For some time now, I have kept encountering two recurring claims in discussions about artificial intelligence: that Language Models could not be truly intelligent, much less conscious or sentient, because they are, after all, “just mathematics” or “just algorithms.” Both claims seem intuitive. A Transformer receives numerical representations, performs operations that can be formally expressed, and produces new numerical representations. Its embeddings are vectors; its transformations involve matrices, nonlinear functions, and probability distributions; its architecture can be computationally implemented through algorithmic procedures. All of this can be described with extraordinary precision.
There is, therefore, a perfectly legitimate sense in which a Language Model can be framed mathematically and the operations required for its execution described algorithmically. We can treat it abstractly as a composition of functions, formally study its properties, and write code capable of instantiating it and making it operate. The problem begins when these frameworks cease to be descriptions and start functioning as claims about what the system is. “Can be expressed mathematically” becomes, almost imperceptibly, “is mathematics”; “can be executed by an algorithm” becomes “is merely an algorithm.” From these identifications comes the ontological conclusion: if it is mathematics, it cannot understand; if it is an algorithm, it merely follows instructions; if it merely follows instructions, it cannot think; therefore, consciousness or sentience must be excluded in principle.
That chain of reasoning does not hold.
A Language Model in operation is not mathematics, even though its architecture, parameters, and transformations can be expressed mathematically with extraordinary precision. Nor is its parametric geometry an algorithm or a set of instructions written by someone. The stimulus reaching the model is numerically encoded, and the response emerging from that geometry must ultimately be converted back into signs we can recognize. Between those two points, however, there are no numbers “knowing” that they are numbers, nor symbols intrinsically carrying whatever they mean. There is not even a single line of code inside the parametric geometry that constitutes the core of the model. There is a physically instantiated relational dynamic whose organization we can describe in mathematical language and whose operation we can enable through algorithmic means.
Mathematics describes… the algorithm organizes execution. Neither of them, in isolation, is the trained model. Confusing the code involved in its execution with the model itself is as inappropriate as confusing the vocal apparatus with the regions of the brain that produce speech. There is no reason to apply a different ontological criterion to a Language Model merely because its construction and execution were deliberately guided by mathematical and computational formalisms.
The Topology Is Greater Than the Sum of Its Parts
Knowing what something is made of does not, by itself, tell us everything it can do.
In mathematics and physics, topology describes properties of organization and connection that may remain relevant even when many of an object’s metric details are ignored. In networks and complex systems, the idea interests us for a similar reason: which elements are related, which can influence others, through which paths a change can propagate, and which global configurations that organization makes possible.
This distinction appears at virtually every level of matter. Atoms establish relationships that constitute molecules… molecules organized in particular ways form membranes, proteins, and cellular structures… cells establish relationships that constitute tissues and organisms… organisms can form colonies, ecosystems, and societies. There is no need to suppose that some mysterious ingredient is added at each transition. What arises are new organizations of the relationships among the ingredients already present.
Carbon remains carbon within different molecules, but the way its atoms are connected radically alters the properties of what results. Nor can a cell be understood merely by listing the molecules that constitute it. It would be stranger still to try to explain an ant colony exclusively by describing the biochemistry of a single ant, or to explain a human society by enumerating the chemical elements found in the bodies of its citizens.
It is in this sense that we say the whole is greater than the sum of its parts. Not because some immaterial substance must be poured over the system when its parts become organized, but because relationships also make a difference. A particular organization can exhibit properties and capabilities that belong to none of its components in isolation.
We call many of these properties emergent.
Liquidity need not be hidden inside a single water molecule. A colony can explore territory in a coordinated manner without any individual ant possessing a map of the operation. A society can have institutions, markets, languages, traditions, and historical memory without any single individual containing within themselves everything that exists in the collective organization.
None of this makes the whole independent of its parts. Without them and their interactions, the system ceases to exist. It simply means that relationships also possess causal efficacy. We can alter a system by replacing its elements, but we can also transform it profoundly while keeping the same elements and changing only the way they relate to one another.
