Beyond Marginalism: What’s Next? [MR #12]

It is time to conclude my series of posts on Tyler Cowen’s monograph, The Marginal Revolution: Rise and Decline, and the Pending AI Revolution (2026). Let’s look at the fourth and final chapter, “Why Marginalism Will Dwindle, and What Will Replace It?” Here’s how Cowen opens it (p. 85):

The underappreciated news is that marginalism is on the way out. Furthermore, this is old news, though the trend is accelerating.

Most of all it is underdiscussed news. As economics continues to evolve, marginalist insights – probably of all different kinds – will lie ever further from the frontiers of research and knowledge.

I find it easy to imagine that – less than 20 years from now – marginalism will be viewed as a historical curiosity rather than a central analytical engine of economics. No one will quite come out and say that, nor will they present marginalism as false or destructive. Rather it will be seen as of limited relevance, much as we might view parts of the earlier classical economists, such as their expositions of the quantity theory of money. New and different analytical frameworks will replace the ones that have dominated neoclassical economics to date.

Think about that, think about it very carefully. When thinking about it remind yourself that Cowen named his blog, his virtual home base for the last two decades, after marginalism.

For a professional academic to say that the world in which they were trained, the structure of ideas within which they have worked, which they have nurtured in students, which they have communicated to the public at large, which they have come to love, to say that that world is slipping away into the past, man, that’s rough. And rare. Not many have been able to do it.

Back in 1946 the great physicist, Max Planck, remarked, “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” Thomas Kuhn referenced that remark in The Structure of Scientific Revolution, and the economist Paul Samuelson gave a compressed version in a 1975 article in Newsweek. It would appear that Cowen has gotten the message and decided that, rather than dropping dead, he’d give the new ideas a boost.

After that sobering opening, Cowen reviews what happened between the late 19th century and now. He lands on price theory. Price theory? – “the view that the basic intuitive economic concepts, as would be taught in intermediate microeconomics, are highly useful and for advanced problems too” (p. 91). There’s that word, “intuitive.” Cowen explains:

Your hypothesis should be intelligible in terms of microeconomic concepts that you can hold in your mind and understand. In most (maybe not all?) cases, you should be able to explain some version of those principles to a well-educated, non-economist onlooker.

A couple pages later we arrive at something called “Topkis’s Theorem” which is very mathy (p. 94). Two pages after that: “Economic intuition, RIP. And marginalism with it.” Whoops! “I am seeing the traditional, intuitive approach to economic reasoning retreating from one field after another. To give one vivid and also important example, machine learning and neural nets are overturning the world of finance.”

Modeling collective action with 360,000 factors

A couple of pages later Cowen gives us a striking example. It’s from something called Arbitrage Pricing Theory (APT) (pp. 99-100). It’s a model that uses machine learning to develop 360,000 factors and does a better job of predicting than traditional models have only five or six factors. However, the factors in the traditional models are derived from marginalist assumptions and make intuitive sense while none of those 360,000 factors are legible. It’s clear to Cowen that, in the current intellectual marketplace for economics, the unintelligible models with superior performance are out-competing the traditional marginalist models. Bye, bye, marginalism!

I see no need to comment extensively on this particular model as I’ve already given it a great deal of attention, generating two different working papers from it. The first, On Method: Computational Compressibility in Complex Natural and Cultural Phenomena, places it in the context of a half-dozen other investigations in a half-dozen fields in the social and natural sciences. The second, Notes on the Collective Valuation of "Thick" Objects: Financial Assets, Movies, and Novels, compares it with work that Arthur De Vany published in 2004, Hollywood Economics, and a more recent study by Matthew Jockers, Macroanalysis (2013), in which he investigated a corpus of 6000 19th century Anglophone novels. I’ve also written a blog post that complements that second paper: Thick Objects, High-Dimensional Models, and the New Intuitions [MR #11].

