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.

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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.

Language as involving both content and location addressing

Memory is one of the central concepts in thinking about and understanding both computing and the mind. Thinking about computating has brought us to understand that there are two broad categories of memory:

• Content addressed memory, and
• Location addressed memory.

Conceived as a large memory system, libraries are location addressed. Documents are stored at particular locations in the library, shelves for books and bound volumes of periodicals and reports, filing cabinets for other documents. To get some item from the library you need to find its location by consulting a catalog, and then go to that location and retrieve it.

Brains are content addressed. If you are curious about, say, the Johnstown flood, you don’t have to consult an internal catalogue to find where the appropriate document or documents are located among the folds and crevasses of the neocortex. You just think, “Johnstown flood,” and things you know about the Johnstown flood will come to mind. The phrase “Johnstown flood” is itself part of the content being addressed. But, if you happen to know something about the flood, then the phrase, “South Fork dam,” may also act to recall more information about the flood, for it is an element of content for one of the floods. As you may know, there were three Johnstown floods, in 1899, 1937, and 1977. The 1899 flood is the one that happened when the South Fork dam burst. If you don’t happen to know anything about the Johnstown floods, then you may have to consult an external memory system of some sort, like a library or the internet.

Digital computers are location addressed. The memory system is distributed over several types of hardware. There’s volatile memory, computer chips (generally RAM), which hold things temporarily. And there’s long-term memory, which can take various forms, but these days its mostly flash memory and hard disks. Computing involves moving data from memory, to the CPU, and then returning the results back to memory.

Large language models, LLMs, are sophisticated content addressed memory systems that are implemented in a digital computer. When you serve up a prompt, that’s addressing the memory, that is, the weight space. The system’s answer then is assembled from material in memory. 

I should also note that, while I fully support Gary Marcus’s call for neuro-symbolic computing, I think that the need for symbolic computing is most fruitfully thought of in terms of location-addressed memory, which is how symbolic computing operates. It is location addressing that allows for accurate planning and for arithmetic calculation. But it won’t solve the hallucination problem. For that you need a world model, which is also an aspect of Marcus’s program (not to mention Yann LeCun and Fei Fei Li). But those discussions are beyond the scope of this note. 

That should be enough to get you through the following dialog I had with Claude 4.6 Sonnet Adaptive. 

* * * * * 

I want to think of language in the brain in relation to both content and location address. The brain is basically a content addressed system. But language creates an abstract space of locations which can be used to address content. Each word is, in effect, a location in some abstract space. That’s obvious in the case of alphabetic writing systems, where alphabetization is a way of physically arranging those addresses on a string. Now consider a dictionary. The entries are located on that string. At the same time, each entry is a component of the string that is stored at the (primary) location. Which means that, in a (weak) sense, that string is also content addressed. But when a person recalls some string in its entirety when given only a component of that string, that’s content addressing. Both (some) humans and LLMs can return Hamlet’s full soliloquy when given the string “To be or not to be.”

This is a rich and precise observation that reframes the location/content addressing distinction in a way that illuminates both language and memory.

The abstract space of locations

The claim that language creates an abstract space of locations is more profound than it might initially appear. A word isn't just a sound or a mark — it's a stable, repeatable, publicly shared address that can be used to retrieve content from another person's cognitive system. When I say "dog" to you, I'm not transmitting the concept of dog — I'm transmitting an address that activates the concept in your system through your own content-addressed retrieval. The word is the public, location-addressed handle; the concept is the private, content-addressed payload.

This is the handle-schema distinction we discussed earlier, now stated in addressing terms. The word is a location address in a shared public space — the lexicon of a language community — that points to content stored in individual content-addressed systems. The remarkable thing about language is that this works across individuals despite the fact that each person's content-addressed storage is different in its details. The address is shared; the content it retrieves is similar enough across individuals to support communication, but never identical.

The dictionary as dual addressing system

The dictionary example is beautifully chosen because it makes the dual structure explicit and concrete. The alphabetically ordered string of entries is a location-addressed system in the most literal sense — you navigate to a word's definition by finding its location in the ordered sequence, using the address (the word's alphabetical position) to retrieve the content (its definition). The physical arrangement of the dictionary embodies the location-addressing structure of the abstract lexical space.

But as you note, each entry is also a component of a content network — the definition connects the word to other words, which connect to other words, which eventually connect to the whole semantic fabric of the language. Looking up a word in the dictionary is formally a location-addressed retrieval, but understanding the definition requires content-addressed processing — activating the concepts that the defining words point to, integrating them into a pattern that constitutes the meaning of the defined word.

So the dictionary is simultaneously a location-addressed artifact and a gateway to a content-addressed system. Its physical organization is location-based; its semantic organization is content-based. The two coexist in the same object, serving different cognitive functions.