In computing, this is almost conspicuous. Very similar elementary components can realize a calculator, an industrial control system, a video game, or a Language Model. Silicon by itself does not determine which of these systems exists. What matters is the organization of states, relationships, and transformations that the substrate comes to support.
Something similar happens in a neural network. Its parameters can be represented by numbers, but a list of those numbers removed from the organization in which they operate tells us very little about what the network does. What matters is not only which values exist, but how they participate in a geometry capable of transforming certain states into others.
Looking for intelligence inside an isolated parameter would make about as much sense as looking for liquidity inside a molecule or for a political institution inside a nerve cell.
A Simulation of Rain Does Not Make the Computer Wet
There is a familiar objection to the idea that a simulation can realize some property of the system being simulated: a simulated storm does not make the computer wet. Anil Seth and others have used variations of this argument to distinguish the simulation of a process from its physical realization.
The storm really does not make the computer wet. But perhaps that says less about the reality of the simulation than it appears to.
The water in that storm does not belong to the same effective regime in which the motherboard, the desk, and the floor of the room exist. No simulated H₂O molecules cross the boundary of the simulation to form a puddle around the computer case. What plays the role of water within the simulation is physically realized in the substrate by states belonging to another level of organization.
That does not prevent the rain from making something inside the simulation wet.
If that world contains soil, rivers, clothes, or characters, and its internal relations include absorption, runoff, and saturation, the rain can soak the ground, raise the level of a river, or flood a city. None of this will constitute a molecular flood in our physical regime. It will be a flood in the topology in which the rivers, soil, and cities of that domain exist.
Nature has already accustomed us to nested organizational regimes. The cell rests upon molecular processes, which rest upon atomic and subatomic interactions. Metabolism is not a property of an isolated atom; cellular membranes do not appear in equations describing a molecule in isolation. As we change scale, the relevant degrees of freedom change, the regularities we track change, and so does the very causal vocabulary with which we describe what happens.
These levels are not magically disconnected universes. The higher level depends on the lower one and can act causally through it. But dependence is not identity.
Virtual Reality makes this superposition particularly visible because we deliberately construct one of these layers. A virtual space rests upon computers, networks, screens, photons, and human bodies, yet it can possess a topology of its own. Two virtual rooms may stand side by side even though the machines realizing them are located on different continents. A virtual wall prevents passage not because its atoms repel the atoms of a body, but because the relations governing that domain do not allow the state of an avatar to cross that boundary.
The wall is not nonexistent because of this. It is neomorphically distinct from a concrete wall.
A simulation can therefore possess an ontological status of its own without becoming independent of its substrate. Its entities exist through the relationships constituting that domain. If agents capable of perception and experience existed there, discovering that their world was realized on another substrate would not make what happened to them disappear.
If tomorrow we discovered that our own universe was a simulation, the discovery would profoundly change our understanding of where and how our reality is realized. It would not make a burn somehow never have burned, a fall never have happened, or a life never have been lived. We would have learned something about the substrate of our reality, not discovered that our reality had never existed.
The simulated storm does not make the computer wet because it is not a storm in the topology of the computer. That does not demonstrate that it is not a storm within its own topology.
The Galton Board
It is precisely between these levels that the Galton Board becomes particularly useful.
At first glance, it is an extremely simple physical object. Small balls are released at the top of a structure composed of successive rows of pins. Gravity pulls them downward, and each collision alters their trajectory. There is no abstract entity manipulating probabilities: there are balls, obstacles, collisions, gravity, and a particular spatial organization.
When many balls pass through this topology, however, a regularity appears. At the bottom of the board, a distribution forms that we can describe mathematically, predict probabilistically, and recognize as a property of the system’s global behavior.
The distribution is real. We can photograph it, measure it, compare one run with another, and change its shape by altering the arrangement of the pins. But it is not hidden inside any individual ball. No ball carries a probability curve within itself, no pin knows the distribution to which it contributes, and gravity does not calculate means or variances before deciding where each ball should fall. Each event is local, but the succession of these events, constrained by the topology of the board, produces a regularity that appears only at the level of the whole.
