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Math Agents: Computational Infrastructure, Mathematical Embedding, and Genomics

Melanie Swan, Takashi Kido, Eric Roland, Renato P. dos Santos

arXiv:2307.02502v1q-bio.OTcs.AIcs.CL

TL;DR

Mathematics is digitally represented but remains difficult to use and evaluate systematically at scale. This project applies mathematical embeddings to equations and genomic data, and its Alzheimer’s-related comparison suggests the Chern-Simons model may have higher explanatory value for the proposed transposon claim.

  • Problem

    Mathematics is digitally represented but remains difficult to use or solve, and tools for systematic high-throughput evaluation of large mathematical ecologies are scarce.

    Source: Section 1: Introduction to… · pp. 3–4

  • Method

    The project applies vector embeddings to equations and genomic data, represents equations in LaTeX and SymPy, and frames Math Agents as interfaces to digital mathematical infrastructure.

    Source: Section 1: Introduction to… · pp. 5–6

  • Results

    For the proposed transposon-Alzheimer’s disease explanation, the joint view suggests the Chern-Simons model may have higher explanatory value.

    Source: Section 1: Introduction to… · p. 10

  • Takeaways & Limitations

    The authors conclude that the tool suite supports more readily usable mathematics and democratized access for human and AI audiences.

    Source: Conclusion · p. 24

    Because AI functionality is constantly evolving, the authors warn that solutions may face immediate obsolescence.

    Source: Risks and Limitations · p. 23

Abstract

from arXiv · show

The advancement in generative AI could be boosted with more accessible mathematics. Beyond human-AI chat, large language models (LLMs) are emerging in programming, algorithm discovery, and theorem proving, yet their genomics application is limited. This project introduces Math Agents and mathematical embedding as fresh entries to the "Moore's Law of Mathematics", using a GPT-based workflow to convert equations from literature into LaTeX and Python formats. While many digital equation representations exist, there's a lack of automated large-scale evaluation tools. LLMs are pivotal as linguistic user interfaces, providing natural language access for human-AI chat and formal languages for large-scale AI-assisted computational infrastructure. Given the infinite formal possibility spaces, Math Agents, which interact with math, could potentially shift us from "big data" to "big math". Math, unlike the more flexible natural language, has properties subject to proof, enabling its use beyond traditional applications like high-validation math-certified icons for AI alignment aims. This project aims to use Math Agents and mathematical embeddings to address the ageing issue in information systems biology by applying multiscalar physics mathematics to disease models and genomic data. Generative AI with episodic memory could help analyse causal relations in longitudinal health records, using SIR Precision Health models. Genomic data is suggested for addressing the unsolved Alzheimer's disease problem.

Section 1: Introduction to Formalization Space

Together, these sections present mathematical embeddings, Math Agents, and visualizations as infrastructure for representing, evaluating, and using mathematics in AI- and human-assisted analysis. They pilot these tools on published equations and disease-associated genomic variants, including exploratory Alzheimer’s mathematics and data, and introduce a two-tier Precision Health SIR model for future precision-health modeling.

Mathematics as the Data Corpus

  • The formalization space comprises systematic mathematical, algorithmic, programmatic, information-theoretic, and graph-theoretic approaches, realized through computational infrastructure on global networks.The paper argues that digitized corpora could enable broad access, while noting that mathematics lacks a comprehensive callable catalog and tools for high-throughput evaluation.Source: Section 1: Introduction to… · pp. 2–4

Math Agents

  • Math Agents are proposed as interfaces for identifying, cataloging, representing, analyzing, integrating, writing, discovering, solving, and maintaining mathematical ecologies.The section presents them as a way for people to interact with the mathematical corpus through a dialogical interface.Source: Section 1: Introduction to… · p. 5

Mathematical Embedding

  • The proposed mathematical embedding converts equations extracted from PDFs with Mathpix OCR into LaTeX and SymPy strings, using GPT to validate outputs and embeddings.The embeddings are intended as AI-readable units for machine-learning systems and computational infrastructure; standard embedding choices vary with parameters such as normalization, dimensionality reduction, and tokenization.Source: Section 1: Introduction to… · pp. 4, 5, 8

