Human beings are capable of remarkable creativity. In the Arts, we express emotion and imagination freely, creating works interpreted through personal and cultural lenses. In contrast, structured systems like language and games are invented with specific rules and purposes. Science, however, seeks to uncover objective truths - laws of nature that exist independently of us. But where does Mathematics fit? Is it a human invention, like language or chess, or is it a discovery, revealing patterns and truths that were always there, waiting to be found? This long-standing philosophical question invites us to reflect on how we understand knowledge, truth, and what it really means to “learn” something.
There are compelling arguments that mathematics is objective - something discovered, not invented. The Pythagorean Theorem describes a relationship between the sides of right-angled triangles. It holds true no matter who observes it or what language is used. This relationship exists whether or not we ever noticed it. Likewise, properties such as divisibility or primality are intrinsic to numbers: 13 is prime whether we call it “thirteen” or “shí sān.”
Students encounter π early on - a peculiar number essential for calculating the measurements of a circle. But π shows up not just in geometry, but in statistics, physics, and even in the rhythms of music and nature. It’s a number that appears everywhere - whether we’re looking for it or not. π wasn’t invented - it was discovered, hiding in plain sight across the universe. The success of mathematics in describing the natural world - across physics, engineering, and computer science - suggests it maps onto something real.
Yet mathematics is also clearly invented in many respects. Humans have created multiple number systems, from Roman numerals to base-60 Babylonian counting. Our modern system of notation that we use throughout schooling (algebraic symbols, functions, matrices) is a relatively recent invention. Even mathematical structures, like statistics or calculus, are built upon definitions and axioms we have chosen. Different axiomatic systems (like Euclidean vs non-Euclidean geometry) can all be logically consistent, even if they contradict each other. This suggests mathematics is not a single, fixed truth, but rather a flexible framework of human-designed tools for understanding different kinds of patterns.
So, was mathematics invented or discovered?
The evidence seems to suggest: both. Some aspects of mathematics, like fundamental relationships and patterns, exist independently of us and are discovered. Others, such as the symbols, systems, and frameworks we use to work with those patterns, are invented. In that sense, mathematics is like a map: the landscape (truths and relationships) exists objectively, but the map (notation, models, language) is created by humans to help navigate that terrain.
This distinction matters, especially in the context of learning mathematics. When students learn mathematics, they are not merely memorising facts or algorithms - they are learning to explore a landscape that is partly discovered and partly designed. They develop fluency in a human-made language for describing patterns, and they also learn to uncover universal truths that transcend culture and context. Recognising this dual nature - mathematics as both invented and discovered - helps students see it not just as a subject to study, but as a way of thinking. It fosters creativity, critical reasoning, and a deeper appreciation for the patterns that shape our world.
Adam French, Head of Department - Mathematics