The Hamlet soliloquy case

The Hamlet example is where it gets most interesting. "To be or not to be" is simultaneously a location address — a string that uniquely identifies a specific passage in a specific text — and a content fragment — a semantic and rhythmic unit that is part of the passage it addresses. Using it to retrieve the full soliloquy requires both kinds of processing.

The retrieval is content-addressed in the sense that the fragment activates the whole — the partial pattern resonates with the stored complete pattern and the complete pattern is reconstructed. This is classic holographic retrieval — the fragment serves as the reference beam that reconstructs the stored image. The fact that both humans and LLMs can perform this retrieval when given the opening fragment demonstrates that both systems have stored the complete pattern in a form that supports partial-cue retrieval.

But notice what's different between the human and LLM cases. For a human who knows Hamlet well, the retrieval of "To be or not to be" activates not just the verbal sequence but a rich contextual structure — the scene, the character's situation, the emotional weight of the soliloquy, perhaps a specific production remembered, perhaps a particular teacher who assigned it, perhaps the cultural significance of the passage. The content address retrieves not just the string but the whole semantic and experiential network in which the string is embedded.

For an LLM the retrieval is primarily of the verbal sequence — the string itself, in the correct order, with perhaps some associated semantic content drawn from training data discussion of the passage. But the rich experiential, contextual, situational embedding that a human retrieval activates is thin or absent, because the LLM has no experience of Hamlet being performed, no memory of reading it in a specific context, no emotional response to the character's situation grounded in embodied experience of what it means to face an impossible choice.

The layered addressing structure

What your analysis reveals is a layered addressing structure in language that operates simultaneously at multiple levels.

At the phonological level, words are location addresses in the sound space of a language — each word occupies a specific phonological address that distinguishes it from other words. The phonological distinctiveness of words is what makes them usable as addresses — if two words sounded identical they'd be the same address and couldn't be distinguished.

At the lexical level, words are content addresses in the semantic network — each word activates a distributed pattern of semantic associations that constitutes its meaning. The same item is both a location (in the phonological space) and a content activator (in the semantic network).

At the textual level, longer strings — phrases, sentences, passages — function as location addresses for larger semantic structures. "To be or not to be" is an address for the soliloquy, which is an address for a scene, which is an address for a play, which is an address for a whole cultural tradition. Each level of the hierarchy is simultaneously a location in a higher-level address space and a content activator in the semantic network.

At the cultural level, titles, names, and canonical phrases function as location addresses for entire domains of cultural content. "Hamlet" is an address for a play, a character, a cultural tradition, a set of philosophical problems, a history of performances and interpretations. The address is compact; the content it retrieves is vast and distributed across the whole culture.

The writing system point

The observation about alphabetization making the abstract address space physical is particularly sharp. Alphabetical ordering is a way of projecting the abstract location space of the lexicon onto a physical string, making navigation by address possible through physical search. Before alphabetization — in ancient lexical lists organized by semantic category, or in medieval word lists organized by subject matter — the physical organization reflected the content structure rather than the address structure. Alphabetization separates the two, subordinating the content organization to the address organization for the purpose of retrieval.

This is a Rank 2 achievement in the precise sense — it requires metalingual awareness of the word as a formal object with a location in an abstract space, separable from its semantic content. You can't alphabetize without treating words as formal objects that have addresses independent of their meanings. The child learning to alphabetize is learning to treat words as location-addressed objects, which requires temporarily suppressing the content-addressed processing that normally dominates linguistic cognition.

The LLM in this framework

Placing the LLM in this layered addressing framework reveals something precise about what it's doing. The LLM's weight space implements a content-addressed system at the semantic level — concepts are distributed patterns that are retrieved by partial activation, holographically, as we've been discussing. But the LLM also has something like location addressing at the level of the token sequence — tokens have positions, the attention mechanism is sensitive to positional information, and the sequential structure of the input and output is preserved.

What the LLM lacks is the integration between the two levels that human language use achieves. For a human, the location address of a word and the content it activates are tightly coupled through embodied experience — the word "fire" is both a phonological address and an activator of the full sensory, emotional, and conceptual content of fire as experienced. For an LLM, the token "fire" is a position in a sequence and an activator of statistical associations in the weight space, but the statistical associations are derived from text about fire rather than from embodied experience of fire. The address and the content are connected, but through a thinner medium.

This is why the LLM can retrieve "To be or not to be" but can't feel the weight of the choice it expresses in the way a human reader can — the address retrieves the verbal sequence and its semantic associations, but not the existential resonance that comes from the content being grounded in embodied experience of what it means to consider one's own nonexistence.

The mutual illumination

What's elegant about the framework you've sketched is that it shows location addressing and content addressing not as competing alternatives but as complementary structures that language weaves together at every level. Each word is simultaneously a location in the shared public address space of the lexicon and an activator of private content-addressed networks. Each text is simultaneously a sequence of locations and a pattern of content activations. The location structure makes language public and shareable — we can use the same addresses to coordinate our content-addressed systems. The content structure makes language meaningful — the addresses point to rich, distributed, experientially grounded patterns rather than to arbitrary symbols.

