Learning-to-Theorize in Science & AI

• Learning-to-theorize is a research program that frames understanding as the acquisition, revision, and deployment of explicit, transferable theories.
• It utilizes diverse representations—from symbolic rules to executable programs—to drive efficient inference and principled generalization.
• The approach emphasizes compactness, parsimony, and domain specialization, enabling progressive theory refinement and enhanced explanatory power.

Learning-to-theorize is a family of research programs in which understanding is treated not merely as prediction, classification, or latent compression, but as the acquisition, revision, or deployment of explicit theories. In science education, it denotes the capacity to reconfigure an existing, coherent understanding of a mature scientific theory so as to productively adopt its successor (Zuccarini et al., 2021). In neuro-symbolic machine learning, it denotes learning compact predictive theories made of logical rules and core facts from observations (Campero et al., 2018). In unsupervised scientific modeling, it denotes learning small symbolic or neural theories together with their domains of validity, often under description-length or parsimony pressures (Wu et al., 2018). In recent world-modeling work, it denotes inferring executable explanatory programs from raw, non-textual observations (Baek et al., 5 May 2026). Related usages appear in meta-learning, theorem proving, causal-epistemic logics of experimentation, and pedagogical studies of how humans learn to use theory itself as a research instrument (Baxter, 2020, Wang et al., 2020, Barbero et al., 2021).

1. Scope of the concept

The phrase does not designate a single formalism. Rather, current usage spans several closely related notions of what a “theory” is and what it means to learn one. In the conceptual-change literature, the object of learning is a successor scientific theory, and the central problem is how learners reorganize ontological, mathematical, and epistemic commitments during theory change; the classical-to-quantum transition is used as the canonical case (Zuccarini et al., 2021). In logical theory learning, the theory is a compact set of rules $R$ together with a small set of core facts $C$, learned so that forward chaining from $R \cup C$ entails the observations $O$ and generalizes beyond them (Campero et al., 2018). In the AI Physicist framework, a theory is a tuple $T=(f,c)$, where $f$ predicts and $c$ represents a domain of validity, so that multiple theories can specialize to different regimes of a world (Wu et al., 2018).

A further shift appears in program-induction approaches. The Neural Theorizer represents a theory as an executable program $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$ in a learned Language of Thought, with learned primitive symbols and a shared executor that composes them into explanations of observed input-output phenomena (Baek et al., 5 May 2026). In formal reasoning, learning-to-theorize is instantiated as generating theorems and machine-checkable proofs, then using these synthetic theorems to train a stronger prover (Wang et al., 2020). In theoretical models of learning to learn, the learned “theory” is an inductive bias, parameterized for example by a family of hypothesis classes $\{\mathcal H_\phi : \phi \in \Phi\}$ or by a hierarchical prior, which is itself learned across related tasks and then reused on novel tasks (Baxter, 2020).

This range of meanings suggests an important commonality: in each case, learning-to-theorize concerns the acquisition of reusable structure that constrains future inference. That structure may be categorical, logical, symbolic, probabilistic, programmatic, or epistemic; but it is always intended to support transfer, explanation, or principled generalization rather than one-off prediction. This suggests that the term functions as an umbrella for research on explicit inductive structure, even when the representational substrate differs substantially.

2. Representations of theory

A central difference among learning-to-theorize frameworks lies in how theories are represented. In the education-focused model of successive scientific theories, conceptual trajectories are analyzed at four representational levels: phenomenological/visual, conceptual/ontological, mathematical/formal, and epistemic/logical. Categorical change is visualized with dynamic frames: hierarchical structures built from a superordinate concept $S$, attributes $C$0, and values $C$1, with classical and quantum values color-coded to make continuity and rupture visible (Zuccarini et al., 2021). This representation is designed to support comparability between paradigms.

In differentiable rule-induction systems, theories are represented symbolically but parameterized continuously. Predicates have embeddings $C$2; rules are tuples of head and body predicate embeddings with variable positions; facts are quadruples $C$3 with soft truth $C$4. Logical inference is implemented by soft unification using cosine similarity, and forward chaining is unrolled for $C$5 steps (Campero et al., 2018). Here, explicit rules remain interpretable, but their acquisition is gradient-based.

