Arithmetic is the first mathematical art that was categorized by the Christian philosopher Boethius. It’s also one of the oldest kinds of math in history, because it deals with something fundamental to any person’s experience in the world: counting. The earliest manuscripts we have regarding math are Babylonian and Egyptian writings, which reveal the use of addition, subtraction, multiplication, and division, as well as more complicated ideas like squaring numbers, taking the square root, using fractions, and finding missing numbers with algebra.
Key to the arithmetic of the Babylonians and Egyptians was their attitude of practicality. When we deal with numbers, we often think of them as perfect ideals or ideas which are imperfectly represented in the physical world. But when these ancient civilizations dealt with numbers, their interest was mostly in practicality and had nothing to do with the “perfection” we ascribe to numbers today. Although these people accomplished incredible mathematical feats like the Pyramids, their arithmetic was relatively speaking quite crude, and estimation over exactness was par for the course.
We get our higher view of arithmetic from the Greeks who, thanks to their philosophers, were very interested in ideals. In fact, for the Greeks, arithmetic was philosophy. The Pythagorean religious/philosophical cult was the first major group of people to focus on the mysterious and fascinating patterns of counting numbers (1, 2, 3, 4, 5…). They saw those numbers as perfect forms which governed the nature of the universe. Fractions were left to the imperfect world of commerce, and negative numbers were rejected entirely. Irrational numbers like pi were also excluded, because they couldn’t be made from the normal numbers with addition or multiplication.
That seems ignorant of them from our modern perspective, but think for a second about, say, pi: we estimate pi as 3.1415…, and we know that it relates parts of circles, but what is it, exactly? We can only estimate it, or else use a special symbol, π, for it, precisely because we can’t quite pinpoint it with numbers that we know exist, like 2 or 3. Does 𝜋 really exist in the real world after all? When you frame arithmetic this way, it’s easy to see how it gets very philosophical! (I should note that, just as philosophy is applicable to everyday life, so also are those mathematical ideals practically useful. The Greeks didn’t reject practicality; they just saw it as a product of their philosophy.)
The zeal with which the Pythagoreans treated arithmetic did not die with Greek civilization. Ever since the Greeks, mathematicians interested in the nature of the world have invented new ways to speak of arithmetical ideas. As that conversation developed, mathematicians gradually expanded the realm of arithmetic to include those fractions, negative numbers, and irrational numbers previously rejected by the Greeks. With arithmetical expansion came scientific advancements, which in turn motivated further arithmetical discovery. Today, we even accept arithmetical notions like numbers between positive and negative (imaginary numbers), infinitely small numbers (differentials), four-dimensional numbers (quaternions), and more. Each of these concepts has a rich history and philosophical conversation behind it.
When you study the (literally) countless patterns and ideas in arithmetic, it’s easy to see why arithmetic was taken so seriously the Pythagoreans and those after them. In many ways, they were right in their thinking—we can’t deny that numbers are part and parcel to what makes the universe tick. For a Lutheran school which seeks to teach the truth, the question is: how do we teach these mathematical ideas to our children? First, we recognize that Christ does indeed govern the universe in such a beautiful way that we can find amazing patterns in creation using numbers. Because of that, we can be assured that in studying math, we are pursuing actual truth about creation, not concepts made up by ancient Greek cults. Second, we carefully consider what philosophies are behind the various mathematical ideas throughout history, determine which philosophies are in line with God’s Word, then decide which ideas are worth handing down to our children as we educate them. There’s a lot to consider, but it’s a fascinating, exciting world to explore!
In Christ,
Mr. Hahn