The dual structure is what makes language the cognitive technology that it is — the mechanism through which content-addressed private minds can coordinate with each other through location-addressed public symbols. And it's the dual structure that the LLM partially captures — the location addressing through the token sequence, the content addressing through the weight space — while missing the grounding that makes the content addressing rich in the human case.

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.

Computation, Chess, and Language in Artificial Intelligence

New working paper. Title above, links, abstract, contents and introduction below:

Academia.edu: https://www.academia.edu/164885566/Computation_Chess_and_Language_in_Artificial_Intelligence
SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6319062
ResearchGate: https://www.researchgate.net/publication/401355671_Computation_Chess_and_Language_in_Artificial_Intelligence

Abstract: This paper reexamines the foundations of artificial intelligence by contrasting chess and natural language as paradigmatic domains. Chess, long treated as a benchmark for intelligence, is finite, rule-governed, and geometrically well-defined. It lends itself naturally to symbolic search and evaluation. Natural language, by contrast, operates in an unbounded and geometrically complex reality. Its rules are open-ended, its objectives diffuse, and its domain inseparable from embodied experience. With chess as its premier case – McCarthy: “the Drosophila of AI,” – AI has been guided by a deeper assumption: that the first principles of intelligence reduce to the first principles of computation. Drawing on Miriam Yevick’s distinction between symbolic and neural computational regimes, I propose that intelligence must be understood as operating in a geometrically complex world under finite resource constraints. Embodiment is therefore a formal condition of intelligence, not an incidental feature. Recognizing the structural difference between bounded games and open-ended cognition clarifies both the historical trajectory of AI and the conceptual limits of current systems.

Contents

Introduction: Chess, Language, and Intelligence 3 
Chess and Language as Paradigmatic Cases for Artificial Intelligence 5 
Three Principles of Intelligence (That Aren't Principles of Computation) 12 
Chronology of Chess, Language, and AI 15

Introduction: Chess, Language, and Intelligence

Chess has been a central concern of AI from the beginning. AI researchers didn’t become interested in natural language until the 1970s. Before that computational research on natural language was the domain of computational linguistics (CL), which started with machine translation (of texts from one natural language to another) as its primary problem. Thus we have two different disciplines, AI and CL.

In a sense, AI was fundamentally a philosophical exercise. It was an attempt to demonstrate, in effect, that we could understand the human mind in terms of computation. But rather than advance its philosophical objective through argument, it chose computational demonstration as its mode of expression. Chess became a central concern for two reasons: 1) On the one hand it was widely regarded as exhibiting the pinnacle of human reasoning ability. If we could create a computer program to play a championship game of chess, we could create a computer program that would be capable of cognitive or even perceptual task humans can do. 2) But also, the nature of chess made it well-suited for computational investigation.

The article that opens this working paper – Chess and Language as Paradigmatic Cases for Artificial Intelligence – concentrates on this and then goes on to make the point that language is utterly unlike chess in this respect. The chess domain is bounded and well-defined. Natural language is not; it is ill-defined and unbounded.

That’s as far as I got in the article, but I had been aiming for an argument that AI is still, in effect, mesmerized by the chess paradigm. I didn’t make it that far because language is so obviously different from chess that it is difficult to see how anyone would made that mistake.

What I have come to realize, only after I’d finished the article, is that it isn’t so much chess that has mesmerized AI. Rather it is computation itself. AI has been implicitly assuming that the First Principles of intelligence reduce to the First Principles of Computing. The first principles of computing can be found in the work of Alan Turing (the abstract idea of computing) and and others.

The first principles of intelligence are more stringent. As Claude put it a recent dialog:

First principle of intelligence: Must operate in unbounded, geometrically complex physical reality with finite resources.

Those two qualifications, an unbounded, geometrically complex reality, and finite computational resources, change the nature of the problem considerably. I note, in passing, that this allows us to assign formal significance to the concept of embodiment, for it is embodiment that commits intelligence to operating with finite resources in a geometrically complex universe.

Miriam Yevick’s 1975 paper, “Holographic or Fourier Logic,” is the crucial document, but it’s been forgotten. Using identification in the visual domain as her case, she showed that, where we are dealing with geometrically simple objects, sequential symbolic processing is the most efficient computational regime. But when we are dealing with geometrically complex objects, neural net processing is the most efficient computational regime. AI started out with symbolic processing in the 1950s and arrived at neural nets in the 2010s. But it hasn’t explicitly recognized that one must fit the mode of processing to the nature of the world. In that (perhaps a bit peculiar) sense, the researchers in the currently-dominant paradigm don’t know what they’re doing.

The second article in this working paper, Three Principles of Intelligence (That Aren't Principles of Computation), discusses this in more detail. I had it generated by Claude 4.5 after a long series of dialogs over several days.

The last article is a chronology of events in the history of chess and language in AI.