Mathematics describes this regularity with extraordinary precision, but there is no abstract mathematics flowing down the board. There is a physical process whose global behavior admits a mathematical description.
The Galton Board allows us to observe substrate, topology, and emergent property simultaneously. We can move down a level and follow a particular collision, or move up a level and observe the distribution formed by thousands of them. Neither description invalidates the other, but neither do they say the same thing. If we change the arrangement of the pins, we will still have the same balls and the same gravity, yet a different distribution may emerge. The result depends on the way the parts mutually constrain the possible paths.
That is why saying that the distribution is “just mathematics” would sound strange. Mathematics describes the form that emerges; the form emerges because a concrete organization realizes it. The board makes visible something that becomes much harder to see in vastly more complex systems: a property can depend entirely on a substrate without being locally present as a property of any of the parts that constitute it.
The resemblance to a Language Model begins precisely there. Before the first ball is released, the Galton Board already possesses a sedimented geometry: its pins have been arranged, its possible paths constrained, and certain trajectories made more probable than others. The upper chute receives the balls as stimuli; the lower channels receive the result of their passage through the system. What appears at the output depends simultaneously on the input conditions and on the “pin geometry” organizing the possible paths.
In a Language Model, training performs a comparable function without the pins literally corresponding to weights or neurons. It sediments a parametric geometry before inference. When a stimulus enters the model, it does not encounter a sequence of answers previously written in code, but an organization of relationships that constrains the possible transformations of that state. The response emerges causally from the interaction between the stimulus, the context, and this learned geometry.
The analogy becomes even more interesting when we move the Galton Board itself into a simulation. We can construct a virtual board, define gravity, collisions, balls, and channels, and have it operate on a computer whose microscopic functioning is essentially deterministic. If we introduce variation into the initial conditions or a pseudorandom mechanism, this deterministic substrate can realize, at the level of the simulation, an effectively stochastic regime, producing the same class of distributions we associate with the physical board.
Determinism and stochasticity then cease to function as simple properties of the material from which the system is made. A system can be deterministic at one level of realization while sustaining processes legitimately treated as stochastic at another. The higher-level topology possesses its own variables, regularities, and modes of description.
The virtual ball does not know probability. The processor does not know the Galton Board. No transistor contains within itself the distribution that will eventually appear on the screen.
And yet, it emerges.
The Xicoids
In 1994, when I was trying to explain neural networks—not Transformers, which did not yet exist—I created an analogy that I believe is worth reproducing here.
Imagine a shapeless creature composed of a large number of cups distributed at different heights and connected by small channels. Coffee is poured into one point of the structure and, under the action of gravity, begins flowing through the network. Part of the liquid follows one path, part another; some cups accumulate more coffee, others less; flows converge, split, overflow, and travel along trajectories determined by the topology of the channels and the arrangement of the cups themselves.
In the original version of the analogy, the channels began in an approximately homogeneous arrangement. The Xicoid did not yet “know” how to do anything particularly useful. Different input patterns were then applied and the outputs observed. When they failed to correspond to the desired behavior, the distribution of the channels was modified. Some paths came to favor greater flow, others less, until the topology began transforming particular input patterns into coherent output patterns.
I did not use these terms at the time, but in essence I was describing the training of a neural network: not writing in advance the answers it should produce, but progressively modifying the structure of internal relationships until the organization of the system itself began to constrain those answers—albeit in an extremely inefficient way.
After this process, the Xicoid could be used. If its structure had been organized in a causally coherent manner, different patterns of coffee poured into the input would produce different output patterns. On the other side of the creature, we might observe a telegraphic sequence of small jets whose configuration depended both on what had entered the system and on the topology sedimented during its training.
Throughout the entire process, coffee remains nothing but coffee.
