The Language of Natural Numbers: Statistical Properties and Neural Learning of Arithmetic Structures

Date and Venue: Wednesday, 28 October, at 14:00 - ICTP Leonardo Building - *Euler Lecture Hall*

The relationship between artificial intelligence and science is increasingly bidirectional: AI offers new tools for discovering and exploring scientific structures, while science provides controlled domains in which to investigate the capabilities and limitations of AI. We examine this interplay in the fundamental setting of arithmetic. Iterated prime factorization turns the integers into a deterministic sequence of words whose vocabulary and admissible combinations are governed by arithmetic. We establish asymptotic laws for vocabulary growth and word frequencies, examine statistical dependencies, and show that a transformer trained solely on such text captures structural regularities, including sensitivity to arithmetically forbidden sequences. This resulting language supports both directions of inquiry: neural models make arithmetic structure accessible through learning, while arithmetic supplies precise constraints against which to examine what these models learn and how they represent it.

About the Speaker:
PhD in Mathematics and MS in Mathematical Physics (ISAS, Trieste), Laurea in Physics (University La Sapienza, Rome). Postdoc at Princeton University and University of California. Visiting Professor at the Tokyo Institute of Technology, at the Institut Henri Poincaré and at New York University. Research interests in Mathematical Physics, Statistical Mechanics and applications (machine learning, data science). Author of Perspectives on Spin Glasses (Cambridge University Press, 2012) and Probabilità Elementare, Teoria ed Esperimenti (Zanichelli, 2008). Publications in Communications in Mathematical Physics, Nature Scientific Reports, Journal of Statistical Physics, Annales Henri Poincare, Physical Review Letters. Writes on scientific debates of public interest in newspapers like Il Corriere della Sera, Il Sole 24 Ore and magazines like Il Mulino and Vanity Fair. Schloessman Award (2000) from the Max Planck Society. Member of the Science Academy of Bologna.