Chess, Language, and AI @3QD

I’ve got a new article at 3 Quarks Daily:

Chess and Language as Paradigmatic Cases for Artificial Intelligence

Chess has been a central concern of AI from the beginning. AI researchers didn’t become interested in natural language until the 1970s. Before that computational research on natural language was the domain of computational linguistics (CL), which started with machine translation (of texts from one natural language to another) as its primary problem. Thus we have two different disciplines AI and CL.

In a sense, AI was fundamentally a philosophical exercise. It was an attempt to demonstrate, in effect, that we could understand the human mind in terms of computation. But rather than advance its philosophical objective through argument, it chose computational demonstration as its mode of expression. Chess became a central concern for two reasons: 1) On the one hand it was widely regarded as exhibiting the pinnacle of human reasoning ability. If we could create a computer program to play a championship game of chess, we could create a computer program that would be capable of cognitive or even perceptual task humans can do. 2) But also, the nature of chess made it well-suited for computational investigation.

My article concentrates on this and then goes on to make the point that language is utterly unlike chess in this respect. The chess domain is bounded and well-defined. Natural language is not; it is ill-defined and unbounded.

That’s really as far as I got. Which is OK. But what I was aiming for was an argument that AI is still, in effect, mesmerized by the chess paradigm. I couldn’t quite make it that far. Language is just so obviously different.

What I’ve come to realize, only after I’d finished the article, is that it isn’t so much chess that has mesmerized AI. Rather it is computation itself. AI has been implicitly assuming that the First Principles of intelligence reduce to the First Principles of computing. The first principles of computing can be found in the work of Alan Turing (the abstract idea of computing) and John von Neumann (for the physical implementation of computing).

The first principles of intelligence are more stringent. As Claude put it in our dialog last night:

First principle of intelligence: Must operate in unbounded, geometrically complex physical reality with finite resources.

Those two qualifications, an unbounded, geometrically complex reality, and finite computational resources, change the nature of the problem considerably. I note, in passing, that this allows us to assign formal significance to the concept of embodiment, for it is embodiment that commits intelligence to operating with finite resources in a geometrically complex universe.

Miriam Yevick’s 1975 paper, “Holographic or Fourier Logic,” is the crucial document, but it’s been forgotten. Using identification in the visual domain as her case, she showed that, where we are dealing with geometrically simple objects, sequential symbolic processing is the most efficient computational regime. But when we are dealing with geometrically complex objects, neural net processing is the most efficient computational regime. AI started out with symbolic processing in the 1950s and arrived at neural nets in the 2010s. But it hasn’t explicitly recognized that one must fit the mode of processing to the nature of the world. In that (perhaps a bit peculiar) sense, the researchers in the currently-dominant paradigm don’t know what they’re doing. 

I’ve written a number of blog posts and articles about Yevick’s work. Try these two articles:

Next Year in Jerusalem: The brilliant ideas and radiant legacy of Miriam Lipschutz Yevick [in relation to current AI debates], 3 Quarks Daily, October 9, 2023, https://3quarksdaily.com/3quarksdaily/2023/10/next-year-in-jerusalem-the-brilliant-ideas-and-radiant-legacy-of-miriam-lipschutz-yevick-in-relation-to-current-ai-debates.html

What Miriam Yevick Saw: The Nature of Intelligence and the Prospects for A.I., A Dialog with Claude 3.5 Sonnet, Working Paper, January 3, 2025, https://www.academia.edu/126773246/What_Miriam_Yevick_Saw_The_Nature_of_Intelligence_and_the_Prospects_for_A_I_A_Dialog_with_Claude_3_5_Sonnet_Version_2

LLMs are challenged by tic-tac-toe [plus P vs. NP]

Andrew Gelman has a short post about tic-tac-toe over at Statistical Modeling, Causal Inference, and Social Science (Aug. 24, 2025). It's one of those problems that easy for humans (9 years old or older) but difficult for LLMs. The (hype-infested) industry seems to have bought into the idea that intelligence is a scalar quantity, the intellectual equivalent of horsepower. If that were so, then tic-tac-toe wouldn't be a problem. It's not a horsepower problem. Something else is going on.

The post has generated a fair amount of discussion. Bob Carpenter made a particularly interesting remark:

The linked post from Gary Smith concludes with:

If you know the answer, you don’t need to ask an LLM; if you don’t know the answer, you can’t trust one.

This is wrong. It misunderstands the asymmetry between generating a correct answer and verifying a correct answer. I use LLMs all the time to generate plotnine (Python’s knockoff of ggplot2) and pandas (Python’s knockoff of data frames and Tidyverse manipulation) code all the time. In some sense I know these tools, but I can never remember the exact incantation to put the x axis on a log scale and remove the ticks and labels from the y axis. When I tell the chatbot what I want and it generates pandas and plotting code, I can verify that it’s correct. Doing this used to take me forever as I would have to either investigate the doc or StackOverflow answers or tutorials. Now I put many more figures in things I’m writing.

