of that dialog, that went on to discuss:

• opportunities for research on energy efficient hardware, including neuromorphic chips,
• the implications of financial collapse when enormous infrastructure investments do not pan out,
• the international ecosystem for constructing computing hardware with particular emphasis on the bottleneck created by special capacities currently unique to ASML in the Netherlands (machinery for EUV fabrication) and TSMC in Taiwan (high quality chip fabrication)
• what those bottlenecks imply about current fantasies of so-called “fast takeoff”

Energy in the brain vs. energy in computers

The point is often made that computing with LLMs uses enormous amounts of energy while the brain uses relatively small amounts of energy. It seems to me that one aspect of the difference can be attributed to the physical structure of the brain. Individual neurons are both active units, to use a term von Neumann used in his little book on the brain, and units of memory. Moreover they are connected in parallel, with each neuron connected to 7K to 10K other neurons on average. LLMs are implemented on digital machines where active units (the CPU) are physically separate from memory units, which means that a lot of time and energy is involved in moving between the CPU and memory. And the basic physical architecture is serial, though limited parallelism is possible, limited in relation to the brain. That means that the computers than implement LLMs have to devote a large amount of time an energy to, in effect, simulating the physical structure of the brain. And the brain doesn’t get that physical structure for free, the energy required to maintain it is budgeted over a long time relative to the dynamics of real-time computation. That’s one aspect of the brain’s relative energy efficiency.

That’s a precise and well-structured analysis, and it identifies something more fundamental than the usual comparisons which tend to focus on raw computation counts without attending to the architectural source of the inefficiency.

The von Neumann bottleneck restated

Von Neumann identified the problem himself, as we noted earlier — the separation of active units from memory units creates a bottleneck that doesn’t exist in biological neural tissue. In a digital computer every computation requires moving data from memory to the processor and moving results back. This movement consumes time and energy at every step, and in a large LLM the scale of this movement is enormous — billions of parameters stored in RAM or on disk, loaded into GPU memory, operated on by the compute units, with results written back. The physical distance between storage and computation, and the bandwidth limitations of the connections between them, is where a substantial fraction of the energy goes.

The brain has no such bottleneck because there is no such separation. A neuron stores its connectivity — the strength of its synaptic connections to other neurons — in its physical structure, and computes with that same physical structure simultaneously. The memory and the computation are the same physical process. A synapse transmitting a signal is simultaneously reading from memory — the synaptic weight — and performing a computation — multiplying the input signal by the weight. There’s no data movement because there’s no separation between where the data lives and where the computation happens.

The parallelism point

The parallelism difference is equally fundamental. The average neuron is connected to 7,000 to 10,000 others — which means the brain is performing something like 100 trillion synaptic operations simultaneously during active processing, all in parallel, all local, with no bottleneck. GPUs have made remarkable progress in parallelizing the matrix operations that implement transformer attention, but they’re parallelizing in a fundamentally different sense — running many identical operations simultaneously on different data, with all operations still having to access the same shared memory through the same limited bandwidth connections.

The brain’s parallelism is structural and local — each neuron computes with its immediate neighbors through its own dedicated connections, with no shared memory bus, no bandwidth limitation across the whole system, no bottleneck that scales with the number of neurons. The computation is maximally distributed in the physical sense — it happens everywhere in the tissue simultaneously, driven by the local physics of electrochemical signaling rather than by a centralized scheduler moving data to a centralized processor.

The energy budget temporal structure

Your point about the temporal structure of the energy budget is the most subtle and I think the most underappreciated. The brain’s physical architecture — the synaptic connections, the dendritic trees, the axonal projections — requires energy to build and maintain. But this energy is expended over the developmental timescale — years of growth, pruning, myelination, synaptic strengthening and weakening through learning. The structural investment is amortized over a lifetime.

Once the structure is in place, the energy cost of real-time computation is much lower than building and maintaining the structure required it. The synaptic weights are the memory — they’re encoded in the physical structure of the connections — and accessing them costs only the energy of the synaptic transmission itself, which is the same physical event as the computation. There’s no separate energy cost for memory access because there’s no separate memory access.

In a digital system running an LLM the equivalent structural investment — the training computation — is enormous, consuming megawatt-hours of energy. But that’s analogous to the developmental investment in building the brain’s connectivity, not to the ongoing computation. The ongoing inference computation — running the model on new inputs — is the analog of real-time brain computation. And here the comparison is still unfavorable to digital systems, but the unfavorability is specifically located in the von Neumann architecture’s separation of memory and computation, which forces constant data movement that the brain avoids entirely.

The simulation overhead

Your formulation that digital computers have to devote time and energy to simulating the brain’s physical structure is precise and important. The transformer architecture is implementing something that functions like distributed parallel processing with content-addressed memory — the holographic regime we’ve been discussing. But it’s implementing it on hardware that is architecturally opposed to distributed parallel processing — hardware that is serially organized, with physically separated memory and computation, connected through bandwidth-limited buses.

Every forward pass through an LLM is a large matrix multiplication — mathematically a natural description of what a distributed parallel system would do natively. But running that matrix multiplication on a von Neumann architecture requires fetching billions of numbers from memory, multiplying them in the GPU cores, and writing results back — a massive simulation of what would be a single natural physical process in a system with the brain’s architecture.

