👉 Critics often argue that the math behind certain mathematical concepts, particularly in areas like set theory and topology, can be overly abstract and disconnected from practical applications, leading to a perceived lack of intuitive understanding. For instance, the axiom of choice, a fundamental principle in set theory, is criticized for its non-constructive nature and the counterintuitive consequences it produces, such as the Banach-Tarski paradox. Critics also point out that some proofs rely heavily on non-standard mathematical tools, like infinitesimals in non-standard analysis, which can be hard to grasp and may not align with the rigorous standards of classical mathematics. Additionally, the use of advanced mathematical frameworks in fields like functional analysis and measure theory is sometimes seen as unnecessarily complex, detracting from the clarity and accessibility of mathematical education.