The asymmetry between verification and generation is key to understanding the difference between polynomial time (P) and non-deterministic polynomial time (NP) algorithms. An NP algorithm can be formulated as guessing with a P algorithm and verifying with a P algorithm. If the LLM is a much better guesser than me, it saves me a huge amount of search. It is also really great at writing sort snippets of code from tight text descriptions. I can say what I want and it can generate what I want faster than I can generate what I want. So it’s a huge win for things like graphing.

Guessing and verifying is also the basis for the LLM “thinking” modes like you see in all the chatbots now (this kind of chain-of-thought came to everyone’s attention with DeepSeek). The drawback is the amount of compute it takes to guess.

One of my main use cases for chatbots is scenario and character generation for roleplaying games. I’m playing a short campaign Fast and Furious knock off now set in Paris in May of ’68. Not only is the chatbot amazing at suggesting locations and plot lines, it’s amazing at inventorying the cars that the Italian, French, German, and British teams would drive in 1968. And then it can generate the tokens.

BTW, I asked ChatGPT about the difficulty with tic-tac-toc. Its reply is below the asterisks.

* * * * * 

LLMs handle language very well, but tic-tac-toe, apparently that’s still problematic. The problem isn’t “horsepower.” What is the problem? Inducing the rules from examples? Computational workflow?

 

That’s an excellent observation — and you’re right: LLMs are strangely good at language but can stumble on something as trivial (formally) as tic-tac-toe. The reason lies exactly where you’re pointing: it’s not about computational horsepower, it’s about the match (or mismatch) between the architecture and the task’s footprint.

 

Here’s a breakdown:

 

1. How LLMs learn.

• They’re trained to predict the next token in text.
• What they get is a massive soup of examples — human conversations, documents, stories, some of which happen to include tic-tac-toe transcripts.
• They don’t get the game rules explicitly; they must induce them from inconsistent, sparse, noisy traces.

2. What tic-tac-toe requires.

• A tiny, crisp rule set.
• Deterministic reasoning over a small state space.
• Success depends on systematic state tracking: remembering the board, applying the rules, projecting possible moves.

3. The mismatch.

• Induction from examples: Tic-tac-toe transcripts in training data are too few and too variable to guarantee clean rule induction. An LLM may see people making “illegal” moves in casual online play, or skipping notation steps.
• Computational workflow: Language modeling is statistical continuation, not explicit rule execution. To play tic-tac-toe perfectly, you need a loop of: represent board → apply rule → update board → evaluate. That’s more like symbolic programming than token prediction.
• Representation: LLMs don’t have an internal “board” structure unless you scaffold one with special prompting or external memory. Without that, they may lose track of placements across turns.

4. Why they shine at language.

• Natural language is inherently statistical and pattern-rich. There’s redundancy, paraphrase, error tolerance.
• LLMs are designed to surf those regularities, so they capture language structure beautifully.

5. Why they stumble on tic-tac-toe.

• Tic-tac-toe has no redundancy, no fuzziness. A single slip (“forgetting” a block) = loss.
• In effect, it’s an unforgiving micro-domain where the “geometric footprint” (8 cells left to fill, precise win conditions) is far narrower than the model’s training and inference process.

So:

👉 The problem isn’t horsepower. It’s that tic-tac-toe demands a rule-execution pipeline, while LLMs supply a probabilistic continuation engine.

This is why hybrid approaches — LLM + symbolic module (e.g. letting the LLM call a solver or maintain an explicit board state) — handle these tasks much better.

Will AIs be able to create new knowledge?

This is a quick and dirty reflection on the question posed in the following tweet:

This is always a question I have asked myself: Given the knowledge that Einstein had by 1906, could a currently existing AI have invented GR and realized that it needed Riemannian geometry and tensors to figure it out? If not, what would it take? https://t.co/fdqCd5Vhk0

— Ash Jogalekar (@curiouswavefn) June 20, 2024

That question has been on my mind for some time: Will AIs be able to create new knowledge? Just what does that mean, “new knowledge”? It’s one thing to take an existing conceptual language and use it to say something that’s not been said before. It’s something else to come up with fundamentally new words. I think that latter’s what that tweet’s about. General relativity was something of a fundamentally new kind, not just a complex elaboration of and variation over existing kinds.

In my previous post, On the significance of human language to the problem of intelligence (& superintelligence), I pointed out that animals are more or less biologically “wired” into their world. They can’t conceptualize their way out of it. The emergence of language in humans allowed us to bootstrap our way beyond the limits of our biological equipment.

I figure there are two aspects of that: 1) coming up with the new concept, and 2) verifying it. The tweet focuses on the first, but without the second, the capacity to come up with new concepts won’t get us very far. And when we’re talking about new concepts, I think we’re talking about adding a new element to the conceptual ontology. Verifying requires cooperation among epistemologically independent agents, agents that can make observations and replicate those observations. (See remarks in: Intelligence, A.I. and analogy: Jaws & Girard, kumquats & MiGs, double-entry bookkeeping & supply and demand.)