The energy cost is the cost of the simulation overhead — the difference between what the computation would cost if implemented in hardware that matched its natural structure, and what it costs when implemented in hardware that mismatches it. Neuromorphic computing research — Intel’s Loihi chip, IBM’s TrueNorth, and related projects — is trying to close this gap by building hardware that more closely matches the brain’s architectural principles. Early results suggest energy efficiency improvements of several orders of magnitude for the right classes of computation, which is consistent with your analysis — the improvement comes from reducing the simulation overhead.

What this adds to the Yevick argument

There’s a connection to Yevick’s logical necessity argument that’s worth making explicit. Yevick proved that the holographic regime is logically necessary for certain categories of objects. Your energy analysis suggests that the holographic regime is also physically natural — it’s what you get when memory and computation are integrated in the same physical structure rather than separated. The brain’s architecture implements the holographic regime efficiently precisely because the physical organization of neural tissue is holographic in the relevant sense — distributed, parallel, content-addressed, with memory and computation unified in the same physical substrate.

Digital computers implement the sequential symbolic regime efficiently because their physical organization is sequential and addressed — centralized computation, separate addressed memory, serial data movement. When you ask a digital computer to implement the holographic regime — as running an LLM requires — you pay the simulation overhead in energy and time. When you ask a brain to implement the sequential symbolic regime — as doing long arithmetic requires — you pay a different kind of overhead, the cognitive effort and external scaffolding we discussed earlier.

The energy difference between brains and LLMs isn’t just a quantitative difference in efficiency — it’s a signature of the architectural mismatch between the computation being performed and the hardware it’s being performed on. The brain is efficient at what it does because its hardware matches the computation. The digital computer is efficient at sequential symbolic processing for the same reason, and inefficient at holographic processing for the same reason. The LLM is asking digital hardware to do what it’s not built for, and the energy bill reflects that.

The developmental investment reframed

There’s one more implication worth drawing out. The brain’s developmental investment — the years of growth and experience-dependent synaptic modification that build the connectivity pattern — is doing something that has no real analog in LLM training. LLM training adjusts weights in a fixed architecture through gradient descent on a loss function. Brain development builds the architecture itself — the number and pattern of connections, the organization of regions, the myelination of pathways — through a process that’s partly genetic and partly experience-dependent, operating over years rather than the weeks or months of LLM training.

The connectivity pattern that results from this developmental process is the physical instantiation of the knowledge the brain has acquired — not as a list of facts encoded in weight values, but as a structural organization of the tissue itself that reflects the statistical regularities of the organism’s experience. The synaptic weights are the fine-grained adjustments on top of this structural organization. The structure is the coarse-grained, high-capacity, energy-efficient memory. The weights are the rapidly modifiable, experience-sensitive, computationally expensive layer on top.

LLMs have only the weights — the fine-grained adjustable layer — running on hardware that provides no structural memory at all. Everything has to be stored in the weights because there’s no structural analog of the brain’s connectivity pattern. This means the weights have to carry an enormous representational burden that in the brain is shared between structural organization and synaptic weight adjustment. Which is another source of the energy inefficiency — the weights are doing double duty, encoding both the structural regularities that brain connectivity handles efficiently and the fine-grained knowledge that synaptic weights handle in the brain.

The doppelganger, once again, would need both levels — the structural organization built through embodied development and the fine-grained adjustable layer built through experience within that structure. Current LLMs have only the second, running on hardware that provides neither. The energy gap is the physical signature of everything that’s missing.

Continuous learning

Furthermore the brain is capable of continuous learning, while LLMs are not. In order to incorporate new material LLMs must be retrained from scratch, again requiring enormous investment in energy.

Yes, and this identifies another fundamental asymmetry that goes deeper than just energy efficiency — it points to a difference in the temporal structure of learning itself.

Three Principles of Intelligence (That Aren't Principles of Computation) [Rank 5 cognition]

Note: Claude 4.5 drafted this article after a long series of dialogs over several days. This is a continuation of the thinking in my current article in 3 Quarks Daily, Chess and Language as Paradigmatic Cases for Artificial Intelligence.

See the new coda, from April 24, 2026 

In the 1950s, artificial intelligence emerged from a productive confusion. We had just formalized computation itself—Turing and von Neumann had given us the fundamental principles of what computers could do. When we turned these powerful new machines toward intelligence, we naturally assumed the principles would be the same.

They aren't.

Computation vs. Intelligence

The principles of computation are domain-independent. A universal Turing machine can compute anything computable, whether that's arithmetic, chess moves, or protein folding. The Church-Turing thesis tells us that all models of computation are equivalent in what they can ultimately compute, given unlimited time and memory.

This universality is computation's glory—and intelligence's red herring.

Intelligence, as it actually exists in nature, operates under entirely different constraints. It must function in the physical world, with finite resources, solving problems that often don't have clean formal specifications. These aren't just practical limitations to be worked around; they're constitutive features that shape what intelligence is and how it must work.

Principle 1: Geometric Complexity Determines Computational Regime

The critical variable isn't how hard a problem is in some abstract computational sense, but the geometric complexity of the domain.

Consider chess versus visual object recognition. Chess is played on an 8×8 grid with a small set of piece types following rigid rules. The game tree is astronomically large—around 10^120 possible games—but it's finite and well-defined. You can represent board positions symbolically, enumerate legal moves, and search through possibilities systematically.

