# Math: Invented or Discovered?

How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of reality?—Albert Einstein

“Hey, I’ve seen this before. You aren’t making this stuff up!”Once while teaching Algebra 1 in a high school classroom, I put a formula on the board. Instantly one of my students interrupted.

Her assumption to that point was that her math teachers came to school each day and invented the content. Although her comment is pretty humorous, it does bring up an important question. If math teachers aren’t making up mathematics, then where does it come from? Did the Egyptians, Newton, Gauss, and other mathematicians gradually invent mathematics over the course of history? Are the theorems and properties we rely on to build bridges and launch spacecraft the creation of the human mind? Or are they already pre-programmed into the universe waiting to be discovered? These questions lie at the heart of a debate: is mathematics a human invention that we have projected onto the universe or has mathematical truth always existed independently?

Proponents of the invention viewpoint consider mathematics to be a human construct. It works when applied to the world because we invented it with specific characteristics so that it would work. It was designed and created by man with the end goal of modeling our world. We are making it up as we go along, tailoring new mathematics to our needs.

Additionally, we cannot ignore the many instances where mathematics does not successfully model the world.1 If math was embedded in all parts of the universe we would expect there to be functional mathematical models for all aspects of our world. However, there are many areas where mathematics fails us. It may be the tool that propelled us into space, but we are still extremely limited in successfully predicting the weather. In almost any situation we must simplify the complex variables involved before creating a mathematical model.
On the other hand, there are many examples where mathematics appears to be ingrained in the world. Principles from a variety of mathematical fields show up in some of the most unlikely places.Additionally, supporters of the invention viewpoint argue that we must consider the innate mathematical ability many humans have. If someone is born with an aptitude for numbers, where did that genetic predisposition come from? If mathematics is ingrained in the universe then why are some humans such masters of this discipline? It is argued that mathematical ability in mankind makes more sense if mathematics is a human invention that has informed our perception of reality.

Take the example of the 365 days in our year. This number can be formed with an interesting sum:

10*10+11*11+12*12=365

It is surprising to find such an elegant number pattern embedded in the days of our year.

Or we can use an example from nature. When observing the number of petals on flowers (daisies in particular) we often find a majority of them have 1, 2, 3, 5, 8, 13, 21, or 34 petals. These numbers are not random; they are part of a famous sequence called Fibonacci’s Sequence. While this sequence is quite famous in mathematical number theory, it is surprising to find it in something as unrelated as flower petals. Fibonacci’s sequence also appears in the branching of plants and the spirals of sunflowers. It seems to be particularly prevalent in plants.
But it does not only appear in geometry, it is naturally occurring in a number of surprising places: for instance, in the winding of a river. If you compare the length of a winding river with its length as the crow flies the ratio of these two lengths is often very close to .Finally, let’s look at an example of one of math’s most famous numbers: .  is a repeating decimal but it can be approximated as 3.14. It is the ratio of the circumference to the diameter of a circle.

As Christians these patterns and mathematical properties are easily explained by our belief in the origins of creation. The world was planned and designed by an eternal being. This creator used mathematics as a foundation of His world, these mathematical laws are just one of the ways He upholds the universe (Hebrews 1:3). It is our joy as humans to discover these laws bit by bit. Because we are human we do not fully understand everything yet, but we understand enough of the design to know that there must be a Designer (Romans 1:19-20).

In response to the argument that the innate mathematical ability points to mathematics as an invention, it is important to remember that humans are made in the image of God. If God is the great mathematician it is not surprising that humans reflect aspects of his character in our ability to create and think mathematically. Indeed, here is the kindness of God’s wisdom. He created a world based on mathematical principles. Because of this we live in an ordered and structured universe. We can have peace knowing the sun will rise, daisies will blossom, and even rivers will wind in predictable pathways because the Lord has ordained that it be so.

Moreover, he also has equipped humans with mathematical ability. It is now our joy to discover. When we see the movement of the planets we know we serve a God of order. In the constancy and patterns of our days, months, and years we experience in a fresh way His faithfulness. And in the patterns of daisy petals we see that God ordains structures so we can experience beauty. As we gradually discover more of the mathematics in this universe we see more and more of God’s nature.

