Inventing New Math: Challenges and Possibilities

The notion of creating a new type of mathematics might seem daunting, but it also holds immense potential for advancement in various fields, from engineering to theoretical physics. This article explores the challenges and possibilities of inventing a novel mathematical system, drawing on personal experiences and insights from prominent mathematicians.

Personal Impulse: A System of n^n

When I was in 8th grade, I developed a unique system based on the pattern n^n. Each digit in the system had a base value that increased exponentially: 1 for the first digit, 4 for the second, 27 for the third, and so on. Intriguingly, I found that the maximum value for the nth place that did not exceed the next base value was the nth digit’s base value multiplied by 3n. This discovery was considerably more complex and profound for a 13-year-old me than it may seem at first glance.

The Potential of New Mathematics

The possibilities for engineering applications and problem-solving with new mathematics are vast and diverse. From Hilbert spaces to fractals, innovative mathematical systems have opened doors to new solutions and imaginative applications. For instance, the invention of Hilbert space or the exploration of fractals can inspire a plethora of engineering ideas, and these could very well be the same individuals who conceived the new mathematics in the first place.

Motivation and Realistic Goals

Robert Harvey’s perspective on the nature of math is correct. Generally, mathematics is a formal language designed to solve problems, not to invent new types of math. The primary goal is to find solutions that address different types of problems and challenges within existing frameworks. However, there is a real issue where traditional math theories and practices are insufficient.

One central challenge is the need to expand and refine fundamental concepts. For example, the understanding of basic arithmetic operations can be imprecise. We often treat addition, summation, and distribution as the same, yet they serve different purposes in mathematical contexts. Similarly, number theory is often vague and lacks clear explanations and meanings. The definitions of measurement and dimension are also inadequate, which hinders the development of theories such as Einstein's relativity and the proof of the Riemann hypothesis.

The problems with geometric constructions persist even today. Euclid’s famous dictum that all geometric problems can be solved with two instruments—ruler and compass—is based on certain assumptions. However, modern interpretations argue that the compass and the straightedge (ruler) are fundamentally the same, as a compass can be extended into a straightedge. This leads to the question of whether we can substitute the straightedge with a more versatile instrument, like a compass with additional functionalities.

Focus on Theoretical Understanding and Practical Fixes

Inventing a new type of math may be ambitious, but it is often more rewarding to focus on fixing the existing theoretical frameworks. We already have a robust mathematical system, albeit with limitations due to a lack of theoretical understanding. The real challenge lies in refining and expanding the theoretical underpinnings of mathematics. This involves addressing gaps in definitions, clarifying fundamental concepts, and ensuring that our mathematical tools are versatile enough to tackle complex problems.

Simply inventing new math without a clear purpose or practical application is often pointless. Instead, we should aim to enhance our existing mathematical toolbox and develop a deeper theoretical understanding. By doing so, we can unlock new possibilities and solve pressing problems in science, technology, and other fields.

Conclusion

The quest to invent new types of math is rich with opportunity and responsibility. While it is exciting to explore new systems, it is crucial to maintain a focus on practical applications and theoretical refinement. By doing so, we can build a more robust and versatile mathematical framework that serves the needs of the modern world.