Finite Cognition and
Infinite Reality

I The Problem and Conceptual Stratification

1.1 Perception–Cognition–Action: The Primitive Cycle of Human Cognition

As biological organisms, humans live in a physical world that is inexhaustible to any cognitive agent. Human cognition originates in the most elemental biological process: affective signals drive perception, perception drives behavior, behavior triggers changes in the physical world, and those changes produce observable results — thus humans acquire a complete informational chain spanning the entire process. Memory systems store these change-data, forming the raw material of cognition. This cycle of “perceive → cognize → act → observe results → update cognition” is the biological foundation of all higher cognitive activity.

However, the complexity of the physical world is inexhaustible to any finite cognitive agent, and human sensory channels are limited. Humans receive approximately 11 million bits per second of sensory information, yet conscious processing capacity is only about 50 bits per second. Even among information that successfully enters consciousness, only a tiny fraction is linguistically encoded; of the linguistically encoded information, only a tiny fraction is further committed to writing. Each step of transformation constitutes an irreversible information compression.

1.2 Conceptual Stratification: Seven Kinds of “Information”

This paper involves multiple tiers of the concept of “information.” To avoid ambiguity, an explicit conceptual stratification is established here:

This stratification serves as the terminological foundation for all arguments in this paper. Subsequent expressions such as “information loss” and “information gain” refer to specific tiers within this hierarchy, not to a generic concept of “information.”

1.3 The Four-Parameter Cognitive Matrix

The cognitive state of a human being can be described using the four quadrants of a two-dimensional matrix. The two axes are “known/unknown” (the epistemological dimension) and “identified dimensions/unidentified dimensions” (the ontological dimension):

The key distinction between this matrix and the Rumsfeld/Johari matrix (known knowns / known unknowns / unknown knowns / unknown unknowns) is: both axes of the latter are metacognitive dimensions, whereas this matrix introduces the ontological dimension of “identified dimensions / unidentified dimensions” — distinguishing between “not knowing the value of an existing variable” and “not knowing whether the variable exists at all.” These are two fundamentally different kinds of ignorance. The latter is precisely the kind of ignorance manufactured by the Level 0 definitional act.

All human cognitive activity can be modeled as movement among the four quadrants: science converts “state unknown × identified dimensions” into “state known × identified dimensions”; metaphysical thinking — the only cognitive activity capable of questioning the matrix’s boundaries themselves — attempts to discover new dimensions from the lower-right quadrant, leaping toward the identified region on the left.

1.4 Finite Cognition Facing Inexhaustible Reality

Past, present, and future variables are finitely known to any finite cognitive agent. Humans cannot be omniscient about all variables, and therefore require regularities expressed quantitatively to construct cognitive systems. But all cognitive systems are built upon finite observations of inexhaustible data and variable relationships.

II The Irreversibility Principle of Bounded Representation

2.1 The Programmatic Proposition

Any finite cognitive system must form a bounded representational space through selection and encoding. Computations performed within the established representational space — no matter how refined or voluminous — cannot guarantee recovery of dimensions excluded when the representational space was constructed. The validity of cognitive outcomes depends not only on the precision of internal reasoning but also on the appropriateness of the variable space and the system’s sustained connection to external reality.

The formal foundation of this principle derives from the Data Processing Inequality (DPI) in information theory: if random variables form a Markov chain X→Y→Z, then I(X;Y) ≥ I(X;Z) — any post-processing of data cannot increase the mutual information with the original source. Equality holds if and only if Y is a sufficient statistic for X with respect to Z. The Pitman-Koopman-Darmois theorem further proves that only exponential-family distributions admit minimal sufficient statistics of fixed dimensionality. Since the storage and processing capacity of both human cognition and AI are finite-dimensional, compression of non-exponential-family data necessarily entails recoverability loss at the Shannon information level.