1 Abbott, Derek. “The Reasonable Ineffectiveness of Mathematics [Point of View].” Proceedings of the IEEE Proc. IEEE 101.10 (2013): 2149.

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# Math and Beauty

This is part of a series of essays I wrote to show how math leads us to know and worship God. Additional essays will be posted in the upcoming weeks.

Math and Beauty

When God created the world he left an imprint of his character. His creation reflects his nature, although imperfectly. When we study nature we can learn about the character of God. Math is meant to reflect and interpret the world around us. Because of this we can catch glimmers of that character of God in mathematics.

It is important to remember that God did not only give us creation. We have been entrusted with the Bible which clearly reveals the nature of God. As we study the world and mathematics we can compare the clues we find about God’s character with what scripture says.

One aspect of God’s character is that he loves beauty. We see this repeatedly in scripture. The descriptions of the tabernacle in Exodus show us that splendor is important to God. Again we see breathtaking glory and majesty described in Revelation. In the Psalms not only is their elegance in the content, but also in the literary form.

In mathematics we see beauty in patterns. One of the great joys of a mathematician is in finding a pattern. Patterns allow us to predict what will come next and they contain an innate elegance. Something in us thrills in recognizing a pattern rather than seeing a random assortment of numbers. Let me demonstrate with some examples.

Look at the resulting products in this list of multiplication facts.

This pattern of sums is formed by adding consecutive integers. The results are called triangular numbers because they can be used to form equilateral triangles.

The triangle above is a famous one in mathematics and is called Pascal’s Triangle. There are many, many patterns in Pascal’s Triangle. One is that the third diagonal row is all the triangular numbers. Another is formed by using an L-shape. If you add all the numbers on a diagonal their sum can be found at the bottom. This is illustrated below.

Explore

Can you determine how Pascal’s Triangle is formed? Then can you find any additional patterns within the triangle? (Hint: try shading in all the multiples of 4).

We can also see patterns in scripture, repetition is a literary feature used to create beauty and meaning. Choose one of the passages below and describe any patterns you find.

Psalm 136, Matthew 5:3-11, Amos 5:4-6

# Where did math come from?

Science, literature, and history all come from concrete examples that we observe in the universe. There organisms, planets, and physical books to analyze. History is the record of things that actually happened. But math doesn’t tangibly exist in the world around us.

What do I mean? Well, you cannot go out and find the Pythagorean theorem or even the number 2. You can find situations where these mathematical concepts are applied,  but you cannot find the actual principle. This has sparked a debate about math that is nonexistent for other subjects.

Where did math come from?

There are two main theories to answer this question. First, some say that math was invented by humans. It exists only in our minds. We created numbers and theorems and then project them onto the universe. There is no number 2, only the concept of 2 that all humans agree upon.

The second theory is that math is embedded in the universe around us, the principals of mathematics are a part of the fabric of our world. Humans didn’t invent these principals, we discover them as we explore our world.

As a Christian, what you believe about math is shaped by what you believe about God and the world. I’ll share some articles and videos on this subject soon but it is worth puzzling over for yourself for a moment. Was it invented or discovered? What do you think?

# God and Math

I am a Christian and I am also a mathematician. During high school I didn’t think those two identities had anything to do with each other (and I was not referring to myself as a mathematician either!). But in college I realized that every academic discipline is affected by our worldview. Even the “unbiased” study of mathematics is based on certain assumptions and those assumptions are based on our philosophy of the world and how it was made.

We cannot separate our faith from any area of our lives. Most of us agree that our worldview should shape our thoughts about every academic discipline…but what might be a rather obvious connection to the humanities and even science can be a bit more elusive for math. Many of us are focused on just getting through whatever math course we have to take, it is an obstacle to overcome so we can get on with our lives and serve God in the church and the workplace.

But I believe God has much more for us than that. Math helps us interpret the world and God created the world, leaving imprints of his character for us to discover. When we discipline ourselves to learn and master mathematics we can expect to gain new glimpses into the character of God. And that is much more convincing reason to get up and study integrals than just wanting to pass an exam.

This summer I am spending time studying the connection between theology and mathematics. I’ll be posting my thoughts and findings here in a series called “God and math.” I hope it serves to stretch your mind and reveal how we can worship God even as we solve equations.