In that second paper and in the blog post I argue that those three cases are about a collective process where a population of human actors – traders and analysts in one case, movie goers in another, and novel readers in the third case – make judgements about “thick” objects. Even before I made an explicit argument, I had an intuition, an intuition that, despite the obvious differences, what De Vany was up to with movies was somehow like what Didisheim et al. were up to with stocks. Just where those intuitions came from, I can’t say, but I’ve been thinking about complex systems for a long time. [As an aside, for what it’s worth, Robert De Vany’s work on movies is perhaps where my interests in culture and cultural evolution come into closest contact with Cowen’s interests in economics and, in particular, in the economics of culture.]

As for the idea of thick objects, the term was suggested to me by either ChatGPT or Claude to characterizes complex objects whose characteristics cannot be fully enumerated because of that complexity. Moreover they are under constant scrutiny by a population of people who are interested in them and constantly evaluating them back and forth among themselves and, in that process, revealing further characteristics. It is not difficult to see that movies and novels are the same kind of thing, each is a mode of storytelling, and that they are complex objects. But what do they have to do with stocks? A remark by the pundit, Scott Galloway, made the connection for me in a podcast with Kara Swisher, “Stocks are like brands and that is they’re part promise and part performance.” Performance is assessed by a wide variety of metrics, metrics which go into the models such as the one by Didisheim et al., while promise is subject to endless speculation, some of which inevitably precipitates into those metrics.

Animal spirits, narrative economics, memes, and a Squid Game market

And that leads me to a conjecture that follows from the analysis that ChatGPT and I undertook in the collective valuation paper. Perhaps those 360,000 parameters are picking up traces left by those “animal spirits” that Keynes talked about. Their effect on asset values is too diffuse and indirect to be detected by those classical models with a half-dozen or so factors, each of which is intuitively legible on its own. But those traces show up distributed across those 360,000 parameters and allow the model to produce more accurate predictions. If that is what is going on, then I wouldn’t expect any of those factors to be intuitively legible, any more than one would expect such legibility of individual weights in a large language model. That’s not the nature of this conceptual world.

While we’re speculating, why not continue on? Those animal spirits can’t work their ways on the market by wafting around like odors in a breeze. They need to be embodied in some form, like gossip and stories. That leads us to Robert Shiller’s 2017 paper on “Narrative Economics” in the American Economic Review. Here’s his abstract:

This address considers the epidemiology of narratives relevant to economic fluctuations. The human brain has always been highly tuned toward narratives, whether factual or not, to justify ongoing actions, even such basic actions as spending and investing. Stories motivate and connect activities to deeply felt values and needs. Narratives “go viral” and spread far, even worldwide, with economic impact. The 1920–1921 Depression, the Great Depression of the 1930s, the so-called Great Recession of 2007–2009, and the contentious political-economic situation of today are considered as the results of the popular narratives of their respective times. Though these narratives are deeply human phenomena that are difficult to study in a scientific manner, quantitative analysis may help us gain a better understanding of these epidemics in the future.

Perhaps those high factor models are picking up the narrative dimension of asset value, which is a product how performance and promise become intertwined in the stories that analysts and traders tell themselves and one another about the assets they’re watching.

That, in turn, leads to the concept of meme stocks, a term that dates back to 2020. Here’s how Wikipedia characterizes them:

...a stock that gains popularity among retail investors through social media. The popularity of meme stocks is generally based on internet memes shared among traders, on platforms such as Reddit's r/wallstreetbets. Investors in such stocks are often young and inexperienced investors. As a result of their popularity, meme stocks often trade at prices that are above their estimated value – as based on fundamental analysis – and are known for being extremely speculative and volatile.

More recently, Owen A. Lamont, a senior analyst at Arcadian, has speculated that we’re in what he calls a “Squid Game market”:

Something’s happening in the U.S. stock market. We see cult stocks and crypto stocks. We see money pouring into leveraged single-stock ETFs and crypto ETFs. And we see dramatic price moves, for example in quantum computing stocks in December 2024. What’s going on?

Here’s one theory: these phenomena partly reflect an influx of Korean retail investors into the U.S. stock market. Last year, I wrote that “the U.S. stock market is Koreafying,” meaning that the U.S. market was starting to behave like the retail-dominated Korean market. What I didn’t realize was that this Koreafying process involves actual Korean retail investors.