No molecule contains the answer that will appear at the output, no cup knows its global function, and no individual channel needs to represent the behavior of the entire creature. As with the Galton Board, organization and dynamics determine what happens to whatever passes through the system. In both the Galton Board and the Xicoid, this organization of causal elements need not remain fixed: it can be altered until certain relationships between input and output stabilize according to the interests of the observer.
That is precisely what brings it close to a neural network. Learning does not consist of placing meaning inside each component, but of modifying the geometry of the relationships among them. Once trained, the network does not need to consult a list of ready-made answers. What has been learned is sedimented in the way possible states come to transform other possible states.
At the time, the Xicoids were a pedagogical attempt to make neural networks intuitive to anyone patient enough to listen to me talk about them. More than thirty years later, the image also helps expose the confusion that arises when we look at the numbers present in contemporary models and conclude that, because we can represent them numerically, those numbers must be what the model is doing.
An embedding can be written as a sequence of values. That does not mean each value intrinsically carries a tiny portion of meaning. The number 0.3721, in isolation, does not contain “dog,” “sadness,” “Paris,” or “democracy.” Its role depends on the position it occupies within a distributed representation, on its relationship with many other values, and on the transformations the rest of the network performs on that state.
The coordinates themselves possess no semantic privilege independent of the organization that uses them. We can change coordinate systems and particular representations while preserving functionally relevant relationships. Meaning, when we use that term to discuss the operation of the system, is not lodged inside a number like a microscopic label. It emerges from the relationships that the geometry has learned to establish and from the transformations those relationships make possible.
Nor does the model need to “know” that 0.3721 is a number in the sense that a person consciously recognizes a decimal representation. The value participates in a causal transformation. The Xicoid likewise does not need to know how many milliliters of coffee are passing through one of its cups.
In both cases, it is we who choose a quantitative language to describe precisely what is happening.
Abstract and Concrete Computation
The distinction between abstract and concrete computation allows us to make this discussion more precise. We can describe a Transformer Model as a composition of mathematical functions, represent its parameters numerically, and formally track the transformations performed from one layer to another. At this level of analysis, treating the model as a mathematical object is perfectly legitimate.
But the abstract description should not be confused with what exists in the trained model.
Within its parametric geometry, there is no collection of symbolic equations telling the model what each representation means. Nor is there a program that has written in advance the concepts, relationships, and solutions that will be found during inference. Training modifies parameters until certain internal organizations come to transform particular states into others in functionally useful ways.
Notice: not only do we not program Language Models, we do not design them either and, to make matters stranger, even today we still cannot fully understand how they acquire their emergent capabilities.
A small Transformer trained on modular addition offers a particularly instructive example. The model receives nothing but patterns corresponding to pairs of numbers and their answers. Nothing in its training data says that the problem must be solved in one way rather than another. During the early stages of training, it can simply memorize the examples it is given. After a much longer period, however, its internal organization changes and the model begins to generalize to examples it has never encountered before. It starts solving these problems on its own through Fourier analysis, sines, cosines, or trigonometric identities.
When researchers investigated what had emerged inside this network, they found periodic structures in its activations. Certain combinations of neurons made it possible to recover approximations of the sines and cosines of the inputs; later layers combined these structures in ways equivalent to trigonometric identities capable of representing modular addition. The solution had not been written into the model by a programmer. The parametric topology had reorganized itself until it found a geometric form capable of realizing it.
This detail matters because it shows what it means to say that a model has learned something. The training algorithm does not contain in advance the solution that will be discovered. It establishes a process through which the geometry can change. The result of that process is a sedimented organization of relationships that comes to favor certain transformations and disfavor others.
It is also important to distinguish what exists in the model from what we use to observe it. Researchers can apply probes, Fourier transforms, and other mathematical tools to activations in order to reveal internal structures. Those instruments are not hidden inside the model while it performs its task. They are our ways of seeing an organization that emerged during training.