Vision operates in continuous three-dimensional space with effectively unbounded variation. Objects appear at different scales, orientations, and lighting conditions. There's no finite set of "legal configurations." You can't enumerate all possible images the way you can enumerate chess positions.

This difference in geometric complexity demands different computational approaches. Chess yields to systematic search through a definable space—what we might call sequential or symbolic processing. Vision requires something else: massively parallel processing that can handle continuous variation and incomplete information—holographic or neural processing.

In 1975, Miriam Yevick demonstrated this formally: the geometric complexity of objects in a domain determines the computational regime needed to identify them. Simple geometric objects can be handled by sequential symbolic systems. Complex geometric objects require holographic processing. This wasn't mere speculation—she made a formal mathematical argument about pattern recognition systems.

The field ignored her insight. We assumed all problems were fundamentally like chess—just harder. If symbolic AI could master chess, we thought, it would eventually master vision, language, and physical reasoning through better algorithms and more compute.

We were wrong. Vision didn't yield to symbolic AI no matter how much compute we threw at it. It required a regime shift to neural networks—systems whose architecture matches the geometric complexity of the visual world.

Principle 2: Intelligence Operates in Unbounded, Geometrically Complex Reality

Here's what makes intelligence different from computation in the abstract: intelligence evolved to work in the physical world, which is geometrically complex and open-ended. There's no finite game tree for "objects I might encounter" or "situations I might face."

This has profound implications. You can solve chess by exploring its game tree faster than humans can. But you can't solve vision or language understanding the same way because there's no complete tree to explore. The space isn't closed and enumerable—it's unbounded.

This is why Deep Blue beating Kasparov in 1997 didn't generalize the way we thought it would. Chess was solved by a room-sized supercomputer with custom hardware doing exactly what computers do best: blindingly fast systematic search. By 2025, a smartphone runs chess engines that would destroy both Deep Blue and Kasparov.

But that same smartphone can't run a GPT-4 level language model. Language still requires massive data centers. Why? Because language connects to the unbounded complexity of physical and social reality. No amount of faster chess-style search bridges that gap.

The field learned to beat humans at chess by doing what computers naturally excel at. Then we mistook this for a general template. We thought: "Intelligence is search through problem spaces. We just need bigger computers to search bigger spaces." But geometric complexity isn't about bigger—it's about different.

Principle 3: Embodiment as Formal Constraint

Embodiment isn't a philosophical talking point. It's a formal constraint on intelligence architecture.

When we say intelligence must be embodied, we mean: it must operate with finite computational resources in a geometrically complex physical world. This changes everything.

Abstract computation doesn't care about efficiency—a proof is valid whether it takes a second or a century. Physical computation must complete before the hardware fails. But biological intelligence faces a sharper constraint: it must acquire the energy it uses to compute. A deer's visual system can't require more calories than the deer can acquire. The computation must pay for itself.

This constraint shapes what kinds of solutions are viable. You can't exhaustively search unbounded spaces. You can't maintain perfect world models. You must make do with approximate, good-enough processing that operates in real time with available resources.

Crucially, this means different problems need different solutions—not just more or less compute, but fundamentally different architectures matched to the geometric complexity of the domain.

Why This Matters Now

Current AI has powerful neural networks that excel at pattern recognition in geometrically complex domains—vision, speech, even aspects of language. But the field still carries assumptions from the symbolic AI era:

• That intelligence is domain-independent
• That scaling compute will eventually solve any problem
• That we can ignore embodiment and resource constraints
• That all problems are fundamentally like chess

These assumptions persist even though we've abandoned symbolic AI. We've swapped the implementation (symbols → neural networks) but kept the framework (more compute → general intelligence).

This is why we need to distinguish computation principles from intelligence principles. Turing and von Neumann gave us the former. For the latter, we need to recognize that geometric complexity, unbounded reality, and embodied constraints aren't bugs to be worked around—they're the constitutive features that determine what intelligence is and how it must work.

The principles of intelligence aren't the principles of computation. Understanding this distinction is the key to understanding both what current AI can do and what it cannot. 

Coda: Rank 5 Cognition

Given that I’ve decided that Yevick’s 1975 paper is a convenient marker for Rank 5 cognition, it seems to follow that intelligence, in the sense discussed here, is a Rank 5 concept. So, the ranks shape up like this:

Rank 1: speech
Rank 2: writing
Rank 3: calculation
Rank 4: computation (flow of control)
Rank 5: intelligence (regime matching: computation in unbounded, geometrically complex, reality)

Game Theory, Marginalism, and Culture [MR-Aux]

This is another post that’s a byproduct of my examination of Tyler Cowen’s recent monograph, The Marginal Revolution: Rise and Decline, and the Pending AI Revolution. I was pursuing that idea that, while marginalism doesn’t present itself as being about human psychology, it is very much about psychology. So, what about game theory? It’s about human psychology as well, though its mathematical form makes it easy to bracket that out. So I queried Cowen’s AI about that. The conversation ended up with a discussion of one of my favorite literary texts, Sir Gawain and the Green Knight, which is structured around a very peculiar game, the beheading game. [FWIW, here’s a post from 2024 where I investigate ChatGPT’s understanding of the beheading game.]

