Artificial intelligence (AI) is prompting every field of human endeavour to reconsider its future: Is human engagement and investment still valuable? What kinds of expertise will remain essential? Where should future effort be directed? This document argues that mathematics provides the foundations of modern AI, and further mathematical innovation will be essential to AI's future development. As AI systems become more capable, more pervasive, and more consequential, progress will depend not simply on larger models, greater computational power, or wider data access, but on deeper theoretical understanding. Questions of reliability, interpretability, optimisation, uncertainty, safety, and robustness are all, in substantial part, mathematical questions. A strong mathematical sciences research base is therefore essential both to the UK's capacity to develop the next generation of AI and to the evaluation, deployment, and regulation of AI—responsibly and for the benefit of society as a whole. This "primer" brings together short contributions from across the mathematical sciences to show why continued investment in mathematics, statistics, and data science is critical if we are to reap the benefits of AI in the future. Each section introduces a mathematical topic, notes some of its established applications, and outlines its relevance to AI. We could have included many more. The reader is referred to the Royal Statistical Society's Policy piece, AI is Statistics, and the SIAM report The Role of Applied Mathematics in a New Era of Artificial Intelligence for further examples of the importance of mathematical science for current and future developments in AI. Taken together, these pieces are intended to provide foundational evidence to support a wider advocacy effort within and beyond the mathematical sciences community: to help policymakers, funders, university leaders, and other decision-makers recognise that sustained investment in the mathematical sciences is vital to the future of AI—and to the public benefit that future AI may deliver.