2.2 Simpson’s Paradox and Related Phenomena: A Symptom Spectrum of Pipeline Information Leakage

If the above principle is correct — that compression of bounded representations under standard conditions leads to recoverability loss — then we should observe numerous systematic anomalies arising from this cause in statistical practice. In fact, these anomalies have appeared repeatedly throughout the history of statistics and have been independently discovered and named:

The six phenomena above share a common superordinate background — recoverability loss caused by representational compression — but their proximate mechanisms differ. Simpson’s Paradox arises from structural collapse when grouping variables are aggregated; Goodhart’s Law arises from a metric losing its measurement validity for the original concept after being socially gamed; the replicability crisis involves p-value selection bias and insufficient statistical power. Unifying them under the umbrella of “information loss” is legitimate, but their distinct mechanisms should not be blurred.

III The Level 0 Definitional Act

3.1 Operationalization: The First Operation of the Pipeline

The intellectual paradigm of statistics can be summarized as a stage-by-stage compression pipeline from definition to prediction: define and collect quantitative data, organize the structural relationships among the data, compress away the surface data layer to access deeper relational structures, strive to inductively summarize underlying structural relationships and express them mathematically, and ultimately produce quantitative predictions about the future.

The pipeline’s five-level structure: Level 0 (definitional partitioning: choosing what to measure / what not to measure) → Level 1 (sampling and collection) → Level 2 (compression and aggregation) → Level 3 (inductive modeling / formalization) → Level 4 (reverse projection / prediction).

Among these five levels, Level 0 is the most fundamental. Before data collection begins, the researcher must perform a precondition act: defining what to measure and what not to measure. “Intelligence” is defined as “IQ test score,” “health” is defined as “BMI,” “economic development” is defined as “GDP.” Whitehead termed such acts the “fallacy of misplaced concreteness” — projecting the ontologically inexhaustible complexity onto a finite-dimensional measurement space, then equating this projection with reality itself.

3.2 The Absolute Irreversibility of Level 0

The definitional act is not a technical flaw of statistics but its ontological precondition. The validity boundary of the statistical paradigm is drawn the moment it decides “what to measure.” The problem does not only occur after data collection — it also occurs before “what is permitted to become data.”

3.3 Internal Correctness Does Not Guarantee Boundary Correctness

The deeper implication of the Level 0 argument is: a model can exhibit extremely high goodness-of-fit within its variable space, pass all internal validation, and still make erroneous inferences about reality due to omissions in the variable space itself. Internal correctness does not guarantee boundary correctness — because the information needed to verify boundary correctness is precisely the information excluded by the definitional act.

IV Prediction and Temporal Structure

4.1 Cross-Sections and Longitudinal Sections: Two Epistemological Strategies

Ontology is a spatiotemporal continuum. Statistics and causal reasoning slice it from two different directions:

Descriptive association tends to proceed from the cross-sectional structure of state co-occurrence — selecting a cross-section on the time axis, freezing physical conditions, and assuming that variable relationships at that moment can be measured in isolation. Causal explanation must introduce the longitudinal structure of change, intervention, or generative mechanisms — tracing along the time axis or intervention axis “how a change in A causes a change in B.”

The two are not strictly dichotomous. Time-series analysis, longitudinal studies, and dynamic models also study change within the statistical framework; Pearl’s do-calculus encompasses intervention, counterfactuals, and structural invariance, extending far beyond temporal transmission. But as ideal types, this geometric contrast reveals a key tension.

4.2 The Temporal Misalignment of Prediction

After statistics completes its formalization, its mission is not to describe the past — but to predict the future. “Predicting the future” is itself a temporal projection act. You take a compressed formula derived from a cross-section and attempt to project it onto a future moment. This projection act forcibly introduces the temporal dimension — which is to say, it forcibly introduces a series of unverified causal assumptions: “past structural relationships will continue to hold in the future” is a causal assumption; “a change in independent variable X will cause dependent variable Y to change according to the formula” is a causal assumption; “no new confounding variables will emerge in the future” is also a causal assumption.

Therefore, predictive validity is not guaranteed by the pipeline itself but is parasitic upon the external ontological assumption of “structural temporal invariance.” A model’s effective time window equals the period during which the difference between the old variable structure and the new variable structure remains small enough to be negligible.

4.3 Causal Inference: An Operation That Crosses Boundaries

Pearl explicitly stated: causal concepts cannot be inferred from statistical associations, nor can they even be defined using statistical associations. Observational statistical data, in the absence of additional structural assumptions, cannot in principle answer interventional questions — this is not a limitation of data volume or model capacity, but a mathematical impossibility.

