Models Can and Should Embrace the Communicative Nature of Human-Generated Math

Abstract: Math is constructed by people for people: just as natural language corpora reflect not just propositions but the communicative goals of language users, the math data that models are trained on reflects not just idealized mathematical entities but rich communicative intentions. While there are important advantages to treating math in a purely symbolic manner, we here hypothesize that there are benefits to treating math as situated linguistic communication and that LLMs are well suited for this goal, in ways that are not fully appreciated. We illustrate these points with two case studies. First, we ran an experiment in which we found that LLMs interpret the equals sign in a humanlike way -- generating systematically different word problems for the same underlying equation arranged in different ways. Second, we found that LLMs prefer proofs to be ordered in naturalistic ways, even though other orders would be logically equivalent. We advocate for AI systems that learn from and represent the communicative intentions latent in human-generated math.

• The paper proposes that AI models should view human-generated math as a form of situated communication rather than purely abstract symbols.
• Empirical evidence from two case studies shows Large Language Models exhibit preferences for natural ordering in equations and proofs, reflecting sensitivity to communicative context.
• Treating math communicatively has practical implications, potentially improving AI's role in math education by recognizing student errors and enhancing collaborative research tools.

Embracing the Communicative Nature of Human-Generated Mathematics in Artificial Intelligence Models

The paper, "Models Can and Should Embrace the Communicative Nature of Human-Generated Math," presents a compelling hypothesis regarding the treatment of mathematics by LLMs (LMs). It advocates for viewing math not solely as a domain of abstract symbolic reasoning but as a form of situated linguistic communication imbued with communicative intents, mirroring the ways language is employed. The authors propose that AI systems should be designed to understand and leverage these communicative dimensions to enhance their mathematical reasoning capabilities.

Overview and Methodology

The document outlines an innovative approach by illustrating two case studies that demonstrate how LMs process mathematical expressions differently depending on their presentation and underlying communicative nuances, reflecting their sensitivity to context and order.

1. Case Study on Equation Asymmetry: This study examines the hypothesis that LMs, like human reasoners, do not interpret equations symmetrically, even when those equations are logically equivalent. The research employs GPT-4o in an experimental setup whereby it generates word problems from varied presentations of equivalent equations. The model's ability to recover these original presentations is tested, revealing a preference for retaining the original order despite possible logical equivalence.
2. Case Study on Mathematical Rules and Proofs: Here, the ordering preferences of LMs in mathematical proofs and rules are explored. Through evaluating different permutation orders of equations and proofs, the study finds consistent patterns where LMs exhibit lower surprisal values (indicative of higher probability or normalcy) for natural ordering over logically equivalent reorderings. Various models, including math-fine-tuned ones, are accessed to ascertain whether training on extensive mathematical corpora influences these preferences, indicating a broader communicative understanding rather than an overfitted pattern recognition.

Implications and Applications

The paper suggests significant practical applications for the findings within the domains of math education and research.

• In Math Education: By understanding the communicative cues in mathematical problems, AI systems can be developed to better recognize student errors and learning points. Such systems can aid educators in crafting problems that not only challenge the learner's symbolic understanding but also their comprehension of the problem's communicative subtleties.
• In Math Research: The potential for AI systems to commune mathematical findings, proofs, and derivations in a way that is interpretable and relevant to human researchers can broaden collaborative opportunities. These systems can provide insights into human understandings of math that transcend purely symbolic solvers, leading to a more intuitive and human-like AI in mathematical assistance.

Conclusion

The hypothesis presented in this paper—that math should be treated as communicative and not merely symbolic—is supported by empirical evidence from both case studies. Furthermore, the implications on practical AI applications echo the calls from previous research to integrate cognitive science perspectives into the development of AI. The Communicative Math Hypothesis aligns AI models more closely with the human approach to mathematics, suggesting a nuanced future for AI that includes contextual understanding as a core component of mathematical reasoning.

Overall, the paper proposes an intriguing reevaluation of how AI models engage with math, advocating for the incorporation of communicative aspects inherent in human-generated math. This could herald the development of AI systems that better emulate human reasoning and interaction with mathematical content.