Is Math a Human Invention or a Natural Phenomenon?
Thoughts on the age-old question of “found vs discovered””
Disclaimer
This is an opinion article. Whether you are team “invented”, or team “discovered”… or perhaps team “I have better things to do than wasting time on philosophy”… Feel free to discuss in the comments your view on this topic. I may update regularly with any interesting thoughts provided on the top of my reflections. I am, as always, happy for your feedback.
There is ongoing debate among philosophers and mathematicians about whether mathematics is discovered or invented. Some argue that mathematics is a human invention, a system of symbols and rules created by people to represent and organize abstract concepts. Others argue that mathematics is a universal and exists independently of human consciousness and that we merely discover and uncover its truths. There is no one “correct” answer to this question, and different people may have different perspectives on it depending on their beliefs and experiences.
First, few questions to mess with your head
- If there is a certain number of trees in a forest, but no one counts them, does that number exist?
- Are numbers real?
- Why does the concept of number exist, but the concept of complex number invented?
- Is God a Mathematician? (This question is only for you in case you are religious. I am definitely not, but it is also a title of my favorite book.)
Is mathematics discovered or invented?
What would a mathematician do when answering a complex question? If I have taken one practical skill from my studies, it is reducing problems to small pieces (definitions or axioms). Let´s do the same here and
break the question down to the smallest granularity.
🧠 What is “mathematics”?
It is not entirely clear how mathematics should be defined. I have previously written an article on this topic, so I will refer readers to that article. (This is not an advertisement for my other article, but my laziness to repeat another complex topic here).
🔍 What is “discovered”
The word “discovered” refers to finding or uncovering something that was previously unknown. In science, discovering something means that it already exists and we are gaining knowledge about it. It involves learning about something that was previously unknown or unseen. This word is commonly used in the context of finding new places and scientific breakthroughs.”
🔨 What is “invented”
The word “invented” means to design something that did not previously exist. We could also say “created”, “found” or “constructed”.
There is an ongoing debate over the nature of mathematics throughout history of mankind.
The question of whether mathematics is invented or discovered has been a central issue in the philosophy of math and has implications for our understanding of knowledge, language, reality, and truth.
Indeed, this question still revolves around central mathematical concepts and themes — such as number. The Pythagoreans believed that numbers were living entities and assigned specific properties to them.
There have been different schools with different views on the “invented vs discovered” problem. This idea has been addressed by many well-known philosophers. Kant’s Critique of Pure Reason concludes that space and time, as understood in Euclidean geometry, are not objective features of the world, but are part of the framework we use to organize our experiences.
Platonism is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. The term Platonism is used because such a view is seen to parallel Plato’s Theory of Forms and a “World of Ideas” described in Plato’s allegory of the cave: the everyday world can only imperfectly approximate an unchanging, ultimate reality.
Intuitionists believe that math is a purely human creation and exists only in our minds. Their motto is that “there are no non-experienced mathematical truths”.
Indeed, as a wannabe mathematician and philosopher, I also decided to share my view.
🔍 Arguments for “Math is discovered”
My thoughts on math being discovered
Why would one argue math exists in nature? I would say it is absolute, compared to other sciences where a lot of truths depend on context. It is a universal truth that 2 + 2 is always 4, regardless of the observer.
Argument 1: Our universe and nature is mathematical
Our universe is mathematical in nature according to laws that exist, everything can be explained by numbers and axioms. Everything inside the system can be explained, governed and quantified by math. Even Einstein [1] argued this school of thinking until his death, believing relativity could explain everything perfectly if it could be extended enough, thinking
How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?
Argument 2: Even if we do not discover mathematics, it still exists regardless.
Consider the headline “Huge new prime number discovered” from 2008. It is unlikely that anyone would argue that this prime number was invented in 2008. Right? We are not just randomly discovering new mathematical facts. We are even actively searching for them, meaning we somehow are aware they already are present.
Similar case is calculation of the value of pi. This has come a long way since its first approximations in ancient times. Isaac Newton, who used infinite series and calculus to compute pi to 15 digits in 1665. This was a major achievement, but it was only the beginning. The use of electronic computers and algorithms in the mid-20th century brought even more rapid progress in the calculation of pi. Today, we have the ability to calculate pi to an incredible number of decimal places, with the current record standing at around 31.4 trillion digits.
Argument 3: Discovery of Non-Euclidean Geometry went againts trying to invent Math
This example is my favorite, because it depicts beautifully how humans got trapped doing math correctly, but creating wrong axioms to start with. But observing the real world led us to the right path in the end.
To understand, keep in mind that axiom = a starting rule that is so clear that we take it as a fact in mathematics. On axioms, we then build mathematical theories and prove facts (theorems).
Euclidean geometry was the dominant form of geometry until the 19th century. Euclid set five rules (axioms) for geometry. However, later mathematicians realized that not all spaces are nice-beahving (called Euclidean) and Euclid´s rules dont apply everywhere. See the picture below. In the middle, there is a nice Euclidean space where you can draw a triangle where sum of the inner angles is 180 degrees. However, on the sphere or hyperbolic surface, this is suddenly not true. Therefore, mathematicians needed to come with other sets of rules for other types of spaces. (It took them several centuries though). Fun fact: These other types of spaces and geometries on them later allowed Einstein to develop his theory of relativity.