Causal inference is therefore not an advanced operation within the same representational space — it changes the system boundary itself: from a closed system to an open system, injecting causal assumptions, background knowledge, or interventional experimental data from beyond the boundary. But even after introducing external information, one can never confirm that “sufficient” external information has been introduced — causal models incorporate known relationships among variables, but cannot guarantee that the known relationships cover all relationships that hold.

V Structural Mapping Across Three Systems

5.1 The Shared Topology of Statistics, AI, and Human Cognition

Statistics is a set of formalized methods; AI is a technological system and training architecture; human cognition is an open system jointly constituted by biological, psychological, and social dimensions. The three belong to different levels. Yet they all involve selection, encoding, compression, and prediction, sharing the basic topological structure of bounded representational compression:

The three share a common bounded-representation topology — this is why they can be described in the same conceptual language. But their specific information mechanisms, degrees of openness, and modes of boundary reconfiguration differ. Statistics operates on entirely closed datasets; human cognition continuously receives new information through body-environment coupling; AI’s boundaries depend on its training paradigm and tool-calling capabilities.

5.2 AI’s Four Validity Boundaries

Taking large language models as the example, AI is constrained by four structural boundaries: the precision boundary (floating-point truncation deletes information beyond the decimal point), the sampling boundary (Top-K/Top-P zeroes out low-probability tokens), the definitional boundary (the tokenizer and corpus selection determine what “exists”), and the temporal boundary (the world after the training cutoff date constitutes unidentified dimensions). These four boundaries are not technical limitations but structural constraints determined by the pipeline’s topology. Increasing compute power merely creates a denser grid within the frame — it does not alter the frame itself.

5.3 AI’s Three-Level Metacognitive Distinction

Does AI possess metacognitive capability? This question requires stratified discussion:

AI can simulate metacognitive behavior within a given dialogue — because its training data contains abundant textual traces of human metacognition. But whether it can autonomously initiate structural questioning of its own representational space without external prompt triggering is a capability that current architectures have not yet adequately demonstrated. Reinforcement learning, by interacting with the environment and autonomously generating data to explore unknown variable spaces, and reasoning models, through chain-of-thought in latent space, may give rise to cross-boundary discoveries — therefore, treating AI’s metacognitive capability as a static upper bound is not warranted.

VI Metaphysical Compression and Description Complexity

6.1 The Uniqueness of Metaphysical Compression

Metaphysics is the limit compression and abstraction of all information known to the self. The philosophical products of human metaphysical compression — ontology, epistemology, methodology, information theory, systems theory, cybernetics — are all essentially exercises in organizing the informational relationships of the entire “perceive → cognize → act” cognitive process. The entirety of both Eastern and Western philosophical traditions are products of limit compression applied to different stages of the same pipeline. The underlying structures are similar because the objects of compression are similar: the structural constraints encountered by humans as finite cognitive agents executing the “perceive → cognize → act” cycle within inexhaustible reality.

Metaphysical compression differs from statistical compression in one fundamental respect:

6.2 The Kolmogorov Complexity Perspective

Kolmogorov complexity (K(x)) is defined as the length of the shortest program that generates object x on a universal Turing machine — it represents the ultimate lossless compression limit of data.

When Newton discovered that the fall of an apple and the orbit of the moon are governed by the same force, he did not recover the color of the apple or the distribution of lunar craters (original information at the Shannon level was still lost), but he found a shorter program to describe both classes of phenomena (K(apple + moon | universal gravitation) < K(apple) + K(moon)). When Einstein discovered E=mc², two concepts — mass and energy — were compressed into one — description complexity decreased.

Therefore, metaphysical compression does not recover already-lost original information — the Data Processing Inequality at the Shannon level is not violated. But by changing the level of description, it discovers structural relationships that were not explicit in the original representation, thereby reducing the system’s Kolmogorov complexity. Statistical compression may increase conditional Kolmogorov complexity (discarding conditional information needed for decoding); metaphysical compression reduces Kolmogorov complexity (discovering a shorter overall description). The two operate in opposite directions at different information tiers.