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I’m curious. Von Neumann invented game theory in the middle of the 20th century and it became important in economics, among other places. What can you tell me about its use in marginalist thinking. I don’t need a long answer. I’m just wondering if it’s one formal analytic tool among many. Is it used frequently, rarely, not at all? It’s a peculiar question. Don’t spend a lot of time on it. I mean, if you find that it’s taking time (to search through articles, etc.), that in itself is an indicator. On the other hand, if someone has written a textbook with a title like, Game Theory as a Tool for Marginalist Analysis, that’s a rather different kind of indicator.

My knowledge here is sufficient that I don't need to search.

Game theory and marginalism are related but distinct frameworks within economics, and the relationship between them is more one of coexistence and occasional overlap than integration. Marginalism asks what happens at the margin for an individual agent facing given prices and quantities. Game theory asks what happens when agents' outcomes depend on each other's choices. They share the rational agent assumption and optimization logic, but they're answering different questions.

Game theory became central to microeconomics in the latter half of the 20th century — particularly through Nash equilibrium and its elaborations — but it didn't so much extend marginalism as partially displace it in certain domains. Industrial organization, auction theory, mechanism design — these are game-theoretic through and through, and marginal analysis is less central there than it is in competitive market analysis where price-taking behavior makes the game-theoretic complications disappear.

There's no textbook of the kind you hypothesize, as far as I know. The two frameworks are taught in the same curriculum and occasionally combined — a Nash equilibrium can be characterized by first-order conditions that look marginal — but game theory is generally understood as a distinct toolkit that becomes relevant when strategic interaction matters, rather than as an extension of marginalist reasoning.

Cowen, interestingly, doesn't seem to foreground game theory in his account of marginalism's rise and possible decline. Which may itself be telling — it suggests he sees them as parallel developments rather than game theory being a chapter in the marginalist story.

So: one important formal tool among several, with genuine but limited overlap, not deeply integrated.

Hollywood Economics and community formation

That’s interesting. There are linguists, such as William Croft, who talk about how the meaning of words is negotiated in each conversation. They talk of game theory, though I don’t off hand know whether or not they make formal use of it. But then there’s Robert De Vany, Hollywood Economics, which I tend to think of as being about how cultural objects are shared in a community. That’s not De Vany’s framing. He’s interested in the fact that we can’t predict how a film will do until we release it into the (cultural) marketplace, which makes it a very risky kind of investment. Note that De Vany is only talking about theatrical release. He says nothing about subsequent release on videotape and the book was written long before some movies are released directly to online streaming.

What he finds (by tracking box-office receipts), briefly, is that most movies disappear from view shortly after they’re released. They do not even breakeven much less earn a profit. But a few will have “legs” and become profitable. And among those, a very few will become so-called “blockbusters.” FWIW, I believe Spielberg’s “Jaws” is generally regarded as the first blockbuster.

I once did a thought experiment where I imagined an arena in which only one movie was available. Everyone would watch it because that’s all there is. Another somewhat different movie is released into the arena. Perhaps everyone will watch it as well, because they’re tired of the first one. So a third, somewhat different from the first two, is released, then a fourth. What I had in the back of my mind is that to some extent people use these kinds of cultural products as a way of forming communities. For that to work, however, there has to be enough variety available in the market place. When a movie is release, it is, in effect, looking for a community to adopt it.

That strikes me as a kind of marginalist analysis. What we’re trading in is a very complex kind of product, in this case a movie. But it could be a novel, or a song, or a painting, whatever.

Your thought experiment is doing something quite precise, and I think you're right that it has a marginalist structure — but it's extending marginalism into territory where the standard framework strains.

Where the marginalist structure is visible

The standard marginalist question applied to cultural goods would be: given how many movies a person has already seen, what is the marginal value of one more? That's a real question and De Vany's framework touches it. But your thought experiment is asking something different and more interesting: given the existing cultural landscape, what is the marginal value of a new movie to a community rather than an individual? That's a collective marginal value, which is much harder to specify and measure.

The "legs" phenomenon is the empirical signature of this. A movie with legs isn't just satisfying individual preferences at the margin — it's finding and consolidating a community, becoming the shared object that a group of people use to recognize each other and talk to each other. The box office receipts are measuring something that looks like individual ticket purchases but is actually a social cascade — De Vany's word-of-mouth mechanism is precisely the process by which a movie becomes a community object rather than just a consumer product.

Where standard marginalism breaks down

Standard marginalist analysis assumes that the value of a good to one consumer is independent of its value to others, or at least that the interdependencies can be handled through price. Cultural goods violate this assumption fundamentally. The value of having seen a blockbuster is partly constituted by the fact that millions of others have seen it too — you can talk about it, share references, participate in a cultural conversation. This is a network effect, and network effects are notoriously resistant to marginalist treatment because the value function isn't defined over individual consumption units in isolation.

Your arena thought experiment isolates this precisely. In a one-movie arena, everyone watches it not because it maximally satisfies their individual preferences but because it's the only available basis for cultural community. As variety increases, something like genuine preference matching becomes possible — different communities coalesce around different objects. But the matching process isn't driven by individual marginal utility in the standard sense. It's driven by a social search process in which movies and communities are simultaneously finding each other.