He then goes on to develop the parallel between the Korean streaming series, Squid Game, and the U.S. retail market over the last few years.

If those high parameter models are picking up the effects of animal spirits embodied in gossip and narratives, then we’d expect their advantage over classical models (based on a handful of fundamentals) to be larger in the case of these meme stocks. So, if we compare the results of a classical model with those of a high-parameter machine learning model, are the assets with the greatest divergence also those otherwise identified as meme stocks? Perhaps some intellectual fishing expeditions are in order. Perhaps we can develop some new intuitions by comparing the results of classical models with machine learning models.

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)

Miriam Yevick now has a Wikipedia Entry

The entry is quite recent. I didn’t notice it until yesterday but, judging by the article’s history, it went live on January 1, 2026. It contains some information I wasn’t aware of, which is not at all surprising. The two most interesting pieces of information are that she began her career at the National Bureau of Standards in Washington, D.C. and that she had had an affair with David Bohm, the physicist. I knew about her extensive correspondence with Bohm I suspected that there had been an affair – judging from a remark she’d made, though I forget just where – but I didn’t actually know that.

What’s important is simply that she’s finally in Wikipedia. She wasn’t when I first mentioned her at New Savanna back in June 3, 2020. I’ve known about her work since 1978, when she made a comment on Haugeland’s article about cognitivism. Her 1975 article on Fourier logic became central to the article David Hays and I published about the brain, Principles and Development of Natural Intelligence, which is cited in the Wikipedia entry. In that article she considers two different kinds of computational regime, which she refers to as Fourier or holographic, and sequential. That distinction is fundamentally the same as the symbolic vs. neural distinction in current AI discourse.

That article is important because, and here I’m quoting from a remark Claude made in a recent discussion I had with it:

She doesn't take one computational system as object. She takes the relationship between two incommensurable computational regimes as object, and proves something about what the structure of reality requires of that relationship. She steps outside both regimes simultaneously and asks: given the kinds of objects that exist in the world, what must any adequate cognitive system contain? The answer — both regimes, necessarily, not contingently — is a proof about the space of possible cognitive architectures rather than a result within any particular architecture.

And that is why I’ve been mentioning her work whenever I have a chance. Until her work has been taken into account, the current debate is poorly formulated and incomplete, to put it charitably. A less charitable formulation would be that the debate isn’t intellectually serious. It’s mostly about intellectual ideology and commercially-oriented hype.

I take the fact that Yevick now has a Wikipedia entry as a sign that her work of 40 years ago may eventually recognized and extended.

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.

* * * * *

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.

Table of Cultural Ranks from August1981, A Five-Fold View

This post consists of a table that David Hays and I drew up in August 1981, nine years before we pubished our basic article on ranks theory: The Evolution of Cognition. That is to say, when we made this, we hadn’t really thought things through. Notice, first of all, that each table has five columns, one for each rank. We were thinking in terms of five ranks. When it came time to publish we couldn’t conceptualize a fifth rank in a way that satisfied us, so we dropped it from our accounts. Through interacting with chatbots I’ve come up with a way of thinking about Rank 5, but I want to set that aside for now.

While I have thought about updating this table with my current thoughts, that would take a lot of work now that I’m working with a much more sophisticated theory. The point of posting this table is siimply to show what a quick sketch looks like. It’s the form of the individual tables that is important, not the specific labels we placed in the cells back in 1981.

Summary by Rank

This table summarizes some, but not all, of the individual tables below. The ranks are columns, from left to right. The entries in the cells indicate what is new at that rank. Those entries are taken from the diagonals in the following tables. Some of the cells contain abbreviations, some of which are obscure. I could expand some, but not all of those abbreviations. 1981 was a long time ago. Don’t worry about it.

