One example of a specific thing that physicists tend to do, which can sometimes draw criticism from pure mathematicians, is the use of mathematical models or approximations that prioritize practicality and predictive power over mathematical rigor and formal proof.
In physics, scientists often develop mathematical models to describe and understand physical phenomena. These models are typically based on mathematical equations that capture the essential features of the system under study. However, in order to make progress and obtain useful results, physicists sometimes employ simplifications, approximations, or heuristic arguments that may not meet the rigorous standards of pure mathematics.
Pure mathematicians, on the other hand, generally prioritize formal proof and mathematical rigor, striving for absolute certainty and logical consistency in their arguments. They may criticize physicists for using mathematical techniques that are not fully justified or rigorously proven.
For example, physicists might make use of "hand-waving" arguments or rely on intuitive reasoning to develop physical theories or derive mathematical expressions. While these methods can often lead to useful and accurate predictions, pure mathematicians may object to the lack of airtight logical justification or the absence of rigorous mathematical proofs in such approaches.
Another area of potential criticism is the use of certain mathematical techniques or concepts in physics that may not be fully explored or understood in the realm of pure mathematics. Physicists sometimes repurpose mathematical tools or borrow concepts from other branches of mathematics, adapting them to describe physical phenomena. This can lead to fruitful results, but pure mathematicians might argue that the full mathematical implications and rigor of these techniques have not been adequately addressed.
It is worth noting that this difference in approach between physicists and pure mathematicians stems from the distinct goals and requirements of their respective disciplines. Physics aims to understand and explain the natural world, often prioritizing practicality and empirical verification, while pure mathematics focuses on rigorous logical reasoning and the exploration of abstract mathematical structures. Both approaches have their strengths and contribute to the advancement of knowledge in their respective domains.