Mathematics: Invention or Discovery?

The debate of human creation or objective truth.

Posted May 10, 2025

There is a true beauty to mathematics, not only in the feeling of fulfilment one often gets as everything falls into place, creating a flawless solution, but in the way that mathematics is used in society and without it, we would not be able to live the way we do. This raises the question of whether Mathematics is an invention of the human race, or simply a placement of nature to which we are discovering. Both of these perspectives contain compelling arguments, invention suggested by with the absence of humans there would be a lack of the concept; discovery supported by the complex mathematical patterns found in nature, notably the golden ratio (φ).

Discovery

Mathematics is “the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organising them”. To truly answer the question of invention or discovery, the initial concepts of mathematics need to be understood. Key mathematical discoveries, noting the word discovery, would include Pythagoras’ theorem, Newton and Leibniz’s development of calculus and Gödel’s incompleteness theorems. These are just a few examples showing how mathematics evolves over time. The reliability and the application of these ideas convey the idea that mathematics is more than a human construct but instead a fundamental part of the universe to which we are discovering.

Building on the argument that mathematics is an inherent part of the universe there exists numerous unsolved mathematical problems – one such is the Riemann Hypothesis, which a proof to which would shed light on prime number distribution. The very existence of having a distribution of prime numbers that we do not know would be incompatible with the concept of invention of mathematics. If mathematics were an invention of society these concepts simply could not exist independently of our knowledge. The existence of previously unsolved mathematical problems later proved correct further supports this argument. One such example is the proof of Fermat’s Last Theorem by Andrew Wiles. This beautifully demonstrates how the theorem’s truth was inherent to nature, and thus mathematics itself, even after 350 years since its first introduction by Fermat.

Invention

An alternative perspective may argue that the world itself, and the society that we have built is simply a human construct. Without human perception, no such concept of mathematics would exist. The construct of mathematics can be traced back to around 18,000 BCE, where etchings, similar to what we would call tally markings, can be seen upon an Ishango bone. This viewpoint would pose the argument that numbers, formulae and shapes are just mental tools created by humans and with that, mathematical statements are just syntactical strings and manipulated symbols, pieced together with no inherent meaning beyond their application in our society.

Furthermore, throughout the world there are many cultural differences in how mathematics is approached, not just in the discrepancies in notations and methods there are other differences. Notably, the Mayans used a vigesimal, or base 20, number system; the Babylonians, a sexagesimal, or base 60, number; and the Egyptians a duodecimal, or base 12, number systems. These contrast from the denary, or base 10 number system that is commonly recognise today. The reasoning behind these changes in bases is down to cultural differences within societies. It is suggested that the Mayans used base 20 as they lived in a warm climate and thus counted with both their fingers and toes. This is just one example that would indicate the variations within mathematics and whilst there may be many objective truths, such that 1+1 is indeed 2, the way that humans conceptualise and communicate these truths is subjective.

Further, the introduction of smart transportation systems within recent years evidences the integration of engineering design with technology. Autonomous vehicles would ideally use advanced algorithms to optimise fuel consumption and reduce traffic congestion per journey, returning to the idea of every journey contributes to sustainability and innovation.

Objective Truths

To counter this, the phrase objective truths is important to consider, and the meaning of why these truths in mathematics are objective. Whilst many can be argued to be of human construct, other mathematical ideas such as the Fibonacci sequence, appear throughout nature in the world in which we inhabit. The Fibonacci sequence shows itself in sunflowers, pinecones and even rabbits. The golden ratio (φ) presents itself in art, architecture and at the very core of nature, biological structures. To push this point further, physical phenomena are governed by the laws of mathematics. From waves on Earth all the way to planetary orbits within the solar system, we can predict cycles with the use of mathematics, suggesting its existence within nature independent of humans.

TAlbert Einstein is famously quoted as asking, “how is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?”. At the very basic level, this quote reiterates the argument of objective truths, and how it would be improbably, arguably impossible, for every aspect of mathematics to align with the universe’s own interpretation. This paradox leads to questions of whether humans have free will, or whether our cognitive abilities are moulded by the mathematics of the universe which exist outside of our society.

Einstein himself was able to predict natural events with the use of his field equations. Einstein predicted the existence of gravitational waves, which were later confirmed by the LIGO observatory, once again reiterating a point of discovery as opposed to invention.

A Philosophical Take

Reverting to the idea of objective truths, Platonism is the metaphysical view that argues mathematical constructs exist in an abstract realm, and thus their existence is independent of humans and our language. The most important argument for Platonism is by Gottlob Frege who argued that numbers exist independently of us, and they are neither psychological constructs nor linguistic conveniences. Frege further showed that a great number of mathematical theorems are true but only as the factors to which they are dependent on are true. Hence, since the theorems are true, the mathematical constructs behind them must exist, and that is in this abstract realm with Platonism describes.

Opposing this is the argument of Nominalism, or anti-Platonism, which suggests that mathematical constructs, such as numbers and sets do not actually exist. For instance, there is an idea that there is 3 of something and a number within the mind; but it does not exist outside of the human mind, and truly the number 3 does not exist. This viewpoint is argued by Jody Azzouni who suggests that whilst mathematical concepts seem to refer to objects, it is complete with the same effect as describing a fictional character. Azzouni rejected the idea of objective truths and requiring reference to real objects, this allows mathematics to function without committing to a realm where numbers rule first.

To Conclude

It is evident that mathematics is embedded into the universe, from planetary movements to fractals and the golden ratio in biological structures, mathematics is all around us. The patterns suggest a discovery to mathematics and its objective truths which exist independently from humans. However, the mathematics that the human cognit is familiar with is shaped by human invention within the symbols, notation and methods that we use on a daily and date back to early civilisation.

Ultimately, mathematics appears to be discovered, not invented, but a synthesis of both could be a compelling conclusion, and whilst the underlying principles of mathematics exist outside of our control, we have developed the tools and understanding to manipulate ideas and bridge a connection between human society and the fundamentals of the universe we reside in.