6.3 The Ultra-Long Validity Span of Metaphysical Knowledge

Empirical knowledge has an extremely short half-life: psychology approximately 7 years, engineering approximately 10 years, medicine approximately 45 years. Metaphysical knowledge, by contrast, has a validity span measured in millennia: Aristotle’s logic has remained valid for 2,400 years; Laozi’s “Dao” has remained valid for 2,500 years. The reason: metaphysical compression reaches the cognitive hardware layer — human biological cognitive architecture has not significantly evolved in the past 50,000 years. The closer knowledge is to the hardware layer, the longer its half-life. Metaphysical knowledge endures not because it is “profound” but because it describes the lowest-level structures within the human cognitive system — structures with the slowest rate of change.

VII Boundary Reconfiguration Mechanisms: Paths of Paradigmatic Rethinking

7.1 Paradigmatic Rethinking: Reconstructing the Variable Space

“Paradigmatic rethinking” — redefining the variable space itself — is the core cognitive operation for transcending representational boundaries. It is not a finer-grained compression within old boundaries but a questioning and redrawing of the boundaries themselves. The essence of every cognitive revolution is not the discovery of new relationships within the old variable set, but the discovery of entire dimensions that the old variable set’s definitional act had omitted.

7.2 Trigger Paths for Paradigmatic Rethinking

Boundary reconfiguration can occur through multiple paths:

Anomaly detection: Observing data that cannot be explained within the existing variable space — such as the anomalous precession of Mercury’s perihelion relative to Newtonian mechanics. Anomalies are signals from reality directed at representational boundaries.

Cross-domain analogy: Borrowing a variable framework from another field — such as Darwin importing Malthus’s population theory into biology, forming the concept of natural selection.

Tool / experimental extension: New instruments open new perceptual dimensions — such as the microscope discovering microorganisms, or LIGO detecting gravitational waves. This is a physical expansion of the Level 0 definitional space.

Art and metaphor: Bypassing logical derivation in a nonlinear fashion, intuitively touching dimensions not yet named.

Collective cognition and social collaboration: Different individuals’ bounded representational spaces may cover different dimensions; at their intersection, structures that no individual could discover independently may emerge.

Philosophical / metacognitive reflection: Identifying the above changes as boundary problems, and subjecting them to deliberate organization and scrutiny.

7.3 The Unique Role of Philosophy

Philosophy is the principal constituent of human metacognition. Its unique contribution lies not in being the sole trigger mechanism for boundary reconfiguration — experiments, tools, art, and collective collaboration can all trigger dimensional discovery. Philosophy’s irreplaceability lies in this: it enables the cognitive agent to identify boundary changes themselves as cognitive problems, thereby making the reconfiguration process a conscious, repeatable, and transmissible cognitive capability.

The historical conditions for philosophical flourishing — Gottlieb observes that Western philosophy’s achievements are concentrated in two bursts of roughly 150 years each, the Athenian period and the Enlightenment, both occurring before industrialization — are directly related to the biological cost of deep thinking. Metaphysical thinking requires four resources to be simultaneously satisfied: energy, state (emotional stability, ability to concentrate attention), total knowledge parameters (a sufficient quantity of known information as compression raw material), and sustenance. The four are in a multiplicative relationship — if any one is zero, the output is zero.

VIII The Theory’s Self-Boundaries

8.1 Does This Theory Obey Its Own Principle?

This paper claims: all finite cognition is compression, all compression is subject to representational boundaries, and all operations within boundaries cannot guarantee recovery of dimensions outside. The theory itself must therefore acknowledge:

The “compression pipeline” itself is a compression of reality — a specific metaphor that selects the “pipeline” as a linear topology while excluding other possible cognitive structures (such as networks, cycles, or fields). The “four-parameter matrix” itself is a Level 0 definitional act — selecting these two axes while excluding other possible cognitive dimensions. The structural mapping across three systems may have omitted other forms of cognition — embodied cognition, distributed cognition, tool-mediated cognition, and others. The concept of “paradigmatic rethinking” itself establishes new boundaries — it assumes that reconstruction of the variable space is the core form of cognitive progress, but progress may also occur in other forms.