I’ve colored the rows for material that either or both of us, Hays and me, have more or less covered in one of our publications. The first row became the central argument in our cognitive evolution paper. “PlaceNo” is place notion, for the arithmetic we argued was central to Rank 3. “Code” is computer code, for the computation that is central to Rank 4. By “Network” we probably meant cognitive or semantic networks, which was the notation we used for knowledge representation, as it was known back then. These days it could almost, but not quite, be artificial neural network (ANN).

Marginalism is a Rank 4 idea, along with thermodynamics and biological evolution [MR #2]

Yesterday I made a post about a passage that occurs very near the end of Tyler Cowen’s new monograph, The Marginal Revolution: Rise and Decline, and the Pending AI Revolution. At that time I suggested that I might have more to say about the book, but I made no promises. Well, I’m saying more, and it looks like I’ll be doing a series of posts about the book, though I can’t say how long that series will be, perhaps only one more post, but maybe two, three, or even four more. Who knows.

The monograph is of particular interest to me for two reasons: 1) I’ve just posted a small monograph of my own, on the rise of the contemporary academic discipline of literary criticism: The Discipline of Literary Criticism: A Quixotic Essay about Thinkers, Methods and Authority. Literary criticism is a very different beast from economics. Still, now I have two case studies in the rise of an academic discipline. More deeply, 2) I now have a case study for the theory of cognitive ranks that David Hays and I developed in the last quarter of the previous century. That’s what I want to pursue in this post.

The theory of cognitive ranks, as readers of New Savanna may know, is about the development of ever more sophisticated cognitive architectures over the long haul of human history. Rank 1 cognition arises with the emergence of language between, say, 100 thousand and who knows how many years ago. Rank 2 thought precipitated around written language and Rank 3 around the introduction of arithmetic calculation into Europe in the Late Medieval, Early Modern period. Rank 4 began emerging late in the 19th century and began consolidating around, first the ideas and then the machinery, of computing in the 20th century.

My hunch was that marginalism in economics is a Rank 4 idea. Rather than work out the details myself, I put the question to Claude Sonata 4.6 extended. I’ve appended that conversation below. Just to be clear, I’m using my Claude account, not the AI facility that Cowen has made available with his book. One final note: You’ll see that at the very end of this conversation Claude mentions Rank 5. While Hays and I talked about a possible Rank 5, we never published about it because we couldn’t figure out how to conceptualize it. That changed for me in the last few weeks and I have been talking about it with both Claude and ChatGPT.

The rest of this post consists of my interaction with Claude, which was a simple one: I posted a long prompt to get things started and Claude responded.

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I have uploaded a PDF of a book by Tyler Cowen, The Marginal Revolution: Rise and Decline, and the Pending AI Revolution (2026). In it he recounts the rise of marginal thinking in economics starting roughly in the late 19th century up though 20th and on into the 21st, where it seems to be dissipating. I’m interested in understanding it in terms of the cognitive rank theory that David Hays and I have developed (see our paper, The Evolution of Cognition, in the project materials). The timing suggests it is a Rank 4 concept, along with thermodynamics and Darwinian evolution.

Though we don’t mention Piaget’s term, “reflective abstraction,” we use the idea, which is: the process by which the operations of one cognitive level become the objects of the next. Piaget was mostly interested in cognitive development in individuals, but he also applied his ideas to the history of ideas (e.g. Genetic Epistemology). That’s what I want to do with Cowen’s argument about the marginal revolution. My intuitive understanding is that the technical development of that idea presupposes a technical understanding of supply and demand. I want you to vet that idea.

I think that supply and demand is a Rank 3 idea. Here are some remarks I made about supply and demand in a blog post from 2023:

But how do we get to supply and demand? I'm not prepared to give a detailed answer to that question. [...] Here's the question I'd ask myself: What does double-entry bookkeeping have in common with supply and demand? Double-entry bookkeeping dates back to the late 15th century and is a practical discipline. What becomes visible by 'going meta over that practice?

Double-entry bookkeeping is a system for maintaining closure over a set of transactions that grow over time as transactions are added to the list. We've got closure and change over time. Supply and demand change over time. Where's the closure, the dynamic closure? How do supply and demand complement one another in a say similar to the balance of debit and credit accounts in double-entry bookkeeping? How was Adam Smith able to get "on top" of that in the way that previous thinkers could not? How does the famous metaphor of the invisible hand do its work?

