What are some math concepts that have a disagreement between mathematicians and physicists?

One area of disagreement between mathematicians and physicists is the use of infinitesimals in calculus. Infinitesimals are quantities that are infinitely small and were originally used by early mathematicians like Newton and Leibniz. However, these infinitesimals lacked a rigorous definition, leading to skepticism among mathematicians. To address this, mathematicians developed the concept of limits, which allowed for a more precise and rigorous formulation of calculus without relying on the notion of infinitesimals. Today, most mathematicians prefer using limits because they provide a solid foundation for calculus that avoids the ambiguities associated with infinitesimals.

On the other hand, physicists often continue to use infinitesimals because they are intuitive and practical, especially when dealing with physical phenomena. In physics, the emphasis is on solving real-world problems, and the use of infinitesimals often simplifies the process of setting up and solving differential equations. For example, when calculating the motion of particles or the flow of fluids, infinitesimals can be a convenient tool. Physicists are generally less concerned with the rigorous foundations of these concepts as long as the results are consistent with experimental observations.

Another significant point of contention is the Dirac delta function, which is widely used in physics, particularly in quantum mechanics and signal processing. The Dirac delta function is not a true function in the traditional mathematical sense because it is not defined at all points and does not satisfy the usual rules of calculus. Instead, it is treated as a "distribution," which acts like a function that is zero everywhere except at a single point, where it is infinitely large in such a way that its integral over all space equals one. While physicists find the Dirac delta function extremely useful for modeling point charges, impulse forces, and other localized phenomena, mathematicians initially viewed it with suspicion due to its lack of rigor.

However, over time, mathematicians developed the theory of distributions, which provided a more rigorous framework for understanding objects like the Dirac delta function. This theory reconciled the mathematical concerns with the practical needs of physics, though the way the delta function is introduced and used in physics still differs from the more formal mathematical approach. This divergence reflects the broader difference in priorities between the two fields: physicists prioritize utility and alignment with physical intuition, while mathematicians focus on formal rigor and logical consistency.