This paper cannot prove from within itself that it has discovered all critical dimensions — this is precisely the self-demonstration of the paper’s core principle.

8.2 Compression Is Not the Totality of Cognition

Cognition is not only compression. It also includes simulation (constructing virtual replicas of external processes internally), combination (assembling fragments of existing representations into new structures), external recording (extending working memory through pen and paper, databases, and AI), exploratory action (generating new information through bodily interaction with the environment), and social collaboration (transcending individual bandwidth limitations through collective cognition). Equating cognition with compression is a simplification made in this paper for analytical convenience — it captures a core dimension of cognition but should not be mistaken for a complete definition of cognition.

IX Social and Human-AI Collaboration Inferences

The following judgments are sociological hypotheses derived from the core theory, not the core theory itself. They are heuristic but await independent verification.

9.1 The Seven-Stage Compression Pipeline: From Physical World to AI Output

9.2 The Dual Role of Affective Signals

Affective signals play not a single but a dual role in the cognitive system. Low-level affective signals — fear responses, social anxiety, immediate desires — do constitute a competitive occupation of resources against long-range abstract reasoning, locking attention onto “this moment, this place, this body” and running orthogonal to the direction of dimensional ascent. But high-level affective signals — curiosity, aesthetic intuition, awe, sustained motivation — may serve as catalysts and directional guides for paradigmatic rethinking. Damasio’s somatic marker hypothesis demonstrates that emotion plays an irreplaceable role in deep cognition. Therefore, affective signals have different value for different cognitive tasks.

9.3 Social Trust and Attention Bandwidth

Through social activity, individual cognition is converted into exchangeable information. But information is actively stratified through social relationships. Dunbar’s research has revealed the precise structure of this stratification: 1.5 persons (closest intimates) → 5 → 15 → 50 → 150 (Dunbar’s number) → 500 → 1,500. The greater the social distance, the higher the proportion of affective signals in information exchange. The Eastern “idle chat” and the Western “small talk” are essentially the same behavior — using affective signals as trust probes within a low-trust architecture. Rational information exchange is locked behind the trust verification of affective signals.

Ancient humans had extremely narrow information acquisition bandwidth but relatively ample cognitive processing bandwidth — the bottleneck was on the input side. Modern humans have an explosion of information acquisition bandwidth but the same attention bandwidth as their ancestors — the bottleneck has flipped to the processing side. Social networks have not expanded Dunbar’s number; they have merely accumulated, beyond the 150-person cognitive limit, a massive quantity of connections that consume attention but produce no rational information value.

9.4 The Value Point of Human-AI Collaboration

AI bypasses the trust-verification cost of affective signals, dedicating its full bandwidth to rational cognition. Humans retain direct connection to the physical world, value judgment, and metacognitive capability. The limitations of the two are complementary: AI expands the internal search space; humans reconstruct the goal and variable space. AI discovers anomalies within existing frameworks; humans judge whether anomalies signal that boundaries need to be redrawn. External experiments reconnect with reality; new information re-enters the model.

X Unified Framework and Conclusions

10.1 The Shared Topology of Three Systems

10.2 The Core Principle

10.3 Final Conclusions

On statistics: The validity boundary of statistics is determined not by its computational precision but by its definitional act at Level 0, which preemptively constrains the ontological scope of what can be cognized.

On AI: AI’s capability ceiling depends not on compute, parameter count, or data scale, but on the topological structure of the compression pipeline. Yet AI’s boundary is not static — new architectural paradigms (reinforcement learning, reasoning models, tool calling) may progressively expand its structural metacognitive capability.

On human cognition: The bottleneck of human cognitive evolution is not compute, not data — it is metacognitive capability: the ability to examine and redraw one’s own representational space. Exercising this capability requires specific biological and social conditions (ample attention bandwidth, sustained deep processing time, a sufficient knowledge base), and these conditions face new pressures in the modern information environment.

On human-AI collaboration: The limitations of AI and humans are complementary — this may be the most effective path for transcending each side’s validity boundaries.

A Appendix: Chronological Literature Review of Simpson’s Paradox and Related Research

B Appendix: Key Concept Cross-Reference Table