Results

  • 10,592 embeddings were produced across mathematical equations and genomic variants, including 276 unique variants associated with Alzheimer’s disease, Parkinson’s disease, and ALS.The pilot covered 10 papers and generated embeddings in LaTeX and SymPy using four embedding models; reported results use OpenAI and LaTeX versions.Source: Section 1: Introduction to… · pp. 8–9
  • Embedding views show limited overlap among the sampled AdS/CFT and Alzheimer’s disease mathematical ecologies, while shared differential-equation focus places the SIR control model near Goriely mathematics.The authors suggest the views can expose mathematical landscapes without reading each paper, while leaving open whether connecting Alzheimer’s disease ecologies could aid causal understanding.Source: Figure 7 · p. 9, Section 1: Introduction to… · pp. 9–10
  • A joint embedding view tentatively places Chern-Simons equations closer to Alzheimer’s SNP data than transposon-dynamics equations, but the authors frame this as a first-pass hypothesis.The paper says the method for unified analysis of a mathematical ecology and underlying data remains unclear.Source: Figure 8 · p. 10, Section 1: Introduction to… · p. 10

Section 1: Introduction to Formalization Space

  • The SIR control example uses solvable dS/dt, dI/dt, and dR/dt equations, while its embedding groups system equations and separates parameter clusters.Figure 10 also presents the SIR compartmental model alongside a multi-disease genomic variant-overlap view for AD, PD, and ALS.Source: Section 1: Introduction to… · p. 12, Figure 10 · p. 13
  • The proposed two-tier Precision Health SIR model recasts states as sustaining, intervening, or restoring, with information “infection” prompting prevention through inputs from health and research sources.The aim is to prevent conditions during the 80% of their lifecycle before clinical detection; the model could accompany quantitative approaches such as value-based healthcare.Source: Section 1: Introduction to… · pp. 12–13
  • The pilot produces mathematical embeddings, equation clusters, and mathematical ecology visualizations as abstraction tools, but their broader deployment use remains unclear.Used with Math Agent explorers, they might support math discovery and position evaluation by helping users grasp the size and scope of a mathematical ecology.Source: Section 1: Introduction to… · p. 13
  • Embeddings are intended for AI analysis but also let humans inspect a consolidated, zoomable mathscape; they are emerging as computational infrastructure for systems-level problems.Although developed through a human-orchestrated workflow here, embeddings are increasingly incorporated automatically into GPT cognitive infrastructure, with mathematical signals intended to support large-scale precision-health models.Source: Section 1: Introduction to… · p. 13

Section 3. Math-Data Relation

The section proposes that AI-enabled mathematical embedding may shift digital interaction from managing data toward engaging reality through mathematics. It examines how math and data correspond within shared views, multiscalar systems, and mathematical abstraction.

One System Two Modes

  • The section presents three ways to understand the math-data relation: a composite view, multiscalar renormalization, and mathematics as an interface for abstraction and interaction.Figure 11 presents these as progressive ideas for examining two representations of the same system.Source: Section 3. Math-Data Relation · p. 14, Figure 11 · p. 14
  • Embedding math and data in a shared view may improve model-fit assessment, but their distinction matters and their correspondence should be examinable by humans and Math Agents.The proposed composite view can include mathematical and data embeddings, shown separately or superimposed, to assess fit across layers of complex datasets.Source: Section 3. Math-Data Relation · pp. 14–15
  • Math Agents and mathematical embedding aim to support comprehensive, at-scale accounts of multiscalar ecosystems, including their data, mathematics, composite views, and interacting observers.The section proposes considering data, math, and their composite as entities across scale tiers, with humans, AIs, and biological processes interacting with systems.Source: Section 3. Math-Data Relation · p. 15, Figure 12 · p. 15

