The Origins and Nature of New Mathematics
The question of where new mathematics comes from is a fascinating one, as it delves into the creative and logical processes that drive the development of mathematical concepts. Mathematics, in essence, is a language that allows us to map and describe the relationships in our physical world. Throughout history, new branches of mathematics have emerged from the need to solve complex problems, often leading to revolutionary new ways of viewing objects and their interactions.
Conjectures and Mathematical Pioneers
Some of the greatest minds in mathematics, such as those of Fermat and Euler, have contributed to the development of new mathematical concepts through what are known as conjectures. These conjectures are often the starting points for new branches of mathematics. Other mathematicians tackle these challenges by examining special cases or sub-problems. Occasionally, these efforts lead to interesting results that divert their focus into studying related but distinct problems.
With a combination of luck and extraordinary brilliance, an entirely new branch of mathematics may emerge, which could be tangentially or entirely unrelated to the initial problem.
Defining New Types of Mathematics
The concept of 'new types' of mathematics is rather broad. To clarify, I will first ask: what do you mean by 'type' of math? Additionally, what is mathematics itself? Math can be seen as a language that enables us to represent and manipulate physical relationships symbolically, with the ultimate goal of aiding computational processes.
For me, a 'new type' of math involves a fundamental change in the way we view basic objects and their interactions. It's a new language that allows us to articulate and explore novel relationships. Let's explore some examples to illustrate this idea.
New Mathematical Languages
From the basics, we started by representing physical quantities as simple symbols like 1, 2, 3, etc., and developed the four basic operations to manipulate these symbols. This progression from nothing to something represents a new type of math.
The introduction of algebraic expressions like y2x allowed us to represent more abstract relationships, effectively moving from specific instances to general principles. This shift required a new way of thinking and communication, qualifying as a new type of math.
Calculus, another groundbreaking development, introduced the concept of 'change' as a new object, along with operations to understand how this change interacts with other objects. This opened up the possibility of modeling physical phenomena in new and powerful ways, exemplifying a new type of math.
Linear algebra also represents a new type of math, as it encourages us to view a collection of similar objects as a single 'vector' and the relationships among these objects as a 'matrix'. The invention of matrix-vector multiplication rules further extended the language of mathematics in a way that was previously impossible.
Similarly, complex numbers and their operations are a new type of math, allowing us to describe two-dimensional attributes that real numbers cannot capture. Even the operations on complex numbers, such as cross and dot products, were born out of the need to represent real-world interactions symbolically.
The Role of Physical Reality
While it's clear that much of mathematics is abstract and developed through symbolic manipulation, I believe that any truly new mathematical language must be rooted in our interaction with the physical world. The need to represent certain relationships that our previous mathematical symbols and operations cannot capture is the driving force behind new types of math.
For example, physicists and applied mathematicians often pioneer new types of math by inventing languages that allow us to describe new phenomena. For pure mathematicians, proving new theorems does not necessarily constitute a new type of math, as these theorems typically contribute to understanding within an existing framework.
Thus, new types of math arise from a symbiotic relationship between the language of mathematics and the physical world. They enable us to explore and model new relationships that were previously beyond our grasp, potentially leading to entirely new branches of mathematics.
Conclusion
The process of discovering new types of math is a blend of creativity, logical reasoning, and an understanding of the physical world. It is a continuous journey of uncovering new ways to describe and manipulate mathematical objects and their interactions, thereby expanding the boundaries of mathematical thought and application.