Infinite Creativity: How Mathematics Expands Our Imagination

Our God is creative. He is, of course, the Creator. Being made in His image, we have the ability to reflect His creativity. Usually, we associate creativity with artists, writers, musicians, or other so-called “creatives,” and for good reason. Creativity, however, is not limited to certain fields. Even someone working in “pure” mathematics, that is theoretical or abstract math, also reflects the creativity of God.

To start, we need a definition of creativity. “Creativity is the ability to form novel and valuable ideas or works using one’s imagination” (Creativity, 2025). For some, it still might be unclear how doing math could possibly satisfy this definition. Perhaps this is because most people do not have a clear idea of what a mathematician actually does nor what mathematics truly is.

For many people, the way math is presented in K-12 schools, where it often feels like a list of rules and facts to memorize, is what causes them to see math as dull, rigid, and disconnected from imagination. However, this kind of math is a far cry from what a professional mathematician does. While learning basic mathematical facts and practicing their use is essential for a mathematician, it’s only a small piece of the picture. To compare, learning basic mathematical facts and practicing their use is similar to learning and practicing the basic elements of art and design, or learning and practicing the use of different types of paint, brushes, or canvases. A painter cannot produce a breathtaking work of art without first practicing the basics, just as a mathematician cannot produce a brilliant new mathematical idea without first learning the basic mathematical tools available to them.

But as I said, learning the basic mathematical facts is only part of a mathematician’s job. What else do they do? Well consider all those facts that you learned in school. Where did they come from? Why do we know they are true? We didn’t just make them up or have them handed to us out of nowhere. In fact, each one of those mathematical facts you learned, and many others that you’ve probably never learned, were proven to be true by a mathematician using a logical argument called a proof. Creating these logical arguments might not sound especially creative, but the ideas mathematicians have proven and the methods they've developed to prove them are often incredibly imaginative and original.

As an example, let’s explore a mathematical idea not often taught in schools: the idea that some infinities are different sizes, a concept developed by mathematician Georg Cantor. One fact he showed was that the “amount” of counting numbers, (i.e. 0, 1, 2, 3,…), is the same as the number of positive rational numbers, (i.e. numbers that can be written as positive fractions, such as ½). By “the same amount” I mean that every counting number can be paired with exactly one positive rational number, or in other words, you can “count” the rational numbers—a rather counter intuitive fact as there seems to be many more rational numbers than counting numbers. His proof comes down to a very clever method of arranging the rational numbers in a precise way in a grid, allowing you to then “count” the rational numbers (see Figure 1). Using a similarly creative argument he showed that the real numbers (i.e. all numbers, including decimals, fractions and irrational numbers like ) cannot be counted, so the infinite set of real numbers must be “bigger” than the infinite set of positive rational numbers!

It required a great deal of imagination to consider that there might be different sizes of infinity—and even more creativity to devise a way to demonstrate it. Discoveries like this show that mathematics isn’t only about rules and calculations, but also about uncovering surprising truths through careful reasoning and creative thinking.

Mathematicians are tasked with proving new mathematical facts that have never been discovered before. They often need to exercise creativity in determining what avenue to explore next. They need inspiration from somewhere, just as a painter needs to find inspiration for their next work. A popular source of inspiration for a mathematician is asking questions about what is already known—or thought—to be true.

As an example, consider the following geometrical idea: if you have a straight line and a single point not on the line, then there is only one straight line through that point that is parallel to the first line. This is what is referred to as “Euclid’s fifth postulate” and was presented in 300 BC by Euclid in his book Elements as an obvious geometric truth. Mathematicians eventually started asking whether this postulate could be proven to be true using only very basic geometric facts, (Euclid’s first four postulates), but for hundreds of years, they had no luck. Eventually, some mathematicians asked a new question: could we imagine a space where the fifth postulate was not true?

This creative thinking eventually culminated in 1823 when mathematicians Bolyai and Lobachevsky realized that it was possible to do geometry in spaces where Euclid’s fifth postulate did not have to be true, so called “non-Euclidean spaces.” In these new geometries, it is possible either a) for there to be no distinct line parallel to a line through a particular point, or b) for there to be an infinite number of lines through a point parallel to a particular line. And in these new geometries, it is possible for things to happen that cannot happen in our standard, “Euclidean,” geometry, like an equilateral triangle with three right angles (see Figure 2).

The creativity exhibited by Bolyai and Lobachevsky resulted in entirely new branches of mathematics. This is this same type of creativity used by mathematicians every day to push the field of mathematics forward.

Another way to express creativity is by pushing boundaries of resources. As an example, some painters use unconventional tools like a kitchen utensils and blowtorches create their art. Mathematicians can do the same. Utilizing tools from areas of math that seem, at first, to be unrelated to the problem they are trying to solve can present an opportunity for creativity in math.

My favorite example of this is recent work towards solving the following open problem: if you have a sufficiently smooth path in 2D space that starts and ends at the same place (i.e. a loop), can you draw a square that has all four corners on this loop (see Figure 3 for examples)? This problem was posed in 1911 by Otto Toeplitz, and it wasn’t until 2020 that Greene and Lobb solved (a slightly tamer version of) this problem. They found that, yes, there must be not only a square, but also a rectangle of any aspect ratio with all four corners on the loop. The brilliant and creative part of their solution: they translated the question to the seemingly unrelated mathematical field of topology, where they used very basic properties of funky shapes called Möbius strips and Klein bottles (pictured in Figure 3) to solve their difficult problem. Though not an exact solution to the original 1911 problem, which allowed slightly weirder loops, Greene and Lobb’s creativity resulted in the biggest step towards solving that original problem yet.[1]

Mathematics is, at its core, a deeply creative field. Creativity isn’t just a rare spark in the discipline—it’s seen through the daily work of mathematicians and has driven some of the most significant breakthroughs in the field. Without creativity, the field of mathematics would be stagnant. But instead, it is vibrant, rich, and full of beauty.

As mathematicians, we have the great privilege of uncovering some of the amazing “invisible” aspects of our world (cf. Colossians 1:16) and reflecting God’s creativity in the process.

Whether you are a professional mathematician or not, I encourage you to find ways to explore how the field of mathematics can stretch your thinking in new and imaginative ways—a practice vital to our lives as Christians.

Footnotes:

[1] To learn more about their work, I recommend watching this video.

References:

• Creativity. (2025, July 11). Retrieved from Wikipedia: https://en.wikipedia.org/wiki/...
• Vsauce. (2015, August 3). Retrieved from X: https://x.com/tweetsauce/statu...