Theoretical vs. Applied Mathematics: An Artist vs. Craftsman Analogy
Mathematics can be broadly categorized into two streams - the theoretical and the applied. Each stream has its own distinct characteristics, much like how an artistic endeavor differs from a craft-based one. Just as an artist and a craftsman approach their work differently, mathematicians working in the realms of pure and applied mathematics also have their unique focuses and philosophies.
Theoretical Mathematics: The Artistic Pursuit
Theoretical mathematics can be thought of as the artistic side of the discipline. Here, mathematicians pursue the beauty and elegance of general theorems and their proofs. They seek to explore abstract concepts with a view to express the essence of mathematical truth through innovative aesthetic insights. The primary goal is to delve into the underlying structures and principles that govern the universe, and to do so with a sense of open-ended exploration. As mathematicians in this field, they are not bound by practical constraints and can revel in the intricacies and complexities of pure ideas.
Applied Mathematics: The Practical Craft
On the other hand, applied mathematics can be likened to a practical craft. Mathematicians who specialize in this area tackle specific, well-defined problem spaces with the aim of finding solutions that can be applied to real-world scenarios. The focus here is on precision, simplicity, and efficiency. While these mathematicians also appreciate the beauty of mathematics, their primary objective is to solve practical problems and to create techniques and products that can be used for beneficial purposes.
Comparing the Two: Beauty and Utility
One argument presented in this analogy is that beauty in mathematics has evolved. In the past, simplicity and elegance were considered the epitome of beauty. However, in our current era, beauty is more complex and often intertwined with practicality and innovation. This can be seen as a shift from the uncomplicated "beauty" of a craftsman to the more nuanced and multifaceted beauty of an artist. The craftsman strives for perfection in a narrow and well-defined space, while the artist explores broader and sometimes unpredictable contexts.
The inherent friction between these two approaches lies in their differing perceptions of beauty. The theoretical approach values the aesthetic expression of mathematical ideas, whereas the applied approach values the practical solutions and techniques derived from mathematical principles. This contrast further accentuates the difference in approach, where the artist may focus on innovation and novel insights, while the craftsman concentrates on achieving flawless execution.
No Deep-Trenched Difference in Approach
Despite these differences, it is important to note that the distinction between theoretical and applied mathematics is not as neat as the initial comparison might suggest. In reality, there is often an overlap where one mathematician might engage in both theoretical and applied work. A true artist may occasionally produce quick, impromptu pieces for fun or to showcase their versatility, and a skilled craftsman may occasionally step out of their comfort zone to create art that transcends their practical focus.
From a methodological and formal standpoint, there is no deep-trenched metaphysical or Platonic difference between the two approaches to mathematics. All mathematics, regardless of its theoretical or applied nature, is ultimately a blend of abstract thought and practical application. Whether a mathematician is engaging in pure theory or applied practice, the core essence of their work remains the same – the exploration of mathematical truths and the application of those truths to solve problems and create new knowledge.
Both theoretical and applied mathematics contribute to our understanding of the world in unique and valuable ways. While they serve different purposes, they are interconnected and complement each other, providing a comprehensive view of the mathematical landscape. In the end, the distinction between them is not absolute, but rather a flexible and contextual one, reflecting the dynamic and multifaceted nature of mathematical inquiry.