WRITING MATHEMATICS

It was very difficult to do anything with mathematics before you could write it down. Even illiterate societies often came up with a way of storing mathematical ideas in some fashion. Thirty thousand years ago, people whom we know nothing about carved notches in wolf bones to figure out something or other. The Inca people who didn’t have a written language used a set of knotted strings to record numbers and perhaps even more complex ideas. Without something to store mathematics permanently, it’s very hard to progress, it’s very hard to transmit your ideas from generation to generation.

Throughout history mathematicians and scientists have invented new ways of writing down numbers and writing down equations and writing down mathematical objects to describe their ever-increasing understanding of the universe. Without a good mathematical notation, it’s hard to do something as simple as addition, much less modern science like relativity. The Romans had to translate from their written system of M’s and C’s and L’s to an abacus,, do the addition on the abacus and then go back to the written system. For example, what they would do for addition like DCVI plus CCCII would be DCVI plus CCIII equals DCCCVIIII. It’s a very cumbersome notation, and a very cumbersome method. Arabic numerals made it much much simpler to do things like addition, subtraction, multiplication, division, and even more complicated things.

Though the abacus could, after a lot of training, do these calculations, a lot of things were still extremely difficult and out of reach - things like long division, square roots. Arabic numerals put those all within reach and made it all reasonable for a scientist to do these things. Before you had Arabic numerals scientists couldn’t do square roots easily; they could barely do long division. It took a lot of training. With Arabic numerals you could do it on a piece of paper, fairly quickly with only a little bit of training.

ARABIC NUMERALS

Modern science hinges upon a very nice notation for writing mathematics down called Arabic numerals. It was invented around 500 to 800 AD in India and, through a quirk in history, it became known as Arabic numerals. It started early about 300 AD in Babylonia in the Fertile Crescent with a new notation which didn’t use different symbols to represent different numbers. It used the same symbols over and over and over again to represent different numbers, and the way that scientists and mathematicians distinguished the values of those numbers was through their position value. It first was created by Indian mathematicians and then it traveled east and west through conquest. As the Arab lands expanded, Arabic notation expanded throughout the Middle East, and eventually transferred to Europe in about 1200 to 1300 AD through trade. Merchants found Arabic numerals so wonderful to use that they grabbed it immediately, even though the government officials at the time were very suspicious of them. In fact, in 1299, the Florentine government banned the use of Arabic numerals because they thought it was some sort of secret code to use to send messages or to cheat costumers.

Despite the resistance, Arabic numerals caught on, the merchants loved it, it made their calculations much easier. And even Einstein couldn’t have been able to do what he did without Arabic numerals and a good notation for writing down numbers. An equation is simply a relationship between two separate things in letters and numbers. These letters and numbers describe elements of the physical world, and scientists are always amazed that their numbers work so well to describe the physical world.

MATHEMATICAL NOTATION

Modern science couldn’t have existed without mathematics and mathematical notation, without a way of writing down the ideas of science. Often, throughout history, scientists have been inventing mathematical notation as they went along. Newton invented calculus, and Einstein invented something called tenser calculus with fellow mathematicians. Tensers are things like this T sub-mu sub-mu, which describe the curvature of space. Without some way of writing down ideas, science is ephemeral. It disappears into the ether. Archimedes himself, his last words were “Don’t disturb my circles” as he inscribed circles into the sand, trying to protect his ideas from a Roman soldier who was trying to wipe them out.

And there were battles in history over notation. For example, scientists came up with two different ways of expressing calculus; a fundamental mathematical idea that’s used in physics. The continental Europeans and the British fought for decades over which was the better notation for political reasons. On the British side, they thought that Newton had invented calculus, so they used Newtonian notation. On the European side, they thought Liebniz had invented calculus, so they used a different notation, which was actually slightly superior. In fact, the lack of a good notation hampered British calculus for about a century, until everyone came to agreement that the Liebnizian notation was in general better and more flexible to use.

Modern science couldn’t have existed without numbers and a way of writing down mathematics. Without a notation mathematics is ephemeral; without ways of describing the relationship between different objects in science, like energy and matter, there is no way of realizing that these relationships existed. Without writing down these equations, these relationships scientists missed the big picture. Without equations, without numbers, without ways of writing these things down, Einstein could never have realized that energy and matter are interchangeable, that you can convert matter into energy and vice versa. This equation, this relationship between numbers and symbols is a way of showing how much energy is released when you annihilate a bit of matter, and E=mc2 predicts beautifully how much yield a nuclear weapon will create.

ZERO

Considering how fundamental mathematics is, it’s surprising how recently these ideas came out. Zero, for example, is only 1500 years old. Before mathematicians had an idea of zero or negative numbers, there were so many ideas that they couldn’t express. Scientists couldn’t use mathematics to describe the natural world in the way they can today. They couldn’t come up with equations to describe the curvature of space the way Einstein did.

