Die Mathematik, die wir brauchen.

Lim Woong, a professor teaching an introductory machine learning course during an international summer program, has observed a recurring theme among his students: the need for stronger foundational mathematics skills. This year’s class included students from over ten countries, mostly first- or second-year undergraduates eager to understand how machine learning algorithms function and how artificial intelligence intersects with their academic disciplines. At the conclusion of the course, Lim asked students to compose a brief memo for future participants. Almost all mentioned the necessity of improving their Python programming abilities. However, many also expressed a renewed determination to deepen their understanding of mathematics. While the emphasis on Python was expected, Lim found the interest in mathematics particularly intriguing. In South Korea, mathematics holds a unique significance due to its role in the competitive college admissions process. For many Korean students, serious engagement with mathematics typically concludes once they secure university acceptance. This trend is understandable given the intense focus on high-stakes exams throughout their secondary education. As a result, many students associate mathematics primarily with the goal of passing these tests rather than exploring its broader applications. Despite the country's reputation for producing top-performing students, Lim noted that those with the highest examination scores do not always find the mathematics underlying artificial intelligence straightforward. This phenomenon highlights a common issue in Korea, where high-achieving students often encounter difficulties in university-level mathematics despite their success in standardized testing. Exploring the mechanics of artificial intelligence exposes the limitations of an educational system focused solely on test performance. While programming is the visible aspect of AI, mathematics provides the structural foundation for the code. It dictates what calculations an algorithm performs, the assumptions it makes, and the reasons behind its effectiveness in certain contexts versus others. Mathematics transforms abstract notions of “learning” into sequences of operations executable by computers on real-world data. Take, for instance, principal component analysis (PCA). If a dataset characterizes individuals, objects, or events using numerous features, many of which overlap, PCA identifies a reduced set of new directions that capture as much variability as possible within the data. To achieve this, the data is first centered, and feature relationships are encapsulated in a covariance matrix. PCA then identifies the matrix’s eigenvectors and eigenvalues. Each eigenvector outlines a new directional path through the data, while its corresponding eigenvalue quantifies the amount of variation captured along that direction. PCA retains only the directions associated with the largest eigenvalues, potentially revealing patterns obscured in the initial representation. The mathematical expressions used here are relatively simple: Av = λv, where A represents the covariance matrix, v denotes an eigenvector, and λ signifies its eigenvalue. This equation implies that multiplying A by v alters the vector’s magnitude without changing its direction. In PCA, this direction becomes a new axis, while λ indicates the level of information retained. The challenge lies not in the individual mathematical concepts themselves, but in connecting these ideas. Students must navigate transitions between centering, covariance, vectors, matrices, projections, and changes of basis, while simultaneously linking equations, geometric intuition, and functional code. Understanding the symbols is just the starting point. They must interpret what these symbols represent, convert them into code, analyze outputs, and determine if the outcomes align with expectations. Students who demonstrate the greatest progress often persist with challenging concepts. They question why an equation takes a specific form, sketch the geometry, write a few lines of code, review the output, and attempt again. These students do not equate immediate confusion with an inherent lack of mathematical ability. Instead, they frequently return the following day with additional queries. These experiences foster productive mathematical exploration. Students formulate hypotheses, conduct experiments, identify points of failure, and refine their understanding accordingly. Mathematics evolves into an area of investigation rather than a fixed body of knowledge to be memorized.