Approaches to Solving Mathematical Problems in a Complicated Way
In the realm of mathematical problem-solving, it is often more efficient to utilize established formulas and techniques that have been refined over time. However, there are instances where a more deliberate and time-consuming method may be desired. This article explores various approaches to solving mathematical problems, specifically in the domains of differential calculus and algebraic topology, by delving into first principles and foundational definitions.
Differential and Integral Calculus
Mathematical problems, such as those encountered in differential and integral calculus, can typically be addressed using various established formulas and techniques. For instance, derivatives of implicit functions can be found using the chain rule and other known formulas. However, a solution can equally be pursued through first principles, which offer a more foundational and complex perspective.
Consider the derivative of an implicit function $y f(x)$. From first principles, the derivative is defined as:
[ frac{dy}{dx} lim_{h to 0} frac{f(x h) - f(x)}{h} ]This method, while providing a rigorous and fundamental approach, can be notably cumbersome and time-intensive for particularly complex functions, leading to many messy and intricate steps.
Algebraic Topology: Homology via Simplicial Complexes
In algebraic topology, homology is a fundamental concept. Traditionally, homology is computed using various tools such as the Mayer-Vietoris sequence, homology sequences, and functors like Ext or Tor, fiber bundles, among others. These methods provide a streamlined and efficient approach, allowing for a more accessible and manageable calculation. However, as noted by mathematician Albrecht Dold in his Lectures on Algebraic Topology, there is also the traditional approach of computing homology through simplicial complexes.
Dold emphasizes the elegance and utility of the simplicial complex approach, describing it as akin to computing integrals using Riemann sums. Just as Riemann sums provide an intuitive but inefficient method for practical integration, solving homology through simplicial complexes is a more foundational and less practical method, despite its theoretical importance.
“Computing homology with simplicial chains is like computing integrals $int_{a}^{b} f(x)dx$ with Riemann sums where $f(x)$ is a complicated function!! - Albrecht Dold, Lectures on Algebraic Topology, 1980
Dold’s analogy underscores the distinction between foundational understanding and practical application. The former is essential for deeper theoretical insight, while the latter is more efficient for solving real-world problems.
Conclusion
The solutions to mathematical problems can vary widely depending on the context and the desired complexity. While established methods offer efficiency and accuracy, exploring first principles and foundational definitions can provide a deeper understanding. Whether you are working in differential calculus, integral calculus, or algebraic topology, choosing the right approach can significantly impact your journey towards finding a solution.