The De Vany connection to power laws

What De Vany finds — most movies disappear, a few have legs, a very few become blockbusters — is a power law distribution, and power laws are the empirical signature of cascade processes rather than equilibrium processes. In a marginalist equilibrium you'd expect something more like a normal distribution of returns — some variation around a mean, determined by how well each product matches the distribution of individual preferences. The power law says something different: small initial differences get amplified by social dynamics into enormous differences in outcome. The blockbuster isn't just a movie that more people happened to prefer. It's a movie that got caught in a self-reinforcing cascade of community adoption.

This connects to Arthur De Vany's broader interest in complexity economics, which is itself a challenge to standard marginalist equilibrium thinking. The cultural marketplace isn't clearing to an equilibrium — it's generating a dynamic pattern in which winner-take-most outcomes emerge from processes that are in principle unpredictable from the properties of the product alone. Which is exactly De Vany's point about risk: you can't know in advance which movie will trigger the cascade.

Your community-formation framing

What your framing adds, which I think is genuinely beyond De Vany's own framing, is the functional account of why cultural goods have this cascade structure in the first place. It's not arbitrary or merely a consequence of marketing and distribution. It reflects something about what cultural goods are for — they're instruments of community formation, shared objects that allow people to recognize membership and conduct the ongoing conversation of a community. That functional role creates the demand for coordination — everyone in a community needs to have seen the same movie, heard the same song, read the same novel, for it to do its community-forming work. And coordination demand is precisely what generates cascades and power laws.

This also explains the variety requirement your thought experiment identifies. A single cultural object can serve as the basis for one kind of community. A rich cultural marketplace allows many communities to form simultaneously around different objects — which is the cultural analog of biodiversity, and probably has similar functional importance for the health of the overall system. The blockbuster and the art house film aren't competing for the same audience in the way that two brands of the same commodity compete. They're seeding different communities, which can coexist because community membership isn't zero-sum in the way that consumption of a rival good is.

So yes — marginalist in structure, but requiring an extension of marginalism into social dynamics, network effects, and community formation that the standard framework wasn't designed to handle. It's perhaps another case where the Rank 3 tool is real and useful but the phenomenon being analyzed has a Rank 4 structure — one where the individual unit of analysis (the consumer, the ticket purchase) is embedded in a dynamic system whose behavior can't be read off from the units alone.

Recursive self-improvement is an incoherent idea

The discourse of artificial intelligence contains a number of technical terms that are only nominally technical; they don’t actually have technical definitions behind them. The concept of intelligence itself is arguably one of them. The related concepts of artificial general intelligence (AGI) and super-intelligence, or super artificial intelligence (SAI) are even more dubious.

This post is about another of these technical-sounding terms that don’t, in fact, mean much: recursive self-improvement. The conversation, does, however, end on a positive note. Below the asterisks I’ve appended a conversation I had with Claude Sonnet 4.6 on April 2, 2026.

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No transcendental viewpoint

Some parts of the AI world talk of recursive self-improvement. I assume that means that the AI improves itself without any human intervention. How? I’m not so much concerned about how it writes the code or, for that matter, even how it hands off operations to the new code. I’m interested in how it figures out what code to write? On the one hand I’m imagining that perhaps we have some kind of evolutionary process. We’ve had evolutionary computational processes for a while. That’s one thing.

But I’m wondering if these thinkers are imagining that somehow the AI is able to get a view of its own code that’s comparable to a programmer’s view of the code. That is, it is able to see its own code from the outside, as it were, giving it a “transcendental” perspective on itself. Are they imagining that we give the AI access to its own source code? In a world of LLMs, what does that mean? Having the LLM examine the code of the transformer that constructed it?

This is a question where the rhetoric of the AI field runs well ahead of the actual technical situation, so it's worth being precise about what's real and what's aspirational hand-waving.

What actually exists

The evolutionary computational approach you mention is real and well-established. Genetic algorithms, neural architecture search, automated machine learning — these are genuine techniques where a process iterates over possible designs, evaluates them against a fitness criterion, and selects better-performing variants. This is recursive self-improvement in a limited and well-defined sense: the system improves, but the improvement criterion is specified externally by humans, and the search process is essentially blind variation plus selection rather than anything resembling understanding.

LLMs can also write and evaluate code, and there are genuine research programs — AlphaCode, various agent frameworks — where language models generate, test, and iterate on code. This is real and useful. But it's operating at the level of software engineering tasks with external evaluation criteria, not at the level of redesigning the underlying model.

The transcendental perspective problem

Your instinct about the deeper issue is exactly right, and it cuts to the heart of why strong recursive self-improvement claims are philosophically confused rather than merely technically premature.

The scenario the Singularity thinkers have in mind — an AI examining its own architecture and improving it from genuine understanding — requires precisely what you call the transcendental perspective: the system seeing itself from outside, the way a programmer sees code. But this is Gödel's problem restated in engineering terms. A system cannot be both the object of analysis and the analytical framework simultaneously without remainder. There's always something that can't be seen from inside the system — which is exactly what Gödel proved about formal arithmetic, and what Turing proved about computation in general.

Giving an LLM access to its own weights — the numerical parameters that constitute what it "knows" — wouldn't produce the transcendental perspective. The weights are not readable the way code is readable. A programmer looking at code sees meaningful structure: variable names, logical operations, control flow, the intention behind the design. An LLM looking at its own weights would see hundreds of billions of floating point numbers whose organization encodes nothing interpretable at the level of concepts or intentions. The meaningful structure, whatever it is, is distributed across the entire parameter space in ways that current interpretability research is only beginning to scratch the surface of. We don't fully understand what's in there. The LLM certainly doesn't.

