Why Mathematicians Complicate Things While Physicists Simplify: A Closer Look at Differential Geometry and Beyond

The perception that mathematicians unnecessarily complicate mathematical concepts, such as differential geometry, while physicists make them simple and elegant is rooted in the different goals and methodologies of the two fields.

Goals and Focus

Mathematics

The field of mathematics is driven by a quest for abstract foundations and rigorous understanding of underlying structures. Mathematicians often seek to develop complex theories that lay the groundwork for deeper insights and broader applications. Here are the key aspects:

Abstract Foundations: They aim to understand the intrinsic structures and principles governing various mathematical systems. While this can lead to intricate and often abstract theories, these theories can provide profound insights and serve as a basis for new developments. Rigorous Proofs: Mathematics emphasizes formal and rigorous proofs to ensure that theorems are universally valid. The complexity can arise from the need to address every detail and edge case. However, this thoroughness ensures the robustness and reliability of mathematical theories. Generalization: Mathematicians frequently generalize concepts to broader contexts, such as differential geometry, which applies to numerous fields including physics. This process can result in intricate theories but also provides a more comprehensive understanding of the underlying principles.

Physics

Physics, on the other hand, prioritizes practical applications and usability. Here are the key aspects:

Practical Applications: Physicists often develop models and tools that can be applied to real-world problems. This can involve making simplifying assumptions to make complex systems more manageable while still capturing the essential features of the phenomena under study. Intuitive Understanding: Physicists strive to convey concepts in an intuitive and visually appealing manner. This can make the underlying mathematics seem simpler and more accessible, aiding in the development of a deeper, more intuitive understanding of the subject matter. Effective Theories: In physics, effective theories are developed to describe phenomena without needing to delve into the full complexity of the underlying mathematics. This approach can lead to more practical and immediately useful models and tools.

Interplay Between Fields

Mathematics as a Tool

Physicists often use mathematics as a language to describe physical phenomena. They select the most effective mathematical tools for the specific problem at hand, which can lead to the perception that they simplify complex mathematics. This interplay highlights the complementary nature of the two fields and how they can benefit from each other's expertise.

Mathematical Beauty

While many mathematicians also appreciate beauty in their work, it often manifests in different forms. For instance, the elegance of proofs or deep connections between different areas of mathematics. This pursuit of beauty can drive the development of new and innovative mathematical concepts.

Conclusion

While it might seem that mathematicians complicate things, their work often lays the groundwork for deeper understanding and broader applications. Physics, on the other hand, tends to emphasize usability and intuitive understanding. Both approaches are valuable, and the complexity in mathematics can lead to profound insights that eventually benefit physics and other fields.

Ultimately, the choice between complexity and simplicity depends on the context and objectives of the work being done. Whether you need the robustness of rigorous proofs or the simplicity of effective models, both perspectives offer unique insights and solutions to the challenges of the natural world.