Mathematical Factuality: Proofs & Computation

• Mathematical factuality is the study of objective mathematical statements, distinguishing provable, unprovable, and observer-actualized facts.
• It integrates formal logic, model theory, and computational verification to reveal nuances like independent statements and non-standard models.
• Recent advances apply algorithmic methods and chain-of-thought verification to enhance the precision and coherence of mathematical reasoning.

Mathematical factuality refers to the status, nature, and identification of mathematical facts—their objectivity, their relation to proof or verification, the ways they differ from empirical or physical facts, and how they manifest in contemporary mathematics, logic, philosophy of science, and machine reasoning. Recent research addresses mathematical factuality via formal logic, epistemology, computational verification, model theory, and the practical performance of reasoning systems.

1. Defining Mathematical Factuality

Mathematical factuality is the notion that mathematics makes factual claims—statements about mathematical objects, structures, and relationships that can, at least in principle, be true or false. The epistemic and ontological status of these facts is debated. The status of mathematical facts can be characterized as follows:

• Provable Facts: Statements derivable from a fixed set of axioms (as in Peano Arithmetic, PA).
• True But Unprovable Facts: Statements true in the intended model (e.g., the natural numbers $\mathbb{N}$), but not provable from the chosen axiomatic basis.
• Independent Statements: Propositions undecidable in the theory (e.g., Paris–Harrington Principle in PA is true in $\mathbb{N}$ but unprovable in PA) (Valle, 2019).
• Observer-Actualized Facts: Statements actualized or recognized by mathematical observers, which become factual in the "World of Structures" upon observation (Burgin, 2017).
• Physical Embodiment: Facts as tied to the physical limits of mathematicians and computers—provable only if realizable within the constraints of finite resources (Wu, 2023).
• Model-Dependent Facts: Statements may be true in some models and false in others (e.g., non-standard models of arithmetic, geometric propositions true or false “on parts”) (Kovács et al., 2018, Valle, 2019).

This spectrum goes beyond binary truth/falsity, incorporating nuances such as partial truth (true on some components, false on others) and the impact of formalization, physicality, or context.

2. Logical and Model-Theoretic Perspectives: Completeness, Independence, and Non-Standard Models

In mathematical logic, formal systems such as PA attempt to capture the structure of arithmetic through recursive axiomatizations. However:

• The complete theory of $\mathbb{N}$ ($\mathrm{Th}(\mathbb{N})$) is not recursively enumerable and cannot be finitely or effectively captured (Valle, 2019).
• Gödel's Incompleteness Theorems establish that any sufficiently strong, consistent, recursively enumerable formal system is incomplete: there exist concrete mathematical facts true in the intended model but unprovable in the system (Valle, 2019, Wu, 2023).
• The existence of independent statements such as the Paris–Harrington Principle demonstrates that even combinatorially "natural" facts can transcend provability (Valle, 2019).
• Non-standard models: First-order theories like PA admit non-standard models, revealing that some facts (true in $\mathbb{N}$) are not uniquely determined by the first-order axioms, complicating the notion of absolute arithmetical factuality (Valle, 2019).

A plausible implication is that, while arithmetic seems determined as mathematics, factuality there is not exhaustively coextensive with provability, nor with model-theoretic satisfaction in the first-order setting.

3. Computational and Algorithmic Approach to Mathematical Factuality

Recent developments expand mathematical factuality to encompass algorithmic and computational verification:

• Computational Algebraic Geometry: In geometry, the truth of a proposition may hold on some components of a solution space and fail on others. The paper "Detecting truth, just on parts" provides an efficient criterion using elimination ideals (via Gröbner bases) to automatically classify whether an implication is "generally true," "generally false," or "true on parts, false on parts" (Kovács et al., 2018). For hypotheses $H$ and thesis $T$, the precise criterion is:

$$\begin{aligned} &\text{“Generally true”}:~~\langle H, f\cdot t - 1 \rangle \cap K[Y] \neq \langle 0\rangle \ &\text{“Generally false”}:~~\langle H, f\rangle \cap K[Y] \neq \langle 0\rangle \ &\text{“True on parts, false on parts”}:~~\langle H, f\cdot t - 1 \rangle \cap K[Y] = \langle 0 \rangle = \langle H, f\rangle \cap K[Y] \end{aligned}$$

This enables automated systems (e.g., GeoGebra) to provide nuanced factuality feedback.

• Finite Resource Physicality: The countable/uncountable dichotomy shows that the number of possible proofs (finite symbol strings) is countable, while the number of mathematical statements is uncountable. Therefore, some true mathematical facts are necessarily unprovable (as a brute fact of physical limitation) (Wu, 2023).
• Factuality in LLMs and Machine Reasoning: Mathematical factuality is now also addressed in AI, where systems must both generate and verify multi-step, logically valid mathematical arguments. These can suffer from hallucinations and errors, necessitating segment-level or stepwise factuality checks rather than outcome-only evaluation (Chen et al., 2023, Li et al., 30 May 2025, Chen et al., 7 Aug 2025).