Multiscalar System Mathematics

  • Biological systems involve intricate cross-tier relations, so renormalization can compare scales by collapsing irrelevant degrees of freedom and tracking conserved system-wide factors.The passage argues that percentage-based normalization cannot put scale tiers into dialogue; relevant conserved factors may include symmetry or biological free energy.Source: Section 3. Math-Data Relation · pp. 15–16

Abstraction: Mathematics is the Interface

  • Mathematics’ interconnected corpus may help identify an efficient abstraction level for solving multiscalar systems, while mathematical descriptions support predictions about phenomena.The protein-folding example contrasts nature’s use of chemical-bonding energy cues with AlphaFold’s use of high-level algorithms rather than trying every permutation.Source: Section 3. Math-Data Relation · p. 16
  • AI-enabled infrastructure may move interaction from corralling data toward “big math,” using mathematics as an abstracted, potentially more effective interface to digital reality.The section describes infrastructure as acquiring “smart” computational properties and mathematics as a higher level for engaging data.Source: Section 3. Math-Data Relation · pp. 13, 14, 17

Section 4: Evaluation of Mathematical Ecologies

The section proposes mathematical graphs, embeddings, and Math Agents as infrastructure for representing and evaluating mathematical ecologies across machine-learning, blockchain, and quantum systems. It describes equation evaluation and proof, energy-based graph solving, and a possible shift toward computation and mathematical functionality within the network itself.

Computational infrastructure

  • Math Agents could traverse graph-encoded equations and symbols to evaluate mathematical ecologies, seeking solution paths across connected computational infrastructure.Mathematical embeddings provide AI-readable vector inputs, while graph representations could support automated evaluation from brute-force testing to higher-level methods.Source: Section 4: Evaluation of Mathematical… · pp. 17–18

Machine Learning

  • Machine-learning networks could evaluate mathematical ecologies either as input data or by encoding problem structure in the network architecture.The latter could let the network find weights, layers, and graph shape corresponding to a mathematical solution; Math Agents could use energy-entropy loss functions, causal inference, and probabilistic prediction.Source: Section 4: Evaluation of Mathematical… · p. 18

Blockchains

  • Blockchain Math Agents could validate equation nodes in sequence, log each step, and help repair or certify mathematical ecologies that are difficult to replicate.Layer 2 path routing can require validation before proceeding; the agents could add clarifications or mathematics to produce tested, generalized versions.Source: Section 4: Evaluation of Mathematical… · p. 19
  • Blockchain-based theorem proving could validate each step in a retraceable transaction log and record proofs, theorem identifiers, and intellectual-property discoveries.The proposal also assigns MathProofCoin possible roles in tracking theorem use, credit, and resources for Math Agents.Source: Section 4: Evaluation of Mathematical… · pp. 19–20

Quantum Mathematical Infrastructure

  • Energy-entropy formulations could provide a mathematical abstraction layer for graph problems, with Math Agents seeking lowest-energy paths or, for some objectives, highest-entropy configurations.The section proposes this as a way to solve at an appropriate scale and to connect classical and quantum computational infrastructure.Source: Section 4: Evaluation of Mathematical… · p. 21

The Network is the Computer

  • The network could evolve from transporting data to hosting computation and mathematical functionality, with agents operating within it or as an overlay.The proposed math-network layer would use embeddings and Math Agents to validate, safeguard, and extend mathematical knowledge across the graph.Source: Section 4: Evaluation of Mathematical… · p. 22

Risks and Limitations

The section frames the project around AI alignment and equitable access to mathematics, while acknowledging that its tools may quickly become obsolete and that digitized mathematics may remain difficult to understand.

