The question of mathematics being an invention or a discovery has been debatable since ages, not just in mathematical community, but also in general public. Are mathematicians creating mental models and constructions, which have no actual reality but are so consistent with reality, elegant and accurate enough to explain the physical universe that those constructions can be considered ‘real’? Or that these constructions have always been there, waiting to be pondered upon and discovered by the great minds of ancient and modern civilizations? The truths that have always been there and are getting uncovered by mathematicians. There are certain cases in mathematics like the Mandelbrot set and structures of complex numbers which might make us think and believe in the latter than the former. Also, there are some other cases where it seems like it is more an invention rather than a discovery. A supporting instance would be when a mathematical structure does not have a compelling uniqueness and the mathematician finds a need to introduce some contrived and far from unique construction in order to achieve some very specific end and make the construction consistent in explaining the physical reality or phenomena.
Mathematics has been the most successful and is the most mature of the sciences. Its first great master work — Euclid’s ‘Elements’ — which helped to establish the field and demonstrate the power of its methods, written in 300 BC; and it served as a standard text in the mathematics curriculum well into the twentieth century and the twenty first century.
There is probably no other science which presents such different appearances to one who cultivates and one who does not, as mathematics. To [the non-mathematician] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple of bloom of vigorous youth, everywhere stretching out after the “attainable but unattained,” and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan’s road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness.
– C. H. Chapman, Review of Sophus Lie’s Theorie der Transformationsgruppen (1892) Bulletin of the New York Mathematical Society 2, p. 61.
The question still remains the same. Is mathematics an invention or a discovery?
Not everyone had the number zero
The Romans were great engineers but they had a terrible number system. It didn’t even have zero. The number system used in ancient India had zero- it was invented by the Indian mathematician and ancient astronomer Aryabhata However, it was known by other very old cultures like the Mayans in Central America and the Babylonians (from ancient Iraq). And ancient Arab mathematicians not only knew about zero but also really spread the idea of algebra after the 9th century. The word “algebra” itself comes from a text by a famous mathematician called al-Khwarizmi.
People in the middle ages in Europe thought fractions were the hardest math ever One 11th century monk reportedly said:
After spending months working hard and studying, I finally grasped this thing called fractions!
And in the 16th century, people thought negative numbers were pretty evil. They had other names for these numbers, like “absurd” or “defective”.
Before human evolution, before the Cambrian explosion, before the Earth took shape, before any heavy atom formed in the universe — the number 7 was already prime, the number 1729 was still the smallest number which could be expressed as the sum of two different cubes in two different ways, the exponential function already had a period of 2πi, the concept of zero “nothingness” still existed and there were no bijections between any set and its powerset, even then.
Does this mean that we discover rather than invent? Is it as straightforward as it sounds? Well, quite NOT!
The fact that 1 plus 1 equals 2, or that there’s an infinite number of primes, are truths about reality that held even before mathematicians knew about them. As such, they’re discoveries — but they were made using techniques invented by mathematicians. For example, according to Pythagoras’ theorem, the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. This is true for all right-angled triangles on a level surface, so it’s a discovery.
Patterns and numbers are everywhere and have always been there
We live in a mathematically-driven and mathematically-understandable world. Mathematical numbers, figures and patterns can be found everywhere from hexagonal honeycombs made by bees to the Fibonacci sequence in pinecones, seashells, trees, flowers, and leaves. The number π helps us understand the sun, the moon, and has led to many scientific and technological advancements.
Discovery vs. Invention
Discovery consists of the first finding of that which exists but is unknown. It includes what may be newly inferred either by deduction or by induction. What has been discovered by others can still be discovered by someone else as new knowledge. The discoverer, however, does not impact what is discovered. Discovery is always about finding. Invention, on the other hand, is always about creation. It creates something (material or immaterial) that did not exist before. The act of invention requires purpose and always results in a construct. The inventor defines and creates the invention. Once invented by someone, something may be discovered by others.
Mathematics is a tool to understand the Universe!
Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning
— Richard Feynman
Mathematics has been the most successful and is the most mature of the sciences. It has a large aesthetic content. But first and foremost, it is a language to describe the universe around us and the relationships and patterns we discern. The pure mathematician practices his art, if art it is, using language of mathematics to express his discovered thoughts. He discovers wonders expressed in the invented symbology and grammar of mathematics. Mathematics is part of our language to describe reality. There are many mathematical descriptions of reality, called mathematical models, and some of these models are pretty good, so we use them. That’s what science is about. Science studies reality, constructs models of reality, and evaluates those models. Science overlaps engineering and technology which use those models. Mathematics, science, engineering, and technology are closely intertwined. Mathematics is a language like all other invented language, albeit with a much more stringent and rigorous grammar than conventional language. It describes the relationships and patterns we find in the world around us.