Category Archives: history of math

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

Young boy writes math equations on chalkboard
“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.

daisy-2

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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Proving Pythagoras

When I was in college I learned that proving the Pythagorean theorem is a favorite hobby of mathematicians.  There are dozens of proofs for the Pythagorean theorem, both geometric and algebraic.  Even President Garfield discovered his own method.

My favorite proofs are visual because they really help students understand where the Pythagorean Theorem came from.  And if they understand that they will be less likely to apply it to triangles that do not contain a right angle (a very common mistake!).

We’re talking about the Pythagorean Theorem today in my algebra class and I’m showing this video…a proof in 60 seconds.  Pretty cool.