With that in mind here’s Cowen’s basic statement: “Marginalism is the economic doctrine that we can best understand value by considering the question of how many units of a good or service an individual has, and using that starting point to ask how much an additional – or marginal – unit would be worth in terms of other goods and services.” Intuitively it seems to me that that presupposes that an understanding of supply and demand has become thoroughly routine.

Cowen’s paradigmatic example of marginal utility is the diamonds-water paradox:

If someone had to choose between having no water and having enough water to avoid dying of thirst, they would be willing to pay a very high price for water (the exact final price still would depend on how many suppliers were competing to sell you water and the cost of getting you the water, questions which also can be addressed using marginalism). But that is not the choice for most people. Most people have a fair amount of water, and they are paying for more water, or you might say water at the margin. And the value of water at the margin just isn’t that high. In American restaurants, many people don’t even drink the glass of water they get for free.

To consider diamonds, if you could receive plenty more diamonds by turning on your kitchen tap, diamonds probably wouldn’t sell for much in the market, just as water doesn’t. Of course, that is not the case and real diamonds remain relatively scarce, thus boosting their value. That said, circa 2025 competition from high-quality artificial diamonds, which can be made in the lab, is threatening to further depress diamond prices. Again, that is consistent with marginalism.

Your turn. Is my intuition valid? Either way, unpack it for me.

Your intuition is valid, and I think it can be made quite precise. Let me work through it carefully, using Cowen's text and the cognitive ranks paper together.

The surface complication

Cowen himself says something that might seem to cut against your claim. He writes (lines 68–79) that supply and demand "rely very directly on the ideas of marginalism" — that the derivations of supply and demand schedules "refer repeatedly to how much of a resource people currently have." So in the logical order, after the fact, supply and demand depends on marginal analysis.

But this conflates the logical order with the developmental/historical order, and the Piagetian point is about the developmental order. Historically, supply and demand came first — Adam Smith (1776), the classical economists — and the Marginal Revolution arrived nearly a century later (1871). Cowen himself acknowledges that the burst of economic progress following Smith was a real thing, and that progress from the ancient Greeks to the late 19th century was "glacial" before it suddenly accelerated. The takeoff point is 1871, not 1776.

The reflective abstraction argument

The Piagetian structure you're looking for is this: what were the operations that Rank 3 economists performed with supply and demand, such that Rank 4 could take those operations as objects? 

The Rank 3 economist works with supply and demand curves as given primitives. She asks: given a supply curve and a demand curve of such-and-such shape, where does the market clear? What happens to equilibrium price if demand shifts? How do taxes or subsidies alter the equilibrium? The curves themselves are taken as data — they're the tools of analysis, not the objects of analysis. This is exactly what you'd expect at Rank 3: algorithm and calculation applied to theoretical constructs, the constructs themselves unquestioned.

Marginalism makes precisely the Rank 4 move: it refuses to take the curves as given and asks what generates them. The demand curve has a characteristic downward-sloping shape — why? Because of diminishing marginal utility. The supply curve slopes upward — why? Because of increasing marginal cost. The market clears at a particular price — what does that mean at the individual level? That each buyer's marginal valuation equals price, and each seller's marginal cost equals price. Marginalism takes the supply-and-demand apparatus that Rank 3 thought with and turns it into an object of theoretical construction. In the exact language from your paper: "The new thinker can think about what the old thinker thought with."

The diamonds-water paradox is the clearest illustration of this. A Rank 3 economist can use supply and demand to explain why diamonds are expensive (scarce supply, high demand) and water is cheap (abundant supply, though high demand). That's a workable, practically useful analysis. What the Rank 3 economist cannot explain is why demand for water, despite its essentialness to life, doesn't translate into a high price in ordinary circumstances. The answer requires going meta on the demand curve itself: it's not about total value, it's about marginal value at the position where the individual currently stands. That's the Rank 4 move — taking the demand curve as object and asking about its internal structure.

