Creating New Mathematics: Is It Really Creation?

The question of why we say mathematicians create new mathematics, when mathematics seems to already exist, delves into the depths of the philosophy of mathematics. It's a fascinating topic that touches upon the nature of mathematical objects, our understanding of them, and the language we use to describe their discovery. Are mathematicians inventors who conjure new concepts from thin air, or are they explorers charting a pre-existing, immutable landscape? This article explores these ideas.

The Eternal Existence of Mathematical Truths

One perspective, often associated with Platonism, posits that mathematical objects and truths exist independently of human thought or activity. In this view, numbers, shapes, and theorems are real entities residing in an abstract realm. Mathematicians, therefore, don't create mathematics but rather discover it. They uncover pre-existing relationships and structures, much like a scientist discovers laws of physics. This perspective aligns with the feeling many mathematicians have when they stumble upon a particularly elegant proof or a profound connection between seemingly disparate areas of mathematics. It's as if they've glimpsed a small part of a larger, eternal truth.

Consider the Pythagorean theorem. Did Pythagoras invent the relationship a² + b² = c² for right-angled triangles? Or did he discover a truth that was always there, waiting to be revealed? Platonists would argue the latter. The relationship existed regardless of whether anyone had conceived of it or proven it. The act of discovery simply brought it into our awareness. Similarly, the properties of prime numbers, the intricacies of calculus, and the complexities of topology are all seen as existing independently, awaiting exploration. This view emphasizes the objectivity and universality of mathematics. It suggests that mathematical truths are not merely human constructs but are fundamental aspects of reality itself. It's a powerful and compelling idea that has resonated with mathematicians and philosophers for centuries.

Mathematics as a Human Construct

However, another school of thought argues that mathematics is, in fact, a human construct. This perspective, often associated with constructivism or formalism, emphasizes the role of axioms, definitions, and logical deduction in the creation of mathematics. According to this view, mathematical objects and truths do not exist independently but are instead brought into being through our definitions and proofs. Mathematicians, therefore, are not simply discovering pre-existing truths but are actively constructing them.

From this perspective, the axioms we choose to build our mathematical systems upon are arbitrary. We could, in principle, choose different axioms and create entirely different mathematical structures. Euclidean geometry, for example, is based on a set of axioms, including the parallel postulate. By changing this postulate, we can create non-Euclidean geometries, which are equally valid mathematical systems but have different properties. Similarly, the development of set theory by Cantor led to the creation of transfinite numbers, which expanded our understanding of infinity in ways that were previously unimaginable. These examples highlight the creative power of mathematics, the ability to invent new concepts and structures that were not previously conceived of. This view emphasizes the subjectivity and relativity of mathematics. It suggests that mathematical truths are not absolute but are instead dependent on the underlying axioms and definitions.

The Role of Abstraction and Generalization

Regardless of whether one believes mathematics is discovered or created, there's no denying the importance of abstraction and generalization in its development. Mathematicians often take concrete examples and extract their essential properties, creating abstract concepts that can be applied to a wide range of situations. For instance, the concept of a group arises from abstracting the properties of symmetry operations, such as rotations and reflections. By focusing on the underlying structure, mathematicians can develop general theories that apply to many different areas of mathematics and science.

Similarly, the process of generalization involves extending existing concepts to more general settings. For example, the concept of integration, initially developed for functions of a single variable, has been generalized to functions of multiple variables, to complex-valued functions, and even to abstract spaces. This process of abstraction and generalization allows mathematicians to create new tools and techniques that can be used to solve a wide range of problems. It also leads to a deeper understanding of the underlying connections between different areas of mathematics. It's through these processes that mathematics continues to evolve and expand, pushing the boundaries of our knowledge and understanding.

The Interplay of Discovery and Creation

Perhaps the most nuanced view is that mathematics involves a combination of discovery and creation. Mathematicians may start with a set of axioms or definitions, which are human constructs, but then discover surprising and unexpected consequences that follow from those assumptions. The process of proving theorems can be seen as a form of exploration, where mathematicians navigate the logical landscape implied by their initial assumptions and uncover hidden relationships and structures. This view acknowledges the role of both human creativity and the inherent logic of mathematical systems. It suggests that mathematics is not simply a matter of inventing arbitrary concepts but of exploring the consequences of those concepts and discovering the underlying truths that emerge.

Consider the development of fractal geometry. Mathematicians like Mandelbrot did not simply invent fractals out of thin air. They were inspired by patterns observed in nature, such as the branching of trees and the coastline of Britain. However, they then developed a mathematical framework for describing and analyzing these patterns, creating new concepts and tools that have had a profound impact on many areas of science and technology. Similarly, the development of string theory in physics has led to the creation of new mathematical concepts and structures, such as Calabi-Yau manifolds, which are now being studied by mathematicians in their own right. These examples highlight the interplay between discovery and creation in mathematics, the way in which human creativity can lead to the discovery of new and unexpected truths.

The Significance of Mathematical Language

The language we use to describe mathematics also influences our understanding of it. The term "creating new mathematics" may simply be a convenient way of expressing the idea that mathematicians are actively involved in developing and expanding the body of mathematical knowledge. It doesn't necessarily imply that they are inventing something that didn't exist before, but rather that they are bringing new concepts and relationships into our awareness.

Furthermore, the term "creation" can also refer to the process of organizing and structuring mathematical knowledge. Mathematicians don't simply discover isolated facts; they create coherent theories that connect different ideas and provide a framework for understanding them. The development of a new branch of mathematics, such as category theory or algebraic topology, involves creating a new language and a new set of tools for exploring a particular area of mathematics. This process of organization and structuring is itself a creative act, one that shapes our understanding of the mathematical world.

Ultimately, whether we say mathematicians create or discover mathematics may depend on our philosophical perspective. Regardless of our viewpoint, it's clear that mathematics is a dynamic and evolving field, driven by human curiosity and ingenuity. The act of exploring mathematical concepts, proving theorems, and developing new theories is a creative endeavor that expands our understanding of the universe and our place within it.

In conclusion, the debate over whether mathematicians create or discover mathematics highlights the profound and enduring questions about the nature of mathematical reality. While Platonism suggests a pre-existing mathematical universe waiting to be explored, constructivism emphasizes the human role in building mathematical systems. A balanced view acknowledges both aspects, recognizing that mathematicians both uncover inherent truths and shape the landscape of mathematical knowledge through abstraction, generalization, and creative problem-solving. Regardless of one's philosophical stance, the ongoing process of mathematical inquiry remains a testament to human ingenuity and our relentless pursuit of understanding the fundamental structures of the universe.

For further exploration of the philosophy of mathematics, consider visiting the Stanford Encyclopedia of Philosophy.