This presentation investigates the pivotal role of mathematics in providing explanation and scientific understanding, grounded in Immanuel Kant’s anti-metaphysical philosophy of mathematics and my own research on abduction, diagrams, and cognitive affordances. Kant’s framework casts mathematics as a transformative tool that organizes empirical phenomena into a coherent system of relations, enabling advanced cognitive processes aimed at making the universe intelligible. I extend this perspective to explore three key aspects:

1) the generalization of Kant’s “Aesthetics” and “Logic” to support a naturalized and historicized view of mathematics, emphasizing the interplay between mathematical concepts and their historical development;

2) the critical role of mathematical modeling as a powerful method for generating explanatory scientific understanding, serving as a counterbalance to the overemphasis on big data; and

3) the importance of mathematically sound cognitive schemata to prevent superficial modeling mischaracterized as scientific.

Incorporating my own focus on abductive reasoning and diagrammatic thinking, I argue that mathematics functions as a dynamic explanatory framework that structures scientific knowledge and enhances our capacity to comprehend the world. This analysis offers a fresh perspective on the cognitive and epistemic contributions of mathematics to scientific inquiry and intelligibility.