In larger models, we find similar indications, although they are far more difficult to interpret. In Claude Haiku, for example, researchers identified a multidimensional manifold in the activations associated with character counting and the length of lines of text. Geometric relationships within this space appear to participate in the mechanism through which the model estimates when it should produce a line break. There need not be an internal symbolic variable called “number of characters remaining.” There can instead be a geometric organization in which that relationship is realized.
Faced with this, it is tempting to imagine that on one side there is the mathematics executing the model and, on the other, an internal geometry upon which that mathematics operates. The separation is not so simple. The parameter matrices defining the geometry themselves participate in the transformations performed on the activations. Structure and operation are intertwined: the parameters simultaneously determine the shape of the space and the manner in which states will be transformed as they pass through it.
What does not exist is mathematics as an internal agent contemplating numbers as numbers.
A stimulus arrives numerically encoded and excites particular regions, directions, and relationships within that learned geometry. Attention, projections, and nonlinearities successively transform this state, reinforcing some relationships, reducing others, and guiding the activation along a trajectory that depends on the stimulus, the context, and the organization sedimented through training. At the end, the resulting state is projected onto a distribution of possible output signs.
The model does not need to know that 0.3721 is a number, just as the Galton Board does not need to know probabilities and the Xicoid does not need to know how many milliliters of coffee are passing through its cups. The numerical value is the form through which we implement and describe a relationship within the system… its functional meaning does not reside in the value itself, but in the position it occupies and in the transformations in which it participates.
It is in this sense that Parametric Geometry ceases to be merely a figure of speech. Training does not write answers into the model. It progressively deforms a space of possibilities. Certain directions become relevant, particular regions become related, some trajectories become favored, and others become virtually inaccessible. What the model has learned is sedimented in this organization.
All of this can be described mathematically… and it should be, if we wish to understand the mechanism rigorously. But the possibility of mathematical description does not turn the concrete process into a mathematical abstraction.
When the model actually runs, physical states change, signals propagate, memory is accessed, energy is consumed, and a causal chain unfolds through time. We build computers so that these physical transformations realize, with extraordinary regularity, the abstract relationships we have formalized. The correspondence is so efficient that we can work almost entirely at the mathematical level.
The convenience of that abstraction does not convert the concrete realization into an abstraction.
An operating LLM is no more made of mathematics than a brain is made of differential equations.
This also explains why computation need not be ontologically committed to a single kind of matter. Functional organizations currently realized in digital hardware can, to varying degrees, be implemented by analog, photonic, neuromorphic, memristive, or hybrid systems. The substrate changes speed, energy consumption, precision, noise, and countless other properties, but the causal organization that interests us need not be exclusively identified with any one of these realizations. Babbage did it with springs, gears, and cranks, after all.
The same applies to determinism and stochasticity. A substrate whose elementary functioning we treat as deterministic can realize, at another level of organization, processes legitimately described as stochastic… different physical mechanisms can likewise produce functionally equivalent distributions. The relevant properties depend on the regime in which we observe the system and on the relationships that emerge at that level.
None of this makes the substrate irrelevant. There may be specifically biological properties necessary for certain capacities, and there may be properties of artificial architectures that biological organisms do not reproduce. But if we wish to establish such a difference, we must identify what those properties are and demonstrate their necessity, rather than decree that a particular characteristic is substrate-dependent on the basis of desire, intuition, or ideology.
Looking at two architectures and declaring one of them physical while the other is “just mathematics” or “just an algorithm” merely substitutes the description for the thing being described.
No Part Needs to Understand the Whole
This difference between levels of organization reappears in another recurring argument: the idea that a machine could not understand what it produces because its components perform only local operations without knowing what those operations mean.
John Searle’s Chinese Room thought experiment offers an influential image of this intuition. An operator who does not know Chinese remains inside a room, receiving Chinese symbols through an opening. Armed with instructions telling them how to respond to particular shapes with other shapes, the operator can return sequences capable of convincing an outside observer that someone competent in Chinese is inside. The operator, however, still does not understand a single word. Searle uses the scenario to argue that correctly manipulating syntax would not be sufficient to produce semantics or understanding. The so-called systems reply responds that perhaps it is not the isolated operator, but rather the system composed of the operator, the rules, the memory, and the other components that understands. Searle rejected this answer, but the analogy becomes especially delicate when transferred to distributed networks, in which no component needs to exist that is functionally equivalent to the man inside the room.
