A title like “the art of seeing” might conjure up poetic associations, a consideration perhaps of attention and what one sees when they look at the world. So it might come as a surprise to learn that this essay is about mathematics. That all too familiar anxiety inducing part of the mandatory curriculum most of us have had to endure. We might recognize its importance, give it our implicit approval as another part of life that is necessary or unavoidable in our modern circumstance, like death or taxes. A feature of the world that we’re probably too dumb, too busy or too uninterested to explore. Others might read this words and and feel a tinge of anger for that description. Mathematics is beautiful they think, it’s just hard to explain why.
You see I think math is beautiful as well. Like all manner of beauty it’s a feeling that comes from within and colors one’s view of the moment. It’s usually a small dose of wonderful realization, a way for something deep within to say life is worth living for those short few moments. But math isn’t just beauty, it’s hard work, it’s an edifice built using a million small blocks of stone and concrete each painstakingly carried and placed by individuals over millennia. And those who were forced to learn the names and shapes of those stones at a young age with little luck, might come to associate the edifice with those vulnerable moments of doubt where they couldn’t see how those stones could ever come to build anything beautiful or perhaps even useful.
I don’t know if I can convince you that math is beautiful, I’m not sure this is my place to attempt that fit. But I do want to try and show what math is for me, without the clutter and the formal language that is both indispensable and cryptic for the uninitiated.
To that end, I want to explore one idea in a sub-field of computer science called complexity. I want to show in a somewhat detailed way, how simple ideas and logical progression lead to the realization that no matter how advanced and capable the human race is, nature will forever keep her deep secrets hidden.
Our story, and it is a story begins with numbers. Those things for which we’ve grown so accustomed to in our modern living that we might not realize that human being don’t really seem to have a built in sense of numbers greater than three. This can be witnessed by putting three coins on one’s hand and counting them, and repeating the experiment with five coins. The three coins are automatically accounted for, in a way that might not be best described as counting. But the five coins need a conscious attempt to account for them. This rather simple test with its surprising result shows the extent to which we’re unaware of our relationship with numbers.
Numbers dictate our actions in the form clocks and watches that tell us where we might need to be. They give us a sense to interpret temperature readings and change our mood accordingly. They organize our thoughts and the way we understand information (3 people showing up to a gathering or a 1000). In a way it’s very hard to imagine life without them, and surprisingly there are human communities like the Piraha in Brazil or Warlpiri in Australia who have no use for them.
What we consider trivial computation without which we wouldn’t be able to lead a normal life, like multiplication and subtraction have had long developmental histories. Some of those histories had the greatest minds of their time on the losing side of an argument. But as this isn’t a history lesson. I won’t detail the fascinating way in which the ideas that many people associate with a decree by a teacher were hard fought to be accepted.
So what are numbers? That’s a hard question to answer. They are an intangible property that things have. If we have three apples in a basket, and we pick another one from the three, we’ll have four apples in basket. The realization that makes numbers so unique in the vast avenue of human thought is that if we replace the apples with oranges that relationship wouldn’t change. In fact we can replace the apples with cars, flowers and even people. The numberness of something seems to not lie in the object itself, instead it’s a description of something that all physical objects have. In mathematical language this is called an abstraction. Different things have something in common and for counting that thing is numbers. In fact even numbers have numberness, we can have 3 number 3s which we call multiplication.
And that’s a hint as to why it can be difficult to learn and to explain mathematics. Simple generalizations make room to more generalizations of the generalizations. Abstraction building upon itself and losing one’s place in the ever increasing tower can cloud what are simple ideas in an air of “unapproachness”. To a large extent it’s a visage created by the attitude of those who are allergic to anything but perfect rigor. Physics which seems much more approachable to a general audience is as dense and almost as abstract as mathematics, yet physics doesn’t seem to have the same reputation math has.
The counting numbers and the basic operations on them like addition and subtraction are useful. But when one plays with numbers for a while, natural questions arise that feel like they require an answer. One of those question is what happens when we take one counting number and subtract a bigger counting number. What does 3 minus 5 mean? The answer is of course -2. But this obvious answer wasn’t so obvious for most of western history. Counting numbers are positive, by definition, what does a negative number mean? How can you count -2 of something? The intuitive answer is that you can treat negative numbers as something to owe. If I have three apples, and I want to give my friend 5 apples. I don’t have -2 apples, but I do owe my friend 2 more apples. A redefinition in modern math curriculum is the use of the number line. Numbers are points in an infinite continuum. The more right you go the higher and more positive the numbers become. The more left you go the lower the numbers and the more negative they become. In part that’s why this essay is called “the art of seeing”. Math to a large extent is a visual subject, the cryptic alien writings are translated to intuitive images in the minds of those who understand them. To have a grounding in visual intuition that our nature is designed for is part of what makes math so beautiful. In the madness of detail of complicated equations and symbols many times lies a simple image that holds the key to comprehension.
So far we’ve seen examples of two types of numbers. In math speak they are called the natural numbers and the integers. One interesting observation, is that the natural numbers are contained within the integers. And that’s a pattern that will repeat as we see more types of numbers. Each layer of abstraction building on the previous one helping us answer more questions.
But to understand why mathematics tells us we’re fundamentally constrained in our ability to truly know our place in the cosmos, we need more than numbers. We need to grapple with the familiar but slippery concept of infinity. And that’s where our story will go next.