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Is Mathematics Discovered or Invented?

Clara Valtorta

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For a class project in graduate school, I went through a popular algebra textbook and noted each reference to the history of mathematics. In all, there were only about a dozen references. All of them were to specific mathematicians. What had started out as a project on the history of mathematics ended up focused on the people who are credited with “discovering” mathematics. This project, coupled with an interesting geometry class, got me thinking about how we discuss the origin and development of mathematics.

Recently I’ve noticed that the history of mathematics is completely ignored in math class. This leaves out interesting, human aspects of mathematics. It also may prevent students and teachers from addressing the rich question:

Is mathematics invented or discovered?

I know, I know, there is no “correct” answer to this question. But that does not make it a worthless question. This question ties together history, math, and philosophy. It may engage younger learners who are unmotivated by the procedural and repetitive aspects of classroom mathematics.

My goal with this blog post is to encourage everyone - students, mathematics teachers, people who work with maths, and those who do not - to think about this topic. So, I have been asking friends and acquaintances their opinions. Here are some of their responses to think over.

How do you define discovered? Invented?

Nicholas Bartlett, One on Epsilon data scientist:

In my opinion, these two words mean the same thing, but we usually say discovered in physical space and invented in "conceptual space." I think of all ideas as already existing as if buried in the earth, somehow latent in the connected space of concepts, and simply waiting for something to do the work necessary to unearth it. An argument in support of this point of view is that discoveries in physical and conceptual space occur in a certain order, according to "inaccessibility."

James Tanton, founder of the Global Math Project:

If you open your fridge and see a chocolate fudge cake sitting inside, you wouldn't say that you created (invented?) that cake, but instead that you discovered it. That seems clear.

If you open your fridge and you see in it all the ingredients sitting there for you to make a chocolate fudge cake, and then you make the cake, you might likely say that you created the cake, not invented it, but you didn't invent or create the building blocks of it - the butter, the eggs, the chocolate, etc.

Yousuf Marvi, high school math teacher and One on Epsilon content editor:

For something to be invented, it must have an inorganic origin. If the origin of math is in symbolic representation, then we can call the symbolism invented. If it’s in logic, then it’s an adaptation; if it's about finding a way representing numbers, then it is discovered.

Mathematics, is it discovered or invented?

Phillip Isaac, university mathematics lecturer and researcher and One on Epsilon content editor:

I would have to say both. There is certainly a feedback loop of discovery/invention in mathematical research. For example, once results are discovered as a result of something invented like a definition, it is often the case that a mathematician would go back and "tweak" that definition to understand the effect on the results that follow.

Yousuf Marvi:

I think it is mainly invented just like languages are. If I borrow from Chomsky's Language Acquisition Device theory, all brains have the capacity of language hardwired; however, humans have figured a way out to generate several different unique languages. Similarly, reality might exist but humans invented math to understand it.

Yoni Nazarathy, university mathematics lecturer and researcher and One on Epsilon content editor:

As a mathematics researcher working in stochastic modelling (also known as applied probability), I love to invent my own problems. This often involves inventing mathematical models that exhibit “interesting behaviour” or come with built-in “interesting questions” or “challenging problems." Not much discovery in this part of my work, just invention, invention, invention.

However, what makes mathematical models interesting? A model is interesting if you can discover interesting things about it. That is, once done inventing, I’m left standing in front of nature. In front of the mechanics of the model. In front of the delicate intricacies of the beast. From that point onwards, an incredible process of discovery takes place. Making good discoveries is a rare occurrence, but when they happen I feel great joy.

Frederick Powell, middle school mathematics teacher:

I believe that math was invented because I feel that there was a need for a counting and a data system during ancient civilization. In my opinion, when we invent something, we create something. Therefore math is invented to help world systems thrive and make numbers applicable to worldly situations.

James Tanton:

It feels like, to me, there are basic building blocks (eggs, cocoa beans, cane sugar, wheat stalks, etc) and new blocks that have been put together from them (chocolate, butter, ground wheat), and that, when doing math, I put them together in different ways. I usually make fudge cakes - things known to the world - but every now and then I make something not (publicly) known to the world at all. (So very rarely and usually not interesting!). But I never I feel I can say the word "invented." I feel like I simply discover how building blocks can be put together in new ways (and these can become new blocks to play with later on).

I think mathematics is such a hard and demanding (and therefore appealing and intriguing) enterprise that it generates a sense of humility and awe. "Discovering" feels more fitting to say with that sense of awe.

Epsilon Cat

Does it matter?

Coco Bu, high school mathematics teacher and One on Epsilon content editor:

I guess it matters for mathematicians to realize what it is out there that they need to find and what it is they need to come up with themselves in order to find what's out there. But for me, as an educator, I am more curious about how maths knowledge is explained, delivered and imparted to my learners. Whether it's a theory that is discovered or something people invented or put together, as long as I communicate with my students where the origin is, I don't see it is a big matter for them considering their curriculum.

Azzurra Crispino: Associate Professor of Philosophy:

It does matter in terms of whether we say that mathematical truths are truths, or they are merely constructions and that those constructions may be used to model truth more or less accurately. I think the constructivist (invented) view is left holding the bag that we can approach truth but never have certainty that we have attained it, whereas the realist (discovery) view has the potential for attaining mathematical Truth. Of course, if it turns out that Mathematical Truth (tm) does not exist, then attempting to attain it is a lost cause. On the other hand, if it is possible to attain it, and we do not because we believe ourselves to merely be constructing it, that would be a loss, too.

Ultimately, I am unsure whether this has question makes any practical application changes for those researching mathematicians and logic. But I do think that what position you take on this question may very well change the kinds of approaches and assumptions that you make in your mathematical theorizing. I have long advocated that theorists should be required to list all of their theoretical assumptions at the beginning of a science, math or logic paper, in the same way we list key words. That way, those who are not making the same assumptions can easily know that is the difference, and if some of those theoretical assumptions turn out to later be shown false, we have a way of easily indexing the articles to see what previously held truths need to be revisited.

Phillip Isaac:

I'm not really sure how to answer this. In a sense, my opinion is irrelevant. It is what it is.

Yousuf Marvi:

How we define the origin story of Math definitely matters. Often learners think that math is some grand reality that is beyond their reach; hence, math becomes a factor of ability. However, if we consider the idea of the invention, then we can entertain creativity and hence, subjectivity.

Frederick Powell:

Even though I feel that math is invented, it really doesn't matter in today's classroom. Teachers are spending the school day making math applicable to everyday situations. Students are not invested in the origins of math, rather they invested in how to use math that benefits them and their endeavors.

This topic has been debated and written about extensively. Here are some other great resources to further develop your point of view:

Mathematics is often portrayed as being uncreative, stagnant, and mechanical. I believe posing questions like the ones in this blog post help combat that stereotype. Now it is time to share your own opinion. The following is a survey I created for adults and children alike. The goal of the survey is to get you thinking. Please take and share!

Click here for Survey

An incredible celebration of the joy of mathematics awaits us during Global Math Week, where we hope that over ten million people will celebrate a new (invented or discovered?) method of exploring arithmetic and algebra: Exploding Dots. What is your plan for Global Math Week?