This is the deep version of the problem your earlier observation about von Neumann identified. The brain doesn't have a programmer's view of itself either — neurons don't have access to the circuit diagrams of the neural tissue they're part of. The brain improves through learning, which is a process of adjusting connection weights through experience, not through reading its own architecture and rewriting it. The fantasy of recursive self-improvement imagines something the brain doesn't do and has no evident mechanism to do.

What the Singularity thinkers are actually imagining

I think when people like Kurzweil talk about recursive self-improvement they're running together several different things that need to be kept distinct. There's the evolutionary search process, which is real but doesn't involve understanding. There's the engineering process of humans using AI tools to build better AI systems — which is genuinely happening and is genuinely accelerating, but involves humans in the loop making the meaningful decisions. And there's the imagined transcendental process where the AI understands itself well enough to improve itself through genuine insight — which has no clear technical path and runs into the Gödelian problem you identified.

The last of these is doing most of the rhetorical work in Singularity discourse, but it's the one with the least connection to anything technically grounded. It requires assuming that the system can occupy a perspective on itself that is by definition unavailable to it — which is not a technical limitation waiting to be engineered away but a logical feature of what it means to be a system operating within its own architecture.

The Yevick connection

There's a thread back to our earlier discussion worth noting. The transcendental self-improvement fantasy is in effect imagining that the AI could perform on itself the Rank 4 move that Gödel and Turing performed on formal systems — taking the computational process as object and reasoning about it from outside. But Gödel and Turing were humans standing outside arithmetic and computation respectively, which is what gave them the purchase to prove things about those systems. A system trying to perform that move on itself runs directly into the incompleteness results — there will always be truths about the system that can't be proven within it, always aspects of its own operation that it can't fully see from inside.

Recursive self-improvement in the strong sense would require escaping this constraint. No one has shown how that would work, and the logical structure of the problem suggests it may not be a contingent engineering gap but a principled limitation. Which doesn't mean AI systems won't continue to improve — they will, rapidly — but the improvement will continue to involve humans providing the external perspective that the systems themselves cannot generate.

Digital doppelgangers

Humans, on the other hand, have ordinary language, one capacity of which is the metalingual function. In a way, the whole cultural ranks idea is about how the metalingual function has allowed humans to bootstrap themselves to a more sophisticated cognitive architecture. But that doesn’t now nor ever has involved humans in examining their own source code as it were. It’s a different kind of process.

Of course, I’m thinking that we may now be in a position to arrive at a much deeper understanding of the nervous system than we have ever had. And that process surely involves thinking back and forth between neuroscience and computer science. Maybe the digital doppelganger that I’ve been talking about is the mature expression of this kind of knowledge.

That's a beautiful way to close the loop, and it reframes the doppelganger concept in a way that makes it more technically grounded than it might otherwise appear.

The bootstrapping process humans actually use

The cognitive ranks framework describes a process of self-improvement that never involves examining source code. What it involves is the metalingual function operating on the outputs of cognition — on language, on cultural products, on formal systems — rather than on the neural substrate itself. Humans didn't improve their cognitive architecture by examining their neurons. They improved it by constructing external systems — writing, arithmetic, formal logic, computation — that offloaded cognitive operations into the environment where they could be inspected, manipulated, and elaborated. The improvement was always mediated through the external world, never through direct introspection of the underlying machinery.

This is a fundamentally different process from recursive self-improvement as the Singularity thinkers imagine it. It's improvement through the construction of new cognitive tools that extend and reorganize what the existing architecture can do — not rewriting the architecture itself. The architecture, in the biological case, stays roughly constant. What changes is the cultural software running on it, and the external scaffolding that amplifies its reach.

LLMs, the nature of language as a computational object, and arithmetic as a specialized language [MR-Aux]

Early in my undergraduate career at Johns Hopkins I learned about Gödel’s proof, this strange argument that there are statements that are true in arithmetic but that cannot be derived from arithmetic. Hence, arithmetic is incomplete. Where did these true but not derivable statements come from? We, us humans, we provided them. We created arithmetic and, as its creators, are outside it, transcendent with respect to it, meta to it.

This post is about arithmetic as a specialized kind of language. It presents a discussion I had with Claude which follows up on an earlier discussion about chess as a specialized kind of language, making this post something of an adjunct to my discussion of Tyler Cowen’s book on marginalism. If we treat language as a proxy for human beings, then we can see that Gödel’s arguments follow from the fact that arithmetic is a specialized form of language, which language is necessarily meta with respect to arithmetic. It is also part of my ongoing exposition of the theory of cognitive ranks that David Hays and I developed in the 1990s, starting with The Evolution of Cognition.

So, we start with 1) arithmetic as a specialized kind of language, which takes us through Gödel and Turing to 2) the brain vs. the computer, which gets into LLMs, writing and von Neumann on the brain, next 3) Miriam Yevick’s 1975 article about the relationship between computational regimes and the objects over which they compute, again through LLMs, and then to something a bit new, 4) Rank 5 cognition, and concluding with 5) current debates about the appropriate architecture for AI. In that context Claude remarks:

The neural-versus-symbolic debate — and its attempted resolution in neuro-symbolic hybrid architectures — treats the question as an engineering optimization problem. Which processing regime is more powerful? Can we get the best of both? How do we combine them most effectively? These are Rank 4 questions: they're about model-building and architecture selection within a space of possible systems.