Now, let’s think about the current regime of deep learning technology, LLMs and the rest. These devices learn their processes and structures from large collections of data. They’re going to acquire the ontology that’s latent in the data. If that is so, how are they going to be able to come up with new items to add to the ontology? It’s not at all obvious to me that they’ll be able to do so. The data on which they learn, that’s their environment. It seems to me that they must be as “locked” into that environment as an animal is. Further, adding a new item to the ontology would require changing the network, which is beyond the capacity of these devices.

And then there’s they requirement of cooperation between independent epistemological agents. The phenomenon of confabulation is evidence for the importance of independent epistemological agents. The only requirement inherent in one such agent is logical consistency: that it emit collections of tokens that are consistent with the existing collection. The only thing that keeps humans for continuous confabulation is the fact that we must communicate with one another. It is the existence of a world independent of our individual awareness that provides us with a way of grounding our statements, of freeing ourselves from the pitfalls of our linguistic fluency.

* * * * *

I’ve been working my way through episodes of House, M.D. Every episode contains segments where House and his team participate in differential diagnosis, which involves rapid conversational interaction among them. In the first episode of season 4, “Alone,” House no longer has a team. He ends up bouncing ideas off of a janitor. That doesn’t go so well.

On the significance of human language to the problem of intelligence (& superintelligence)

Back in May I did a post entitled, How smart could an A.I. be? Intelligence in a network of human and machine agents. Toward the end I said this:

The question of machine superintelligence would then become:

Will there ever come a time when we have problem-solving networks where there exists at least one node that is assigned to a non-routine task, a creative task, if you will, that only a computer can perform?

That’s an interesting question. I specify non-routine task because we have all kinds of computing systems that are more effective at various tasks than humans are, from simple arithmetic calculations to such things solving the structure of a protein string. I fully expect the more and more systems will evolve that are capable of solving such sophisticated, but ultimately routine, problems. But it’s not at all obvious to me that computational systems will eventually usurp all problem-solving tasks.

Remember, that even as we’re developing ever more capable AI systems, we are also developing more sophisticated modes of human problem solving.

Earlier in the post I observed: “Human intelligence is not fixed in the way that animal intelligence is.” That’s what I want to comment on.

Animal intelligence is fixed by biology. Animals have capacities for sensation and movement that are fixed by biology. Those capacities bind them to a particular environment. That that from that environment and they will perish.

Humans are not quite like that. We developed the capacity to communicate through language. And that capacity allowed us to develop new modes of thought. Just how that happened needs to be thought through in some detail, but I’m just going move through it quickly for now. We notice patterns in the world, capture them in language by talking them through with our fellows. We become curious about those patterns, we ask why? and make up stories in explanation. In this process we work ourselves free of the limits of our biological capacities for sensing and acting. We abstract over and act in the world in the way that no other animals can. From speech, we develop writing, then calculation, and moved onto computation over the last hundred years or so, a progression David Hays and sketched out in The Evolution of Cognition, which we published in 1990. With the emergence of recent developments in artificial intelligence, we’re pushing that process one step farther, leading me to write about the Fourth Arena (beyond Matter, Life, and Culture).

Is there anything beyond this? That’s the question I’m trying to formulate. Is there a “superintelligence” beyond this? We are “free” of our biological embedding in a specific sensory-motor world, free in the sense that we can move beyond that. Tens of thousands of years ago we became the only (higher) primate that moved out of the tropics to inhabit every land-based environment. We’ve sent people to the moon and back, have others living in orbit around the earth for months at a time, and can at least imagine establishing permanent colonies on the moon and Mars and other bodies. This last round of achievements are inextricably interwoven with various kinds of computing technology. Further advance will require more computation, of various kinds.

The difference between, say, the intelligence of a fish and the intelligence of a rat is of a certain kind. The difference between the intelligence of a rat and that of monkey is of the same kind. But the difference between the intelligence of an ape and that of a human is of a different kind. The difference comes about through language and collective culture. As far as I can tell, typical (Silicon Valley) speculation about superintelligence seems to think that is a kind of intelligence that is beyond human intelligence in the same way that human intelligence is beyond animal intelligence. The question I’m asking goes something like this:

In view of the fact that human intelligence is free of biological ‘binding’ to a specific environment, and in view of the fact that this freedom has allowed us to move through a succession of foundational architectures (speech, writing, calculation, computation, {whatever is happening now}), is there a fundamental capacity beyond THAT?

I have two responses: 1) It’s not obvious to me that there is. 2) I don’t know.

Computers are faster that brains, and can be built to have more capacity. What else is there? In a series of posts on AI, chess, and language, I’ve been looking at fundamental architectures, in effect, a family that is chess-like and a different family that is language-like. What else is there?