We have seen that mathematicians were able to develop an entire geometric theory that was largely accepted for centuries, but it was only by looking around the world that this inadequate theory could be expanded. If mathematicians were cut off from the outside world, they might continue to invent something that is not true. But is it really the case that mathematics exists completely in nature and we find it, just like researchers discover new species of plants and animals?
🔨 Arguments for “Math is invented”
My thoughts on math being invented
There are clear indications of human effort in the development of mathematics. Can you simply observe an object and determine its length to be 2 meters? No. Instead, we build up our understanding of math gradually, using axioms as the foundation, much like building with Lego blocks. Of course, these axioms have to be created to make sense, as we have seen on the example of Euclidean geometry. If it is done correctly, this process does not require us to directly observe the external world.
Argument 1: Big parts of mathematics were first invented independent on real world, only using human brain
Some mathematicians even take pride in the fact that math is not directly tied to the real world. Hardy, for instance, developed number theory and claimed that it had no practical use. It was later discovered that his work had applications in fields such as cryptography and genetics. Another example of building math and discovering it only later is Riemann’s work in the 19th century was later utilized by Einstein in his theory of general relativity, demonstrating the practical applications of mathematics.
Argument 2: Some mathematical objects simply do not exist in nature.
For instance, it is not possible for a perfect circle to exist in reality because it has no 3D structure in its coordinates. Limits in calculus cannot exist in reality but help explain the universe in a natural way. Irrational numbers like pi and Euler’s number are also examples of this, as they are infinite numbers that cannot be created from finite energy packets. Limits in calculus cannot exist in reality but help explain the universe in a natural way. Therefore, math can be seen as a tool for explaining reality, but is not equivalent to it.
Argument 3: Is mathematics really absolute truth?
Mathematics doesn´t always exist in nature and theorems are not absolute. When stating something in mathematics, we always claim some conditions ( If X is true, then Y is true). I have said mathematics is absolute truth, but still under some mathematical conditions. Below you can see few very important statements — you don´t have to understand the meaning to see that they all state (at the beginning of the theorems) some conditions for which they apply.
Argument 4: Inventing calculus to treat big and small infinities
- Calculus was invented in the late 1600s by Newton and Leibniz, and it is a technique for understanding continuous motion as a series of infinitesimal steps. This discipline deals with both infinitely large and infinitely small numbers. These fields even deal with infinite series and how to sum them, as well as derivatives (infinitely small changes in a function’s movement) and integration (finding the volume under a curve). Clearly, infinity is a “number” that doesnt exist. You cannot come to a grocery to buy inifnity of spaghetti. It is only a very useful concept invented by humans to approximate what we see in the universe. Without these tools, it would be maybe impossible to describe parts of our world.
Conclusion
As I like to say, it depends. In particular, it depends on how we define mathematics and what we do or dont consider part of it.
Mathematics, like many other things in Science, is a mechanism to describe the world around us. In my opinion, mathematics consists of discovering real world phenomena, modelling them and predicting un-discovered parts of our world. As is usual with models, it sometimes uses heuristics, sometimes is not 100% precise, sometimes we use wrong models and improve them only after some time. Hence, parts of mathematics are invented, parts are discovered.
🔨 Parts of math that are invented
- Mathematical notation — Some people like to call math a language. If you take the language (and hence also notation) as a part of math, then without doubt, this part was invented by humans.
- Axioms — I would say, personally, that axioms are invented, and theorems are discovered (as a consequences of those axioms). Axioms lead us to discover a set of facts we want to be true. This is certainly true for example with imaginary numbers, we invented them so that we can discover the solutions to problems we previously were unable or difficult to solve. Theorems can reflect reality of this world and axioms are our invented tools that help us prove these theorems.
- Models of the world — We invented the rules of matrix multiplication, but the consequences of the way we do matrix multiplications are discovered.
🔍 Parts of math that are discovered
- Mathematical “concepts” — Concepts which are discovered within the universe of an axiom. The concept of numbers exists, but we invent the notation that the glyph ‘1’ and the sound /wʌn/ refers to the concept of singular object that we discovered.
- Theorems — as a “lego” built from the basic components — axioms.
References
[1] https://www.scientificamerican.com/article/why-math-works/
Useful Resources
- Medium Article — https://medium.com/@JacobTheOne/math-discovered-or-invented-c11adfa18f33
- Paralel Postulate — https://en.wikipedia.org/wiki/Parallel_postulate
- Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe — https://theconversation.com/corals-crochet-and-the-cosmos-how-hyperbolic-geometry-pervades-the-universe-53382
- https://www.youtube.com/watch?v=X_xR5Kes4Rs&ab_channel=TED-Ed
- Quanta Magazine — I discovered this magazine while searching for beautiful images to my text. It has loads of interesting articles, worth checking out! For instance this one: https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/
- Mario Livio — Is God a Mathematician? https://www.goodreads.com/book/show/3095024-is-god-a-mathematician