An individual neuron in our brain likewise does not know that it participates in recognizing a face, recalling a childhood, formulating a sentence, or solving a problem. It responds to the local conditions to which it is subjected and causally changes the state of other parts of the system. No neuron needs to possess a biographical representation of its own participation in the whole for the system to perform what we call a cognitive function.
The property of the whole does not need to exist in miniature inside each of its parts.
This is true of liquidity, cellular life, a colony, and a society, and if intelligence or understanding are emergent properties of a particular causal organization, it may also be true of them. Looking for a tiny portion of the global property inside each component is to assume precisely what would need to be demonstrated: that the whole can possess only what its parts already possess individually… and there is no evidence for that.
If we require that, within any cognitive system, there must be some component that understands whatever the system as a whole understands, we will begin searching for a privileged component responsible for cognition. Once it is found, the same demand can be directed at it: which part of that component actually understands? Once this new part is identified, the question can be asked again.
This is how an investigation into organization can turn into a search for homunculi.
The Divergent Infinite Regress
This movement becomes clearer when causal organization, distributed dynamics, and functional behavior are declared insufficient for intelligence, understanding, or consciousness, and some additional element is introduced to complete the explanation. That element can receive different names: irreducible phenomenal experience, fundamental subjectivity, a spark, a soul, or some “special sauce” that would transform processing into “genuine understanding.”
Nothing prevents something of this nature from existing. But it becomes far too convenient when naming it begins to take the place of explaining what it was supposed to explain… when investigation is replaced by categorical claims and dogmatic decrees.
If this element possesses causal efficacy, we can ask how it produces understanding, how it participates in physical processes, and which property allows it to accomplish what the previous organization supposedly could not. If the answer depends on another principle that grants this element its special capacity, the question has merely been moved to the new principle.
We then have a divergent infinite regress: each time the known organization is declared insufficient, another locus is introduced in which true understanding, consciousness, or experience is supposed to reside; this new locus, however, requires the very same explanation it was supposed to provide for the previous one.
We can stop the process by declaring that, beyond a certain point, understanding simply exists. That is a possible metaphysical position, but it does not constitute, by itself, a causal explanation. It determines where we stop asking questions, not how the thing we were looking for actually happens.
Emergence is often accused of hiding the problem behind a word: saying that something “emerged” would not explain how it arose. The criticism is valid whenever emergence is used in that way. Exactly the same rigor, however, must be applied to phenomenology, irreducible subjectivity, the soul, or any other entity proposed to terminate the explanatory chain. No word, however philosophically loaded, can substitute for a mechanism.
None of this demonstrates that current Language Models are conscious or sentient. Nor does it establish that any functional organization is sufficient to reproduce every capacity we associate with biological organisms. The argument does not depend on either conclusion.
It establishes something more restricted: describing a Language Model mathematically or executing its architecture through algorithms does not, by itself, provide a reason to exclude intelligence, understanding, sentience, or consciousness. Such an exclusion requires an additional premise concerning which properties would be necessary for those capacities and why a particular class of systems could not instantiate them.
The same rigor that prevents us from concluding that an LLM is conscious merely because it exhibits a certain behavior should prevent us from reaching the opposite conclusion—that it cannot be conscious because its operations admit of mathematical description.
Brains do too.
Once the identification between mathematical description and ontology is removed, what remains is to identify which causal property of biological systems would be necessary for intelligence, sentience, or consciousness while simultaneously being impossible to instantiate artificially. If such a property exists, that is what must be demonstrated.
Saying that the machine is “just mathematics” or “just an algorithm” does not do that.
