Paradigm Cycles in Mathematics and Astronomy: A Historical Analysis of Scientific Revolutions and the Recurrent Breaking of Established Limits

This paper investigates the cyclical dynamics of paradigm shifts in mathematics and astronomy, tracing how scientific communities have repeatedly defined boundaries that were later surpassed through revolutionary discoveries. Drawing upon Thomas Kuhn’s The Structure of Scientific Revolutions (1962), it offers a comprehensive historical analysis comparing the evolution of mathematical and astronomical thought with the recurring structure of paradigmatic cycles. The study reveals that contemporary science may be undergoing a similar phase of paradigmatic entrenchment, potentially constraining the exploration of alternative perspectives. By examining key historical transitions—including the emergence of non-Euclidean geometry, the development of calculus and infinitesimal methods, and the transformation from geocentric to heliocentric cosmology—this paper demonstrates that scientific progress consistently follows Kuhn’s sequence of normal science, anomaly, crisis, revolution, and paradigm replacement. The findings suggest that modern scientific communities must remain vigilant against the temptation to declare certain problems unsolvable or certain limits absolute, since history repeatedly shows that such proclamations have proven premature.