Tangles A Structural Approach to Artificial Intelligence in the Empirical Sciences

Tangles offer a precise way to identify structure in imprecise data. By grouping qualities that often occur together, they not only reveal clusters of things but also types of their qualities: types of political views, of texts, of health conditions, or of proteins. Tangles offer a new, structural, approach to artificial intelligence that can help us understand, classify, and predict complex phenomena. This has become possible by the recent axiomatization of the mathematical theory of tangles, which has made it applicable far beyond its origin in graph theory: from clustering in data science and machine learning to predicting customer behaviour in economics; from DNA sequencing and drug development to text and image analysis. Such applications are explored here for the first time. Assuming only basic undergraduate mathematics, the theory of tangles and its potential implications are made accessible to scientists, computer scientists, and social scientists.

Part I. Tangles – A New Paradigm for Clusters and Types: 1. The idea behind tangles; 2. The notion of a tangle; 3. The two main tangle theorems: an informal preview; Part II. Tangles in Different Contexts – A Collection of Informal Examples: 4. Examples from the natural sciences; 5. Examples from the social sciences; 6. Examples from data science; Part III. The Mathematics of Tangles – Concepts, Theorems, Algorithms: 7. The formal setup for tangles; 8. Tangle theorems; 9. Order functions; 10. Choosing the feature system; 11. Algorithms; Part IV. Applying Tangles – Back to the Examples: 12. Applying tangles in the natural sciences; 13. Applying tangles in the social sciences; 14. Applying tangles in data science; Notes; References; Symbol index; Subject index.

Reinhard Diestel is Chair of Discrete Mathematics at Universität Hamburg, where he works on structural graph theory and combinatorics. He is the author of the leading graduate-level text 'Graph Theory' (6th edition, 2024), which has been translated into German, Russian, Japanese, and Chinese.