The AI Physicist formalism uses a different factorization. A theory $C$6 combines a prediction function $C$7 with a domain sub-classifier $C$8. With multiple competing theories, the set $C$9 forms a mixture of domain-specialized models, and a “theory hub” stores learned theories, snapped symbolic forms, and unified master theories that vary by a parameter vector $R \cup C$0 (Wu et al., 2018). The representational emphasis is on piecewise laws, symbolic compression, and lifelong reuse.

In the Neural Theorizer, representation is explicitly programmatic. A finite primitive set $R \cup C$1 defines latent symbols; each primitive has an execution function; and a theory is a compositional program whose execution trace passes through latent states. The generative model marginalizes over both programs and traces,

$R \cup C$2

while the variational posterior amortizes theory induction from paired observations (Baek et al., 5 May 2026). This makes the theory simultaneously symbolic in syntax and neural in semantics.

Other literatures use representational choices mainly to render opaque systems more analyzable. DecisiveNets transform a trained DNN into a deep hetero-associative memory by replacing activations with local winner-takes-all competitions, yielding sparse, discrete internal codes that are described as easier to theorize and manipulate (Gripon et al., 2020). In the logic of experiments, causal models are extended with a Hintikka-style epistemic state and then with observables, so that interventions and measurements can be represented in a common formal language (Barbero et al., 2021). Across these cases, representational design is not incidental: it determines what kinds of explanations, invariants, and abstractions can be learned at all.

3. Mechanisms of acquisition, revision, and selection

Learning-to-theorize systems differ not only in representation but also in the mechanism by which theories are acquired. In differentiable rule induction, learning proceeds by forward chaining from learnable core facts, comparing inferred consequences to observed facts, and optimizing a loss that combines reconstruction with sparsity pressure on the initial fact valuations. Soft unification contributes multiplicative cosine-similarity factors, and the learned initial valuations act as gates selecting which facts belong to the core theory (Campero et al., 2018). Predicate invention is handled by adding randomly initialized auxiliary predicate embeddings that acquire semantics during training.

The AI Physicist architecture combines divide-and-conquer, Occam’s razor, unification, and lifelong learning. Theory specialization is driven by a generalized-mean loss with negative exponent $R \cup C$3, and in practice the harmonic loss $R \cup C$4 is used. Parsimony is imposed through a differentiable description-length objective, after which learned numerical parameters are “snapped” to simple symbolic formulas if total description length decreases. Learned theories are then clustered, canonicalized, parameterized, and merged into master theories in the theory hub (Wu et al., 2018).

The Neural Theorizer uses variational learning with a goal-conditioned theory programmer, a shared executor, and a decoder. Minimum Description Length selects the effective explanation length through

$R \cup C$5

and a state-grounding regularizer keeps intermediate latent states on the encoder’s manifold (Baek et al., 5 May 2026). Test-time scaling in the NEO-S variant samples multiple candidate programs and selects a single program by majority voting for transfer.

Search-based mechanisms remain important in more symbolic settings. Exploratory Model Building constructs hypothetical probabilistic contexts $R \cup C$6 by selecting dependencies from a domain-wide storehouse $R \cup C$7, subject to sufficiency, consistency, and minimality constraints, and ranks them by a merit function $R \cup C$8 using a best-first or A* procedure with an admissible bound $R \cup C$9 (Bhatnagar, 2013). In causal-epistemic logic, learning proceeds through intervention followed by observation: the extended semantics with observables filters epistemic possibilities to those whose post-intervention observable values match the actual measurement, thereby invalidating the no-learning principle that holds in the simpler intervention-only system (Barbero et al., 2021).

Automated theorem proving adds a further variant. MetaGen synthesizes one-step proof trees by choosing invocable theorems and substitutions, then extends them by subtree grafting into multi-step proofs. The generator can be trained by imitation learning or by reinforcement learning with language-model or adversarial rewards, after which the synthetic corpus trains Holophrasm’s relevance and substitution networks (Wang et al., 2020). In all of these systems, theory acquisition is inseparable from a selection principle: sparsity, description length, program length, admissible search bounds, or proof utility.

