We lay the groundwork for a formal framework that studies scientific theories and can serve as a unified foundation for the different theories within physics. We define a scientific theory as a set of verifiable statements, assertions that can be shown to be true with an experimental test in finite time. By studying the algebra of such objects, we show that verifiability already provides severe constraints. In particular, it requires that a set of physically distinguishable cases is naturally equipped with the mathematical structures (i.e. second-countable Kolmogorov topologies and $σ$-algebras) that form the foundation of manifold theory, differential geometry, measure theory, probability theory and all the major branches of mathematics currently used in physics. This gives a clear physical meaning to those mathematical structures and provides a strong justification for their use in science. Most importantly it provides a formal framework to incorporate additional assumptions and constrain the search space for new physical theories.
Safe: Enhancing Mathematical Reasoning in Large Language Models via Retrospective Step-aware Formal Verification
School of Computer Science, National Key Laboratory for Multimedia Information Processing, PKU-Anker LLM Lab, Peking University（计算机学院、多媒体信息处理国家重点实验室、PKU-Anker LLM实验室、北京大学）；Huawei Noah’s Ark Lab（华为诺亚实验室）；The Hong Kong University of Science and Technology（香港科技大学）；The Hong Kong Polytechnic University（香港理工大学）
Correctness, Artificial Intelligence, and the Epistemic Value of Mathematical Proof
Department of Logic and Philosophy of Science University of California, Irvine（逻辑与科学哲学系加州大学伊维奇分校）；Department of Mathematics University of California, Irvine（数学系加州大学伊维奇分校）
FormalMATH: Benchmarking Formal Mathematical Reasoning of Large Language Models
The Chinese University of Hong Kong（香港中文大学）；Numina；Westlake University（西湖大学）；M-A-P；University of California, Los Angeles（加州大学洛杉矶分校）；Max Planck Institute for Intelligent Systems, Tübingen（蒂宾根马克斯·普朗克智能系统研究所）
Proceedings of the 4th Workshop on Formal Reasoning about Causation, Responsibility, and Explanations in Science and Technology
University of Konstanz（康斯坦茨大学）；CNRS（法国国家科学研究中心）；Université de Paris（巴黎大学）；King’s College London（伦敦国王学院）；Alexandru Ioan Cuza University of Iasi（雅西亚历山德鲁·伊万·库扎大学）；Western Norway University of Applied Sciences（西挪威应用科学大学）；INRIA（法国国家信息与自动化研究所）；Stevens Institute of Technology（史蒂文斯理工学院）；University of Bologna（博洛尼亚大学）；University College London（伦敦大学学院）；University of Pennsylvania（宾夕法尼亚大学）；Reykjavik University（雷克雅未克大学）；Icelandic Institute for Intelligent Machines（冰岛智能机器研究所）；Université Paris Est - Créteil（巴黎东-克雷泰伊大学）
Information Science Principles of Machine Learning: A Causal Chain Meta-Framework Based on Formalized Information Mapping
Koguan School of Law, China Institute for Smart Justice, School of Computer Science, Shanghai Jiao Tong University（柯 guar 法学院、中国智能正义研究院、计算机科学学院、上海交通大学）