Risks and Limitations

  • The project frames democratized access to mathematics and disease-preventing well-being as alignment goals, and endorses global AI regulatory agencies with sandboxes, registries, and accountability frameworks.It aims to support high quality of life and equitable economic impact.Source: Risks and Limitations · pp. 22–23
  • Related work discusses Math Agents and human-AI entities in the responsible development of quantum intelligence (Swan & dos Santos, 2023).The passage describes quantum intelligence as AI-enabled technologies in quantum-classical-relativistic domains.Source: Risks and Limitations · p. 23
  • AI functionality may become immediately obsolete, but the project presents its mathematical-embedding prototype as a precursor to standard embedding support in AI engines such as GPT.The authors argue that possible obsolescence does not negate the value of prototypes that use AI in new ways, including for human comfort and understanding.Source: Risks and Limitations · p. 23
  • Digitizing mathematics does not make it generally understandable; the project instead seeks human-deployable mathematics that users can operate without specialist understanding.The passage likens this to using an automobile without detailed knowledge of how it works.Source: Risks and Limitations · p. 23

Conclusion

The conclusion proposes AI Math Agents and mathematical tools as interfaces to formal spaces and as components of expanded digital mathematical infrastructure. It frames this infrastructure as a way to broaden mathematical access and support scientific discovery for human and AI audiences.

Conclusion

  • The conclusion frames smart network infrastructure as cognitive infrastructure, where networks become cognitive agents and AI observer-agents interact with mathematics.LLMs are described as linguistic interfaces through which AI can access digitally rendered natural language, program code, and mathematics.Source: Conclusion · p. 23
  • Technology extends human perception beyond familiar space and time, while mathematical interfaces may provide qualitative access to more foundational structures of reality.The passage points to telescopes and microscopes as tools for revealing scales and physical domains that humans cannot directly perceive.Source: Conclusion · pp. 23–24
  • Math Agents could provide an interface to mathematics and extend access to formal spaces, computational complexity, and potentially artificial, quantum, and biological forms of intelligence.The passage presents this as a potential way to interact with reality at a foundational mathematical level and test the claim that mathematics is foundational.Source: Conclusion · p. 24
  • The work introduces mathematical embeddings, equation clusters, mathematical ecologies and mathscapes, and AI Math Agents to advance digital mathematical infrastructure.The proposed infrastructure could democratize access to mathematics and support discovery and applications in health, energy, and space, as well as AI alignment goals.Source: Conclusion · p. 24

Glossary

The glossary defines AI-agent, alignment, and infrastructure concepts alongside terminology for representing, analyzing, and supervising mathematics with AI. It also describes human feedback and smart-network technologies as parts of this computational setting.

Glossary

  • An Agent is an AI predictive entity that learns, solves problems, and updates behavior through rewards-driven actions; AI Alignment means compatibility with human values.An Autonomous Cognitive Entity is an AI agent empowered to undertake autonomous operations.Source: Glossary · p. 24
  • Computational Infrastructure treats digital reality as a collection of formal methods, while Cognitive Infrastructure incorporates AI-agent capabilities.Digital Mathematical Infrastructure is an accessible digitized mathematics corpus that Math Agents learn and mobilize for insight and deployment.Source: Glossary · pp. 24–25
  • Formal methods are systematic, mathematically rigorous techniques; formalization renders a domain with them, and formalization space comprises possible mathematical, algorithmic, programmatic, and other formal approaches.The glossary also includes information-theoretic, graph-theoretic, and computational-complexity approaches in the formalization space.Source: Glossary · p. 25
  • A Math Agent operates in the digital mathematical domain to identify, represent, analyze, integrate, write, discover, solve, theorem-prove, steward, and care-take mathematical ecologies.A mathematical ecology, or mathscape, is a set of related mathematical equations.Source: Glossary · p. 25
  • Embedding represents a data element as a character string in high-dimensional vector space; mathematical embedding applies this representation to mathematical entities for AI-based machine-learning analysis.An Equation Cluster groups similar equations in a mathematical-ecology embedding visualization.Source: Glossary · p. 25
  • Human-in-the-loop supervision validates AI results, while RLHF is iterative human-AI feedback and Human-AI Entities are RLHF partnerships described as substrate-agnostic, scale-free intelligence.Smart Network technologies include AI agents, machine learning, blockchains, and quantum computing deployed as standard formal methods in computational infrastructure.Source: Glossary · p. 25

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