4. Empirical regimes and performance criteria

Because the underlying objects differ, empirical evaluation of learning-to-theorize is heterogeneous. In neuro-symbolic rule induction, success is measured by recovery of observations, ILP convergence, or link-prediction quality. On the Kinship task, the learned compressed theory achieved 100% success runs, 96% accuracy of recovered observations, and about 30.8 core facts learned versus a target compression of 28. On the Animal hierarchy task, the method reported 70% success, 99% accuracy, and an average of 69 core facts versus an optimal 40. On the Countries dataset, AUC-PR reached $O$0 on S1, $O$1 on S2, and $O$2 on S3 (Campero et al., 2018).

The AI Physicist benchmarks emphasize exact recovery, domain discovery, description length, and sample efficiency. On 40 mystery worlds, median $O$3 MSE was $O$4 for the baseline, $O$5 for the Newborn agent, and $O$6 for the AI Physicist; interior-point classification accuracy was $O$7, $O$8, and $O$9, respectively; and the fraction of worlds solved under stringent criteria was $T=(f,c)$0, $T=(f,c)$1, and $T=(f,c)$2. Median $T=(f,c)$3 was approximately $T=(f,c)$4 bits for Newborn and AI Physicist, versus approximately $T=(f,c)$5 for the baseline. Epochs until $T=(f,c)$6 MSE were 6925 for the baseline, 330 for Newborn, and 45 for the AI Physicist (Wu et al., 2018).

For explanation-driven world modeling, the key metric is transferability of inferred theories. On GridWorld at $T=(f,c)$7, NEO attained compositional OOD transfer of approximately $T=(f,c)$8 and length OOD transfer of approximately $T=(f,c)$9; with NEO-S and $f$0, these rose to approximately $f$1 and approximately $f$2. On Arithmetic, NEO reached approximately $f$3 compositional OOD transfer at $f$4 and approximately $f$5 at $f$6, while NEO-S with $f$7 raised length OOD transfer to approximately $f$8 at $f$9 and approximately $c$0 at $c$1. On Image Editing, for $c$2, NEO achieved approximately $c$3 L1 on compositional OOD and approximately $c$4 on length OOD (Baek et al., 5 May 2026).

In theorem proving, synthetic theorization is judged by downstream proof search. On set.mm with 100% human proofs, Holophrasm proved 557 test theorems, while adding MetaGen-IL synthetic data raised this to 600; with only 10% human proofs, the count increased from 454 to 472, nearly matching the 20% human-only baseline of 476. On iset.mm, the baseline rose from 378 to 398 with MetaGen-IL (Wang et al., 2020). In representation-oriented work, DecisiveNets are evaluated by the trade-off between accuracy and compute. On CIFAR-10 with ResNet-18, $c$5 yielded $c$6 accuracy and 5,070,848 multiplications, while $c$7 yielded $c$8 and 3,125,248 multiplications; on CIFAR-100 with ResNet-50, $c$9 gave $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$0 and 1,297,809,408 multiplications, while $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$1 gave $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$2 and 861,601,792 (Gripon et al., 2020).

These benchmarks show that “theory” is not evaluated by a single criterion. Compactness, transfer, symbolic exactness, domain segmentation, proof success, and interpretability all serve as proxies, depending on whether the target theory is a rule set, a program, a piecewise law, a memory-like architecture, or a curriculum for human reasoning.

5. Human theorizing, conceptual change, and pedagogy

A distinct strand of work treats learning-to-theorize as a human cognitive and educational problem. In the model of successive scientific theories, conceptual change is not reduced to replacing misconceptions with correct propositions. The classical-to-quantum transition is analyzed as a coordinated reorganization across phenomenological/visual, conceptual/ontological, mathematical/formal, and epistemic/logical levels, including changes in the meaning of physical quantity, measurement, state, time evolution, vectors, superposition, operators, and the logic/probability framework itself. Dynamic frames reveal recognizable patterns such as categorical generalization, value disjunction, and change in value constraints, and the paper argues that the incompatibility of observables should be introduced early as an organizing theme (Zuccarini et al., 2021).