The double-entry bookkeeping thread

Your intuition about the structural parallel is suggestive. Double-entry bookkeeping imposes closure on a system of transactions through a debit/credit complementarity. Supply and demand imposes closure on a market through price equilibration — the "invisible hand" as Adam Smith's metaphor for how individual decisions produce systemic balance without anyone intending it. Both involve what you might call dynamic closure: the system absorbs change (new transactions, shifts in supply and demand) while maintaining a global constraint (books balance, market clears). The transition from double-entry bookkeeping to supply-and-demand might itself be seen as a Rank 3 reflective abstraction: going meta on the bookkeeping closure principle to ask what maintains closure at the level of the entire market, not just a firm's ledger. [1]

On the relevance of intellectual history for understanding present events (AI)

Jim Olds, The Chronology Problem, Mar. 12, 2026.

We are surprisingly bad at knowing when things began.

I’ve been thinking about this for a while, partly because I lived through several of the transitions we now misremember. In 1987, I used the Internet for early text-based email, file transfers, and reaching colleagues at other universities. In August of 1991, in the face of an impending direct hit of Hurricane Bob, I moved all of my image data from Woods Hole to NIH in Bethesda in a matter of minutes. This was entirely unremarkable at the time. And yet when I mention it today, people often look mildly startled, as if I’ve claimed to have owned a smartphone in 1987. In their minds, the Internet began sometime around 1994 or 1995, when the Web arrived and made it visible to everyone. Before that, apparently, there was nothing.

Olds then goes on to say more about the (deep) origins of the web, artificial intelligence, climate science, and economics. Here's what he had to say about AI:

The field of artificial intelligence may be the most dramatic case study in collective chronological confusion we have. Most people who interact with today’s language models and image generators believe they are witnessing something genuinely unprecedented — a technology that sprang into being sometime around 2017. What happened is more complicated and more interesting.

The mathematical foundations for neural networks were laid in 1943, when Warren McCulloch and Walter Pitts published a paper describing how neurons could, in principle, compute logical functions. Frank Rosenblatt simulated a working perceptron at the Cornell Aeronautical Laboratory in 1958 — a system that could learn from examples. The 1986 backpropagation paper by Rumelhart, Hinton, and Williams, which most practitioners treat as a founding document, was itself a rediscovery and refinement of ideas that had been circulating since the early 1970s. Yann LeCun was training convolutional neural networks to read handwritten digits for the U.S. Postal Service in 1989. The architecture underlying those systems is recognizably the ancestor of what powers modern computer vision.

None of this was secret. It was published, presented, and in some cases deployed in real systems. What happened instead was a kind of institutional forgetting, accelerated by two “AI winters” — periods when funding dried up, interest collapsed, and computer science turned its attention elsewhere. Researchers who had spent careers on neural approaches moved on or retired. Graduate students who might have built on their work were instead trained in other paradigms. When the hardware finally caught up with the ambitions of the 1980s, around 2012, the rediscovery felt like a revolution. In some ways, it was. But the conceptual foundations were not new, and the people who had laid them got less credit than they deserved, partly because so many of the field’s new practitioners didn’t know they existed.

The practical cost here is the same as elsewhere: repeated investment in problems that had already been partially solved, frameworks that were novel mainly to their authors, and a set of origin myths that flatter the present at the expense of the past. The deeper cost is that we don’t understand what was tried and discarded and why — which algorithms were abandoned for reasons of computational expense rather than theoretical inadequacy, and which might be worth revisiting now that the expense has fallen.

To Olds’s list I would add Miriam Yevick's 1975 paper, Holographic or fourier logic, published in Pattern Recognition. Unfortunately that paper got lost as it didn't fit into either cognitive science or artificial intelligence. What she proved was the for one class of visual objects, those with a complex geometry, neural networks provided the best computational regime while for another class of objects, those with simple geometry, symbolic computation provided the best computational regime. That has a direct bearing on the current debate over whether or not new architectures involving symbolic processing are necessary.

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