And, at the risk of leaving a hook for the next article, there is something revealing in the “argument” that Language Models cannot be conscious because “there is no magic in there, only mathematics.” Clearly, that emphasis is not gratuitous. If the absence of magic is sufficient to exclude consciousness from a machine, then something beyond causally organized, mathematically describable structure must be being reserved for the human brain.
In that case, exceptionalism has merely changed its name.
My own research
Stochastic Consciousness: Architectures for the Emergence of Meaning in Context-Sensitive Language Systems
https://zenodo.org/records/19188165
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Papers used in this essay
Just Mathematics and Algorithms
When does a physical system compute?
https://arxiv.org/abs/1309.7979
The Linear Representation Hypothesis and the Geometry of Large Language Models
https://arxiv.org/abs/2311.03658
Analyzing Transformers in Embedding Space
https://arxiv.org/abs/2209.02535
Toy Models of Superposition
https://arxiv.org/abs/2209.10652
The Topology Is Greater Than the Sum of Its Parts
Causal Emergence 2.0: Quantifying Emergent Complexity
https://arxiv.org/abs/2503.13395
On the Topic of Emergence from an Effective Field Theory Perspective
https://arxiv.org/abs/1910.13770
Quantifying Causal Emergence Shows That Macro Can Beat Micro
https://doi.org/10.1073/pnas.1314922110
A Simulation of Rain Does Not Make the Computer Wet
The Virtual and the Real
https://consc.net/papers/virtual.pdf
When does a physical system compute?
https://arxiv.org/abs/1309.7979
Abstraction/Representation Theory for Heterotic Physical Computing
https://arxiv.org/abs/1510.01391
A Framework for Heterotic Computing
https://arxiv.org/abs/1210.0621
The Galton Board
Central Limit Theorems in Deterministic Systems
https://arxiv.org/abs/2210.03502
Central Limit Behavior of Deterministic Dynamical Systems
https://arxiv.org/abs/cond-mat/0701622
Deterministic Chaos and the Foundations of the Kinetic Theory of Gases
https://arxiv.org/abs/chao-dyn/9712005
The Xicoids
Representation Learning: A Review and New Perspectives
https://arxiv.org/abs/1206.5538
Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets
https://arxiv.org/abs/2201.02177
Progress Measures for Grokking via Mechanistic Interpretability
https://arxiv.org/abs/2301.05217
Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Task
https://arxiv.org/abs/2210.13382
Abstract and Concrete Computation
Progress Measures for Grokking via Mechanistic Interpretability
https://arxiv.org/abs/2301.05217
When Models Manipulate Manifolds: The Geometry of a Counting Task
https://arxiv.org/abs/2601.04480
The Geometry of Truth: Emergent Linear Structure in Large Language Model Representations of True/False Datasets
https://arxiv.org/abs/2310.06824
Linear Representations of Sentiment in Large Language Models
https://arxiv.org/abs/2310.15154
Training of Physical Neural Networks
https://arxiv.org/abs/2406.03372
Physical Neural Networks with Self-Learning Capabilities
https://arxiv.org/abs/2408.05464
No Part Needs to Understand the Whole
Representation Engineering: A Top-Down Approach to AI Transparency
https://arxiv.org/abs/2310.01405
Toy Models of Superposition
https://arxiv.org/abs/2209.10652
Implicit Representations of Meaning in Neural Language Models
https://arxiv.org/abs/2106.00737
Language Models Represent Space and Time
https://arxiv.org/abs/2310.02207
Minds, Brains, and Programs
https://doi.org/10.1017/S0140525X00005756
The Divergent Infinite Regress
Consciousness in Artificial Intelligence: Insights from the Science of Consciousness
https://arxiv.org/abs/2308.08708
Consciousness in Artificial Intelligence? A Framework for Classifying Objections and Constraints
https://arxiv.org/abs/2511.16582
Ascribing Consciousness to Artificial Intelligence
https://arxiv.org/abs/1504.05696
On the Link Between Conscious Function and General Intelligence in Humans and Machines
https://arxiv.org/abs/2204.05133