Mathematics is a field that both fascinates and perplexes those who venture into its abstract domains. But one of the most enduring philosophical questions surrounding mathematics is whether it is invented or discovered. In the grand narrative of mathematics, are we merely uncovering truths that already exist, or are we constructing them from the ground up? This age-old debate has profound implications not only for the nature of mathematics itself, but also for our understanding of reality, knowledge, and the role of human cognition in shaping the world around us.
In this post, we will explore the central ideas surrounding the invention versus discovery debate, considering various perspectives from the realm of philosophy, and examining how our understanding of mathematics impacts how we view the universe.
The Argument for Discovery: The Objective World of Mathematics
To claim that mathematics is discovered is to embrace the idea that mathematical truths exist independently of human thought. This perspective holds that mathematical objects, structures, and relations exist in an abstract, objective realm, which we, as humans, are able to uncover through logical reasoning and empirical observation. According to this view, mathematical entities such as numbers, sets, or geometric shapes are not products of our imagination; rather, they are inherent features of the universe, waiting to be discovered.
A key supporter of this viewpoint was the philosopher Plato. He famously posited that mathematical forms, or abstract objects, exist in a non-physical realm, separate from our sensory experiences. For Plato, humans could gain knowledge of these forms by engaging in rational thought. This position is often called mathematical realism, and it suggests that the objects of mathematics have an existence independent of human minds. For example, the number 2 exists whether or not anyone is there to count it, and a triangle’s properties—such as the sum of its angles being 180 degrees—hold true regardless of human perception.
Contemporary mathematical realists, especially those adhering to structuralism, tend to argue that mathematical structures exist independently of us. According to structuralists, mathematics describes the relationships between abstract entities, and these structures are discovered rather than invented. The number 5, for instance, has no intrinsic properties apart from its relationships with other numbers, such as being one greater than 4 and one less than 6. These relationships hold true whether or not we are aware of them.
The appeal of the discovery model is its simplicity and elegance—mathematics describes a reality that is true regardless of our subjective experience or cultural context. The fact that we can continually make new discoveries, such as solving previously unsolved problems or finding new mathematical structures, further reinforces this view.
The Argument for Invention: The Human-Constructed Nature of Mathematics
On the other hand, some philosophers argue that mathematics is a product of human invention—a tool created by us to describe patterns, relationships, and concepts in the world. According to this view, mathematical concepts do not exist independently of human minds; instead, they are human-made inventions, developed over centuries as tools to make sense of our surroundings.
The perspective that mathematics is invented is often associated with the philosophy of nominalism, which asserts that abstract entities like numbers or geometric forms do not exist in any objective sense. Instead, they are linguistic or cognitive constructs. This view is seen most clearly in mathematical formalism, a school of thought that emphasizes the manipulation of symbols according to specific rules rather than the discovery of underlying truths.
For example, in formalism, a mathematician’s work is akin to a game: the symbols and axioms are created for the purpose of problem-solving. Numbers are mere symbols with no existence outside of the rules we assign to them. From this perspective, the structure of mathematics is a human-made construct, a language we have invented to describe and analyze patterns.
The fact that mathematics has such a diverse range of applications—from the natural sciences to economics to computer science—can be seen as evidence for its human-made, functional nature. Mathematics is not “out there” in the world, waiting to be uncovered, but is instead a product of human cognition and ingenuity, crafted to make sense of the world.
A Balanced View: Interplay Between Invention and Discovery
While the extremes of discovery and invention present compelling cases, a more nuanced view emerges when we recognize that both elements may be at play in the development of mathematics. Perhaps mathematics is a blend of both discovery and invention—an evolving interplay between objective reality and human creativity.
Mathematical objects could exist in a certain sense, independent of human minds, but the way we understand and describe these objects is shaped by human cognition. In this view, humans may invent mathematical systems and languages to articulate the underlying truths of the universe, but these systems are successful because they correspond to real, discoverable structures in the world.
The work of mathematicians often begins with intuition or creative insight—humans invent mathematical concepts and structures. However, over time, these concepts are subjected to rigorous logical reasoning, uncovering deeper layers of mathematical truths that were not immediately apparent. This back-and-forth process, where new inventions lead to discoveries and new discoveries lead to further invention, forms the heart of mathematical progress.
For example, consider the concept of negative numbers. At one point in history, negative numbers were an invention—an abstract idea that seemed unnecessary or nonsensical. Yet, as mathematics progressed, it became clear that negative numbers were necessary to fully describe the world, such as in accounting, physics, and geometry. In this sense, negative numbers were invented, but their utility in describing the world suggests that they were also “discovered.”
Implications of the Debate: What Does it Mean for Our Understanding of Reality?
The question of whether mathematics is invented or discovered goes beyond abstract philosophical musings; it has profound implications for our understanding of the universe. If mathematics is discovered, it suggests that the universe is inherently structured in mathematical terms, and our ability to understand it is limited only by our intellectual capacity. This is a view that aligns with the belief in a rational, ordered universe governed by mathematical laws.
On the other hand, if mathematics is invented, it implies that our understanding of the universe is shaped by the tools and concepts we create. This would suggest that the universe may be more mysterious, and less determined by a pre-existing mathematical order, than we might have assumed. The success of mathematical models in describing physical phenomena would then be seen not as a reflection of a deep mathematical reality, but rather as the result of our own ability to create effective models.
Conclusion: An Ongoing Journey of Exploration
Ultimately, the question of whether mathematics is invented or discovered may not have a definitive answer, and perhaps it’s a question that doesn't need to be fully resolved. Mathematics serves as a bridge between the abstract world of ideas and the physical world we inhabit. It allows us to navigate the unknown, to develop frameworks that help us make sense of our surroundings, and to explore the universe on both the smallest and largest scales.
Mathematics may be both a product of human ingenuity and a reflection of deeper truths waiting to be uncovered. Regardless of which side of the debate one leans toward, it is clear that the exploration of mathematical thought is a journey of profound significance—one that continues to evolve, offering new insights into the world and our place within it.
Further Reading:
The Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures by Thanos Kehagias
What is Mathematics, Really? by Reuben Hersh
Mathematics and the Imagination by Edward Kasner and James Newman