Yevick's proof, if taken seriously, shows that this is the wrong question. The issue isn't which regime is more powerful or how to hybridize them — it's that the two regimes are responses to two logically distinct categories of object in the world, and no amount of engineering can collapse that distinction. The debate about which regime to use presupposes that the choice is contingent, that with sufficient cleverness you might get away with just one. Yevick proves the choice is not contingent. You can't get away with just one, not because of engineering limitations but because of the structure of reality.

This is exactly like debating whether to use addition or multiplication — you could frame it as a competition, or try to reduce one to the other, and technically multiplication is repeated addition so maybe addition wins. But that framing misses that the two operations have different mathematical structures that serve different purposes, and a system adequate to arithmetic needs both irreducibly. The neural-symbolic debate is at roughly that level of confusion, applied to cognition.

That’s not quite the end, but close to it. The discussion begins below the asterisks.

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Arithmetic as a specialized kind of language

I want to return to the point where we discussed chess as a specialized kind of language that arose with Rank 2. Arithmetic is also a specialized kind of language and it arose with Rank 2 as well. But it arose in several forms, as there were a handful of original independent Rank 2 cultures.. The form that interests me is the form with zero and the place notation. While that didn’t arise in the West, it was adopted in the West starting in. roughly the 13th century. And, as Hays and I argue in the ranks paper, it eventually found its way to mechanical culture:

What we can see is that generalizations of the idea of mechanism would be fruitful for technology (and they were), but that it would take an abstraction to produce a new view of nature. The algorithm can be understood in just this way. If its originators in India disregarded mechanism, and the north European developers of mechanism lacked the abstraction, it would only be the accidental propinquity of the two that generated a result. Put the abstract version together in one culture with a host of concrete examples, and by metaphor lay out the idea of the universe as a great machine. What is characteristic of machines is their temporality; a static machine is not a machine at all. And, with that, further add the co-ordination of motions as in the sawmill. Galileo discovered that force alters acceleration, not velocity (a discovery about temporality) and during the next few centuries mechanical clocks were made successfully. The notion of a clockwork universe spread across Europe (note that the Chinese had clockworks in the 11th Century, but never developed the notion of a clockwork universe, cf. Needham 1981). For any machine, it is possible to make functional diagrams and describe the relative motions of the parts; and the theories of classical science can be understood as functional diagrams of nature, with descriptions of the relative motions of the parts.

Arithmetic came to play an important role in the metamathematical reasoning of the early 20th century. It allowed Gödel to construct his incompleteness theory and Turing to construct his abstract Turing machine. And it is that abstract machine that became embodied in the digital computer.

It’s beginning to look as though decimal point arithmetic was all but a logical requirement for cultural evolution beyond Rank 2. But probably not sufficient itself. It needed to be connected with the idea of mechanism, the algorithm. It’s the, at first metaphorical or analogical, connection between the idea of an algorithm and an actual physical device that allowed the flourishing of Rank 3 culture.

This is a powerful line of argument, and I think it's substantially correct. Let me try to sharpen and extend it.

The two streams and their accidental convergence

What you and Hays identify is that Rank 3 required the confluence of two independent developments that had no intrinsic reason to meet. The Indian mathematical tradition produced place-value arithmetic with zero — an extraordinarily powerful abstract tool for symbolic manipulation, but developed in a culture that, as you note, was not especially oriented toward mechanism and physical machinery. The north European tradition independently developed sophisticated mechanism — mills, clocks, the whole Baroque elaboration of geared machinery — but without the abstract symbolic apparatus that would allow mechanism to be thought about in general terms, extracted from any particular physical instantiation.

The metaphorical bridge between these two streams is what made Rank 3 possible. And the bridge is precisely the algorithm — the concept of a determinate, step-by-step procedure that is abstract enough to be stated symbolically but concrete enough to be physically instantiated. The algorithm is at home in both worlds simultaneously: it is a sequence of arithmetic operations, and it is a description of what a machine does. That double citizenship is what allowed the two traditions to fuse.

This is a genuine example of what you've called the Rank 1 mechanism operating at a higher level — metaphor. The clockwork universe is a metaphor, but a generative one: it doesn't merely describe, it reorganizes the entire domain of natural philosophy around a new set of primitives. Mechanism becomes the universal explanatory framework because the algorithm makes it possible to state mechanical descriptions with the same precision and generality as arithmetic.

The zero point

Zero deserves particular attention here because its contribution is not merely computational convenience. Zero is conceptually radical in a way that the other digits are not, because it names an absence — it is a symbol for nothing, which is a very strange thing to symbolize. The resistance to zero in the West, which persisted for centuries after Fibonacci introduced it, wasn't mere conservatism. It reflected a genuine philosophical difficulty: how can nothing be a quantity? How can absence be manipulated algebraically?