Studies of expert practice emphasize that theorizing is also a repertoire of research moves. Cognitive Task Analysis interviews with $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$3 theoretical physicists showed that assumptions are used throughout the research process: in setting project direction and goals, establishing model–math interaction, and revising the model while troubleshooting. The same study found that analogies are used deliberately for generating project ideas, overcoming conceptual and mathematical roadblocks, building intuition, and demonstrating feasibility, with experts privileging structural and mathematical similarity over surface similarity (Verostek et al., 2022). This suggests that learning-to-theorize includes assumption management and analogy construction as explicit skills rather than tacit by-products of expertise.

Pedagogical work in biology makes this skill set teachable by design. A graduate-level course for empirically focused biology students was organized around backwards design, active learning, and just-in-time teaching, with two primary objectives: enabling students to extract biological insights from theory papers even while skipping math they could not follow, and helping them understand the diverse roles theory plays in generating and testing hypotheses in population biology (Masel et al., 14 Apr 2026). The course operationalized theory reading through a taxonomy of model purposes, black-box input-output analysis, reading guides, short quizzes, and Mathematica-based assignments.

Another line of research treats theory-like understanding as the acquisition of relational descriptions of behavior and environment. Agents learn RST-inspired relations such as incompatibility, prevention, definition, and the Although family, which expresses surprise at observed behavior and, in some circumstances, presents a justification for it. These learned descriptions are framed as a theory of what actions do, what propositions can co-occur, what “ought” to happen, and why apparent deviations occur (Botelho et al., 2022). Here, theorization is inseparable from explanation, normativity, and discourse structure.

6. Sequencing, limitations, and open questions

One debate concerns when theorizing should occur relative to data analysis. A Bayesian model in economics distinguishes a Darwinian Learning effect from theorizing first and a Statistical Learning effect from examining the data first. The central result is that post hoc theorizing is optimal exactly when Statistical Learning exceeds Darwinian Learning; the paper argues that, in the modern era of mature economic theory and enormous datasets, post hoc theorizing is typically optimal (Chen, 15 May 2025). This does not generalize automatically across fields, but it makes sequencing itself part of learning-to-theorize.

Across computational systems, the main limitations are structural rather than merely engineering details. Differentiable forward chaining can expand facts exponentially in $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$4 and arity, training is sensitive to initialization, exact logical guarantees are not provided, and rule templates remain fixed (Campero et al., 2018). The AI Physicist uses a heuristic bit-length surrogate rather than a full MDL under an explicit probabilistic noise model, can struggle with overlapping domains and transition regions, and incurs computational overhead from multiple networks, annealing, snapping, and unification (Wu et al., 2018). NEO currently uses small, discrete primitive sets and relatively short programs, its learned primitive meanings need not align with human concepts or causal factors, and long-horizon composition benefits from test-time scaling, which increases computation roughly $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$5 (Baek et al., 5 May 2026).

Representation-simplifying approaches expose a different trade-off. In DecisiveNets, larger $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$6 increases sparsity and efficiency but can reduce clean accuracy; near-ties are brittle; and there is no explicit energy function, even though local winner-takes-all makes analysis tractable (Gripon et al., 2020). In causal-epistemic logic, the agent knows the structural equations $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$7, so learning is only about variable values; extensions to uncertainty over $\tau=(z_{i_1}, z_{i_2}, \ldots, z_{i_K})$8, probabilistic noise, multi-agent settings, and causal-structure learning are left open (Barbero et al., 2021). In education research, open questions include how to coordinate personal epistemologies with contested scientific epistemology, how to map exemplar transitions and framing errors more finely, and how to validate frame-based strategies through large-scale teaching experiments (Zuccarini et al., 2021).

Taken together, these limitations indicate that learning-to-theorize remains less a settled method than a research direction. Its recurring ambition is stable: to build systems or curricula that do not merely fit data or solve isolated tasks, but instead acquire explicit structures that can be compared, recombined, justified, and revised. A plausible implication is that future progress will depend on tighter integration among representation learning, symbolic search, experimental update, and pedagogies of scientific reasoning, rather than on any single formalism alone.