Galileo famously stated that the universe is written in a “grand book” whose language is “mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”

Similarly, Einstein asked, “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?”

These thoughts naturally lead to one of the greatest questions in philosophy: Was mathematics invented or discovered? Is mathematics a purely human construct – an invented set of tools, or does mathematics exist embedded in nature and independent of human existence waiting for our discovery?

The school of thought that believes mathematics is invented is called “anti-realism.” Anti-realists believe that if humans never existed, neither would mathematics.  Anti-realists’ view is that we observe the universe and then formulate theories and mathematical equations to reflect those observations. The rules we create are maintained and reevaluated by humans and are subject to change as we learn more about the physical universe. Anti-realists point to examples of mathematics being developed to explain existing physical questions.  For example, Isaac Newton developed calculus to explain physical properties of the universe he observed such as the acceleration of objects due to gravity and why planetary orbits are ellipses. Calculus grew and developed and perfectly explains and predicts changes in motion in our universe.

The flip side of the coin are the realists who believe that math contains universal truths which humans discover over time.  Realism as summarized by Michael Lessel of Lehigh University: “Mathematical realism holds that the universe operates based on a strict set of equations that governs all of its behaviors and that humans simply reveal them.” Support for this view is found where entire realms of math are created with no real world use and later are found to describe some physical phenomenon.  For example, mathematician Évariste Galois developed a field of mathematics known as “group theory” in the early 1800s. Group theory was developed solely to determine whether polynomial equations were solvable. In the mid-1900s it was found that group theory was important in understanding various aspects of subatomic particles. Another example comes with Bernhard Riemann’s work on non-Euclidean geometry in the mid-1800s which was then used by Einstein to create the the general theory of relativity. Another Einstein example is that the mathematics of general relativity predicted that “gravitational waves” exist, but technology didn’t exist to prove the existence of gravitational waves until the 21st century. 

So, which is it? Discovered or Invented? There is no clear answer and mathematicians and philosophers are divided. Physicist Mario Livio, who has spent a lot of time studying this topic, believes that mathematics is both invented and discovered. He summarizes it like this: “A pattern emerges: humans invent mathematical concepts by way of abstracting elements from the world around them–shapes, lines, sets, groups, and so forth–either for some specific purpose or simply for fun. They then go on to discover the connections among those concepts.”