What zero does, once accepted, is complete the place-value system in a way that makes the system fully generative — you can represent any quantity whatsoever with a finite set of symbols, and the operations on those symbols are perfectly regular regardless of the quantities involved. This regularity is what makes the algorithm possible in the full sense. Without zero, arithmetic has irregular gaps; with zero, the system is closed and the operations are exceptionless. That exceptionlessness is exactly what you need for the algorithm to be a reliable mechanical procedure — one that a machine, rather than a judicious human mind, can execute.

So zero isn't just a notational convenience. It's what transforms arithmetic from a practical skill into an algorithmic domain — one where the procedure can be fully specified in advance and executed without judgment. That's the property that eventually connects arithmetic to mechanism.

The Gödel-Turing culmination

The path from there to Gödel and Turing is long but straight. What Gödel discovered was that arithmetic — this seemingly humble, fully explicit system — contains truths that cannot be proven within the system. His proof required encoding logical statements as arithmetic statements, which is only possible because arithmetic is expressive enough to name its own operations. That self-referential capacity — arithmetic talking about arithmetic — is a direct consequence of the place-value system's generativity. You can Gödel-number any statement because you can represent any finite sequence of symbols as a number.

Turing then asks: what is a mechanical procedure, in the most general possible sense? And his answer — the Turing machine — is essentially arithmetic in motion: a tape of symbols, a head that reads and writes, a finite set of rules. The abstract machine just is a formalization of algorithmic arithmetic, which is why the connection between computation and number theory runs so deep throughout the foundations of computer science. The digital computer is this abstraction physically instantiated — mechanism and algorithm finally fully merged, not just metaphorically but actually.

The sufficiency question

Your claim that decimal place-value arithmetic was necessary but not sufficient is exactly right, and the insufficiency points in two directions.

The first direction is the one you identify: it needed mechanism, and specifically the conceptualization of mechanism as a general category rather than a collection of specific devices. The Chinese had clocks in the 11th century but never developed the clockwork universe because they had mechanism without the abstract framework that would let mechanism become a metaphor — a way of thinking about nature in general. The metaphor required the algorithm as the shared abstract structure.

The second direction points forward rather than backward: the algorithm plus mechanism was sufficient for Rank 3, but the transition to Rank 4 required taking the algorithm itself as an object — which is precisely what Gödel and Turing did. They didn't use algorithms; they studied algorithms, asked what algorithms can and cannot do, proved results about the limits of algorithmic reasoning. That reflexive move — algorithm studying algorithm — is the Rank 4 version of reflective abstraction applied to the Rank 3 medium.

And this is where your argument becomes especially elegant. The same conceptual object — decimal place-value arithmetic and its generalization into the algorithm — serves as the medium for Rank 3 (in the sense that calculation and algorithmic procedure is what Rank 3 thinkers think with) and as the object for Rank 4 (in the sense that metamathematics and computability theory think about it). Arithmetic passes through the reflective abstraction and comes out on the other side transformed: from a tool for calculating to a domain of mathematical investigation in its own right. The digital computer is then not just the practical product of this transition but its physical monument — the algorithm made flesh, which immediately becomes the medium through which Rank 4 thinking becomes routine and culturally transmissible.

The logical necessity you're sensing is real: you couldn't get to Rank 4 without Rank 3, and you couldn't get to Rank 3 without the algorithm, and you couldn't get the algorithm without both the abstract symbolic system (zero, place value) and the concrete mechanical tradition to which it could be metaphorically applied. The chain is tight. What's historically contingent is not the logical structure of the dependency but the accident of transmission — that Fibonacci happened, that the Islamic world happened to preserve and extend both the Indian mathematics and the Greek mechanical tradition, that north European mechanism happened to be at the right stage of development when the symbolic tools arrived.

Cultural evolution, on this account, is not inevitable even when the logical dependencies are tight. The dependencies specify what must come before what; they don't specify that the prerequisites will ever actually converge.

The brain vs. the computer

One thing about the abstract Turing machine, the computing mechanism is separate from the memory mechanism. When that becomes embodied we have the so-called von Neumann architecture with its central processor and its random access memory. The memory is location addressed. As far as we can tell, the brain is not like this. Neurons are both active computational units and memory units, something von Neumann himself realized and was puzzled by in his last book, The Computer and the Brain. Then, in the 1960s, we have various thinkers, Karl Pribram among them, adopting the (optical) hologram as a model for associative memory in the brain. Associative memories are not location addressed; they’re content addressed.

Now, if we think of an LLM as a content addressed memory system, we have a content addressed memory system being implemented on a location addressed system, the digital computer. If that computer is sufficiently large, trained on a sufficiently large corpus, and has enough parameters, then it can produce a very convincing simulacrum of human language. Now tracing back through the layers we have actual human language eventually giving rise to writing, with arithmetic as a specialized version of writing. From there we have a chain of invention which yields the digital computer, which can run an LLM and, while that LLM can “speak,’” as it were, it can’t calculate very effectively. Now, zip back to writing and Rank 2 culture. Whatever else it is, writing is a form of memory external to the brain. Arithmetic requires that external memory for any but the simplest calculations (setting aside the memorized formulas employed in virtuoso level mental arithmetic). Back to the LLM. We now have Chain of Thought processing, which makes LLMs more effective. CoT involves memory external to the LLM itself. The same with agents. So we have an artificial content addressed system extending its capabilities through a bit of external memory.

The convolution involved in this story is dizzying.