Farmers Market

Einblick

Farfalline

Bilmek

Lewis & Quark


     It's one of those things. You've won the lottery, and at the stadium where your favorite sports team is playing, you are asked over the intercom to come down and answer a simple question for the benefit of the audience before they give you your money. You smile. This should go easy. "What is 2+2?" That's a simple question. "Four," you answer confidently. A collective gasp issues from the audience. And then the booing begins. "Wrong!" Wrong? The intercom sighs. "Would you like to try again?" They ask again. "Four," you say again. More booing. "One last time." You consider. "...Four?" Then the people are screaming, and they kick you out of the field, screaming and raging. Has the world gone mad? But a worse question bugs you. Does 2+2=4?
Does 2+2=4?
     If you are at all familiar with basic addition, you have some knowledge of mathematics. You know that mathematics is the study of numbers. But that isn't the whole story. The most famous mathematics textbook in history, Euclid's Elements, does not have a single number in it. Mathematics is much more than just numbers--it is also shapes. Even functions and very advanced mathematics are basically the stories of numbers and shapes. And numbers and shapes play a much bigger part in our lives than just physical application, or doing problems. Believe it or not, mathematics extends from your boring textbook from the early 2000s into the whole universe. Becoming adept at mathematics trains your mind in ways other subjects simply can't.
     While other disciplines were created by humans, mathematics exists outside of us. 2+2 equaled 4 before humans began to understand the concept. And although mathematics was understood practically in Egypt and Babylon for purely practical reasons (areas of farmland, volume, simple accounting), philosophy abstracted mathematics, making it valuable for what it was, not just what it could do. Hence, most mathematical discoveries were made by philosophers.
     Mathematics has held a special place in thinkers' minds since classical Greece. Thales, both philosopher and mathematician, was the first to think of numbers for their own sake. He, and his pupil Pythagoras (of Pythagorean Theorem fame) were the first to think of numbers for their own sake. Plato, the most famous of the Pythagoreans, elevated mathematics to a spiritual level, basing his view of heaven in the Republic on mathematical ratios. Eventually, mathematics became a massive proof system, evidenced by the aforementioned Euclid's Elements.
     After the fall of Rome, mathematical knowledge moved to India and the Islamic empires, only returning to the West through European language translations of the Arabic copies of Greek texts. In the 1600s, philosopher Descartes wedded the previously divergent studies of numbers and shapes in analytic geometry. Sir Isaac Newton invented calculus, based on functions, for more complex shapes and numbers. Even as mathematics grew, it remained empirical, something other studies often wished for. 2+2 always equaled 4.
     But in the 19th century, mathematicians discovered that there were logically consistent geometries fundamentally different from ordinary Euclidean geometry--what you've learned in school. Suddenly, the interior angles of a triangle could be greater or less than 180 degrees, and parallel lines could intersect. Try it out. Then Albert Einstein described all of reality as non-Euclidean, the space curve. On large scales, our normative mathematics does not function. Simple groups of objects could be fundamentally inconsistent. Kurt Gödel, and Austrian logician, discovered that even if you perfectly create an axiomatic system, it will be incomplete. Since Plato and Thales, mathematics had always been regarded as unshakable and certain. These discoveries challenged that notion. But do we hold that notion for no reason? Doesn't 2+2 always equal 4?
     And though we have imagined other worlds, where the laws of physics, biology, and chemistry are changed, we simply cannot begin to conceive of a world where 2+2=17. No matter how we look at mathematics, it is always the same. And because mathematics is a reflection of the order we find in God, mathematics is necessarily true. We can know the solution to mathematical problems without checking the world we live in. We can know the answer to 4,356 plus 809 without having to construct groups of 4,356 and 809 objects and then counting the result. The answer (5,165), does not depend upon the world.
     When you are going about reading or trying to understand a text, there are three steps: observation, interpretation, and application. Interrogatively: what does it say? What does it mean? How do I use the answers to the previous questions? Mathematics has only two of these. We only need to "think" the answer. There is no need for interpretation. There is, however, an application. Thales distinguished between the two kinds of mathematics--pure and applied.
     Pure mathematics deals with mathematical terms, while applied mathematics is about physical objects. What we see in the real world can be quantified by physical objects--applied mathematics. But there is no physical way to express the square root of negative two is irrational. And two apples plus two apples equals four apples and √-2 is irrational was true before you were born and will be true after you die.
     But the pure mathematical form of adding apples to apples is not about nothing. 2+2=4 must be about something. Otherwise, it would not exist. And so it would appear that numbers are eternal, and exist outside of us. Plato, both mathematician and philosopher, had his forms, the perfect prototypes of all that existed physically. Platonism would argue that the computers you are all reading this on have a spiritual perfect computer that they contain a semblance of. Mathematical platonism posits that numbers actually exist--somewhere.
     And while it could be argued that 2+2=4 just stands in for "every time you add two things to two other things you get four things," there is no such physical stand-in for 3 is not a square number. Mathematical platonism, firmly entrenched, and in view of the objective nature of numbers, claims qualities for them that we typically reserve for God, i.e., putting them in His place. Alternative philosophies (Empiricism, Formalism, Logicism, Fictionalism, another "ism") offer solutions out of mathematical platonism, but they ultimately fail to account for both the apparent actual existence and infallibility of mathematical terms.
     But even if mathematical terms are infallible, we still aren't. We are fallible. We make mistakes. And even when our mathematical methods are correct, our owl limitations hinder us. The aforementioned seeming mathematical inconsistencies are not a reflection on God--they are an image of our limitations.
     In spite of that, mathematics has tremendous value. When you solve equations, or graph a circle, you're not just reading the base code of the universe. You're not just familiarizing yourself with an essential component of life on Earth. You are also, in a stunted way, thinking God's thoughts after him.
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