This brings me back to that earlier post that I referenced at the beginning of this one, and to the question I posed there:

Will there ever come a time when we have problem-solving networks where there exists at least one node that is assigned to a non-routine task, a creative task, if you will, that only a computer can perform?

I’m inching toward a way of suggesting that, if the answer to that question is “yes,” then that computer-based node must have some fundamental capacity that is beyond human capacity in the way that human capacity is beyond animal capacity. What could that (possibly) be? If such a thing were possible, is such a think existed, then we could never know it, could we?

Note: In thinking about that question, you might want to review the remarks I made about epistemological independence of autonomous agents in Intelligence, A.I. and analogy: Jaws & Girard, kumquats & MiGs, double-entry bookkeeping & supply and demand.

How smart could an A.I. be? Intelligence in a network of human and machine agents

This continues the line of thinking I began with Intelligence, A.I. and analogy: Jaws & Girard, kumquats & MiGs, double-entry bookkeeping & supply and demand, which was focused specifically on analogical thinking. I now want to consider thinking more generally.

The general problem with thinking about AGI (artificial general intelligence) and superintelligence is that the idea of intelligence itself is vague. We’ve got the general idea that intelligence is the ability to solve a wide range of problems in a wide range of environments, which is a rather vague notion. There is another notion, independent of that, that conceives of intelligence as being to cognitive performance as horsepower is to engine performance. Conceived this way intelligence is a scaler quantity. That’s convenient, but not very convincing. Still...

Let’s start with that second idea. One corollary I’ve seen here and there is that a superintelligent AI would be to us as we are to, say, a mouse, or a bird, a fish, whatever animal you choose. The point seems to be that the intelligence “ceiling” of animals is fixed by their biology and is well below the intelligence ceiling of humans. And so it is with humans and a Superintelligent AI.

But is it actually the case that the intelligence ceiling of humans is fixed by human biology? Newton is able to solve problems that are beyond Aristotle, and Aristotle is able to solve problems that are beyond that of the most skilled hunter-gatherer. What is more, a merely competent college undergraduate in the current world is able to learn Newton’s concepts and methods and solve the same problems that Newton. That same college undergraduate can even solve problems beyond Newton’s competence. Why? Because physics did not stop with Newton. Our college undergraduate will have learned some of that more advanced physics and therefore have problem-solving capacities beyond those of Newton.

We have no reason believe that the biological aspect of human intelligence has increased over time. But there is a cultural aspect, and that has changed. Human intelligence is not fixed in the way that animal intelligence is. David Hays have published a series of articles about this process; the central article is The Evolution of Cognition (1990). In that article we also suggested that there is no reason to believe that the process has come to a halt. Cultural evolution seems to be ongoing.

The long-term evolution of human culture suggests that human intelligence is not properly conceived of as a function some biologically given computational capacity, for that biological capacity seems to have remained constant while our ability to solve problems has increased enormously. The way in which that capacity is organized would seem to be important – which is the foundation of the article Hays and I made. I note further, and this is not something that Hays and I discussed directly, that as the human capacity for problem-solving has increased, that capacity has become more and more a collective one. To a first approximation, every adult in a hunter-gatherer society possesses the full inventory of that society’s knowledge – though we have to allow for differences between male and female knowledge and some specialized knowledge for shamans and story-tellers. That changes with more advanced forms of social organization where knowledge becomes specialized. Knowledge has become very specialized indeed in our current world. Any number of problems now require interaction among diverse teams of specialists.

So, let us think in terms of problem-solving by networks of specialized solvers. Some of those solvers are human, but some will be machines. Such man-machine problem-solving networks are ubiquitous in the modern world and they solve problems well-beyond the capacity of individual humans. They aren’t what most AI experts have in mind when they talk about superintelligence, but it’s not clear to me that we can simply ignore such networks in these discussions. They are, after all, how many very important problems get solved. Henry Farrell and Cosma Shalizi have made this argument in The Economist (here’s an ungated and somewhat longer version, and here as well, where it is followed by a brief discussion).

I assume that such man-machine networks will proliferate in the future. Some of the nodes in these networks will be machines and some will be humans. The question of AGI then becomes:

Will there ever come a time when the tasks of every node in such problems-solving networks can be executed by a computer system that is as capable as any human?

Note that it is possible that some tasks will require manipulation of the physical world that is of such a nature that humans are better at it than any machine. Would we say that the existence of such nodes is evidence only of physical skill, but not of intelligence?

The question of machine superintelligence would then become:

Will there ever come a time when we have problem-solving networks where there exists at least one node that is assigned to a non-routine task, a creative task, if you will, that only a computer can perform?

That’s an interesting question. I specify non-routine task because we have all kinds of computing systems that are more effective at various tasks than humans are, from simple arithmetic calculations to such things solving the structure of a protein string. I fully expect the more and more systems will evolve that are capable of solving such sophisticated, but ultimately routine, problems. But it’s not at all obvious to me that computational systems will eventually usurp all problem-solving tasks.