4. Epistemological and Philosophical Analysis

Differences in the treatment of factuality reflect foundational perspectives:

• Structural Realism: Mathematics studies structures that may exist abstractly or be actualized by observation; factuality emerges when a structure is recognized, named, or used (Burgin, 2017).
• Mathematics as Physical: The practice of mathematics is constrained by the physical instantiation of mathematicians and their computational tools, meaning that mathematical facts are inextricably linked to physical reality (as per Landauer's dictum that "information is physical") (Wu, 2023).
• Imagined or Created Nature: Some accounts treat mathematical facts as "specifications of conceptions," rooted in internal human activity and imagination rather than external or platonic existence (Čulina, 2023).
• Symmetry and Invariance: Mathematical factuality can be understood as a consequence of symmetry—facts are statements invariant under all permutations of their domains ("symmetry of semantics"). This symmetry is general enough that physical regularities become special cases of mathematical regularities (Yanofsky, 2015).
• Fact-Making by Theory: The shift from "facts as empirical, atheoretical conglomerations" (Mach) to "facts as universal, mathematically-theorized regularities" (Einstein) marks an evolution in the status of mathematical constructs—mathematical laws themselves can be regarded as facts (Waal et al., 2020).

5. Factuality in Automated Reasoning and LLMs

In applied mathematics and machine learning, mathematical factuality is treated operationally:

Segment-Level and Contextual Factuality

• FELM Benchmark: Assesses LLM factual accuracy in math by dividing outputs into segments (steps, claims, equations) and labeling each as factually correct or incorrect. Even the strongest models like GPT-4 achieve only moderate accuracy in segment-level factuality detection for math, reflecting the complexity of verifying mathematical facts beyond simple retrieval (Chen et al., 2023).

  Stepwise and Coherent Factuality

• Chain-of-Thought Verification: Rewarding models for correct-only final answers increases hallucination and error rates in intermediate reasoning; algorithms such as Factuality-aware Step-wise Policy Optimization (FSPO) dynamically verify and reinforce factual correctness at each reasoning step. FSPO achieves lower hallucination rates and higher accuracy in mathematical reasoning benchmarks by explicitly tying factuality verification to policy updates (Li et al., 30 May 2025).

• Coherent Factuality: Independent claim-wise checking is insufficient in reasoning; factuality must also consider the logical dependencies and substantiations among steps. Conformal prediction on deducibility graphs enables filtering for "coherent factuality," guaranteeing that retained output chains are both locally and globally correct, and substantiated according to context (Rubin-Toles et al., 21 May 2025).
• Multi-Factor Reward Functions: Balancing factual precision, detail/coverage, and relevance leads to state-of-the-art reductions in hallucination and gains in informativeness for mathematical and long-form factual reasoning (Chen et al., 7 Aug 2025).

  Unresolved Challenges

• Factuality evaluation in mathematics remains an outstanding challenge: segment-level verification, multi-step logical consistency, and alternative formalizations (e.g., impossible possible worlds for explanation) are all active areas (Chen et al., 2023, Halpern, 2023).

6. Beyond Classical Binary Factuality: Extended Notions

Contemporary research generalizes mathematical factuality beyond binary true/false:

• Partial or "True on Parts, False on Parts": For instance, geometric conditions may hold on some irreducible components of a variety but not others, leading to nuanced factuality classifications (Kovács et al., 2018).
• Impossible Possible Worlds and Mathematical Explanation: In probabilistic and causal explanation frameworks, mathematical facts must be treated as potentially unknown to allow for genuine explanation, implemented by expanding the agent’s epistemic state to include "impossible worlds" where some mathematical facts can be false (Halpern, 2023).
• Tradeoff Between Abstraction and Factuality: In settings like abstractive summarization, increasing abstractiveness drives down factuality, requiring new evaluation metrics that account for both fidelity and rephrasing (Dreyer et al., 2021).

7. Contemporary Synthesis and Outlook

Mathematical factuality is multidimensional, encompassing:

• The limitations imposed by physical, cognitive, or logical resources (Wu, 2023);
• The capacity of axiomatic systems to capture truths, with undecidability and non-standard models marking inherent gaps (Valle, 2019);
• The role of structuralist, observer-dependent, or imagined conceptions of existence (Burgin, 2017, Čulina, 2023);
• Practical challenges in ensuring, measuring, and calibrating factuality in mathematical reasoning systems, especially LLMs (Chen et al., 2023, Li et al., 30 May 2025, Chen et al., 7 Aug 2025, Rubin-Toles et al., 21 May 2025).

An overriding theme is that, although mathematics aspires to unerring objectivity, the identification and trustworthiness of mathematical facts depend on intricate layers of formal systems, physical limits, epistemic access, computational tools, and human or artificial observation and verification. The field now recognizes that mathematical factuality encompasses more than theoremhood: it includes nuanced partial truths, context-dependent verification, and evidence of factuality in the behaviors of both human and artificial reasoners.