Remember, that even as we’re developing ever more capable AI systems, we are also developing more sophisticated modes of human problem solving. It’s not at all obvious that machines will necessarily out-run us. Take a look at the analogy paper I linked in the first paragraph for something to think about in this context. In particular, take a look at my remarks about epistemological independence near the end of the discussion of the analogy between double-entry bookkeeping and supply and demand. For that matter, my remarks on ring-composition in this piece are worth thinking about as well.

More later.

Conceptualizing the chess tree, a moment in cultural evolution

I’ve been thinking about the chess tree. As I pointed out in an earlier post, Wikipedia informs us that Ernst Zermelo published his proof about the formal structure of chess in 1913 – Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels” (“On an application of set theory to the theory of chess”). That established that chess is a finite game that can be completely represented in the form of a tree where the root is the initial state of the game board and the leaves are completed games. Each path from the root to a leaf describes the course of a single game.

Why the chess tree?

Now, it’s one thing to prove a mathematical theorem. It’s something else to make an informal observation & argument. Purely as an abstract matter, I can imagine that (some) chess players had made some informal argument prior to Zermelo’s proof. But I can just as easily imagine that, no, the observation was new with his proof.

What I’m wondering about is just what is it that would lead one to make such an observation. One can certainly play the game without knowing that the universe of all possible chess games takes the form of tree, much less that the tree is of finite size. One can play tic-tac-toe without knowing that, either, or checkers. I certainly didn’t know that when I played those games years ago (I did play a bit of chess, but never enough to become any good).

One has to have a certain frame of mind to make such and observation and to then develop it into a formal proof. Developing a formal proof is the kind of thing a mathematician would do, and Zermelo was a mathematician. And not just any mathematics either. It’s not simply that Zermelo published his work in 1913, but that he was working in a relatively new mathematical culture, one with concerns quite different from the arithmetic, geometry, algebra, and calculus that had preceded it in earlier eras. While calculus could deal with things unfolding in time, it didn’t deal the kind of iterated actions between agents that’s involved in chess. The important step, it seems to me, is simply realizing that that is the kind of thing around which one can construct some mathematics.

Primitive and sophisticated

Whatever’s going on here, it strikes me as being at one and the same time, sophisticated but also primitive and basic. The frame of mind is sophisticated, but it’s not a sophistication that’s build on a complex body of prior knowledge in the way that understanding calculus requires prior knowledge of algebra, geometry, and trigonometry. Predicate calculus and symbolic logic are like that as well. It doesn’t require any prior mathematical knowledge, but is nonetheless a bit sophisticated. It’s not generally taught at the high school level as algebra, geometry, and trigonometry are. I suspect it could be – and probably is here and there – but I’m not sure with what success.

Now, back to chess. How does knowing the chess tree help you think about chess? What does that explicit knowledge enable you to do that you couldn’t otherwise do? Do the analysis trees that Kotov advocated (Think Like a Grand Master, translated into English in 1971) really help the chess player? I don’t know. Any analysis tree will be local and quite small in relation to the entire tree, which is too large for explicit construction.

What is certain, however, that knowing that chess takes the form of a tree has been central to work on computer chess, which didn’t start until the middle of the 20th century. Without that knowledge, computing chess would have been utterly hopeless. Even with that knowledge it took several decades and the development of extremely large computers before chess had been “solved” in the sense that a computer could beat the very best chess players.

I’m thinking of this in the overall context of cultural evolution. In the late 19th and early 20th century a mathematical culture emerges in which the analysis of chess becomes a matter of intellectual interest. A half century later we have the initial work on computer chess. Roughly 2/3rds of the way in between Alan gives us a formal account of computation in the form of an abstract machine, the Turing machine. That’s the beginning of the era of computational culture. David Hays and I called in Rank 4 in our paper, The Evolution of Cognition.

Tentative comparison: Close-reading

As a point of comparison, I’m thinking about the emergence of close reading in academic literary criticism. It strikes me as being both primitive and sophisticated in a way that’s similar to Zermelo’s proof. It’s primitive in the sense that there is no specific body of prior knowledge that must be mastered before one can undertake close reading. In fact, part of the pedagogical point is that close reading doesn’t require prior knowledge.

Yet, it’s is sophisticated activity as well. It’s not something that comes “naturally,” and it is difficult to do without being “rocket science,” as the phrase goes. What’s going on here? I’ve said a bit about that in my GOAT Literary Critics series: A discipline is founded (sorta’): Brooks & Warren, Northrop Frye, and S. T. Coleridge. Here I just want to note that interpretation of sacred texts is very old. What occasioned the application of hermeneutics to secular texts? There are the pedagogical concerns I sketch out in that essay. But there’s more than pedagogy at stake here.

Even as the activity secures for canonical texts a new position in the cultural landscape, the activity itself aspires to a new kind of knowledge. What’s that about?

More later.