The origins of the modern zero come from Babylonia in the Fertile Crescent, in modern day Iraq. The idea of zero came from a mathematical notation that they used, cuneiform, that used the same symbols over and over again to represent different numbers. The number zero is one of the most important discoveries of the human race. Its discovery opened up a whole new vista for exploration that led to modern science

It’s hard to think of a world without zero, but for millennia, many civilizations didn’t have it. The Greeks didn’t have it, the Romans didn’t have it, the Egyptians didn’t have it; it was only after the Babylonians in their Fertile Crescent took it and it was transmitted to India around the conquest of Alexander in 300 BC that it developed into a number that people knew about. The zero became an object unto itself; it was a new number. Ever since caveman times, people knew what one, two, and three were, but zero was something completely different. You can’t go to the store and buy zero sheep. You can buy one sheep, you can buy two sheep, but zero was something else. Was it really a number? Well, yes. Zero is the center of the number line, it is the number between negative one and one. It is even, it has properties, it is a number like any other even though it has slightly different properties and looks slightly different.

Some of the things that zero can do are astonishing to someone who is used to numbers like one, two, and three. When you multiply zero by anything you will always get zero back: zero times ten is zero, zero times a billion is zero. No other number has the power to suck everything into itself like zero does, other than infinity. In fact, zero and infinity are closely related; their properties are equal and opposite. Zero is the center of the number line, and it is a boundary between negative one and one. Once you have zero, things that were separate all of a sudden come together. For example, when something is moving, it moves at a certain speed like one meter per second. A scientist would see something moving like this and say its moving one meter per second in one direction, or one meter a second in the other direction. But what happens when it stops? Is there a way to describe that? With zero all of a sudden instead of describing things separately as saying moving to the left or moving to the right or stopped. Its just one expression, it’s moving at one meter per second to the left, or one meter per second to the right, or zero meters per second. One, negative one, zero. And all of a sudden you have a way of describing the motion of objects with numbers.

Before zero you had to treat the three cases separately. But once you had zero and once you had the negative numbers they were parts of the same whole moving at one meter per second, minus one meter per second, zero, you all of a sudden had the ability to describe the motion of an object with numbers. Instead of in separate cases that had to be broken down separately. From there you are able to come up with equations that describe how particles move, how objects behave. And that’s the basis of all of physics. Once you can look at the big picture and describe motion with numbers you can deal with them with equations.

But zero was rejected by many, many people. In fact, the Church hated zero, and they rejected its twin infinity. Zero was intimately linked to the void, and the void was what God destroyed when he created the universe. The current Catholic view of hell is an absence of God. Void was hellish, and infinity also unfathomable. It was God; it was the realm of Imperium, and no one else could approach it. Aristotle himself rejected the idea of void, and rejected the idea of infinity. And the Church built upon Aristotle’s ideas; the Church used Aristotle’s ideas to prove the existence of God, the existence of a prime mover - the existence of Heaven and Hell. And without understanding that zero and infinity destroyed Aristotle’s ideas, you had no concept, no basis on which to build your theology.

The Church rejected zero, yet zero was so useful, so important to mathematics, to science, and especially to banking. For we know that money makes the world go ‘round. The bankers won out; zero spread to the west from the east. The ideas of nothingness and infinity became currency for science, and then all hell broke loose. Zero changed society, the idea of nothingness and infinity. It changed art. In the 15th century, artists discovered a way to describe perspective, to render three-dimensional objects in two dimensions. And it required a nothingness in the center of their painting. The vanishing point is a small zero-dimensional point that represents the infinity of space, and art flourished after these flat figurine icons disappeared from art. Similarly, the ideas of zero and nothingness revolutionized science. Evangelista Toracelli, one of Galileo’s disciples, created the first vacuum: a position where there’s nothing at all, no air, maybe even no God.

Science could not progress until Aristotelean ideas were destroyed; zero helped destroy those ideas. Aristotle had the earth at the center of the universe surrounded by stars, moons, and planets which all were subservient to the central earth that we were on. The Church followed that view. We humans were the center of the earth, we were the most important thing, and Christ came down to give man its salvation. However, once observers began to undermine Aristotle’s ideas, and the idea of an earth-centered universe failed, the sun-centered solar system took over. And people began to realize how huge the universe was, what infinities of planets there might be. And Copernicus, Galileo, eventually Newton and Einstein drew together a picture of the universe which is mind-bogglingly larger than anything that the ancients could have thought of. Archimedes tried to figure out how many grains of sand there were in the universe. Modern physicists think that there are an infinite number of grains of sand - couldn’t even start to fill the universe. Basically the idea of infinity of worlds was damaging to the Church. The idea of a universe populated with galaxies the size of our own is much, much more interesting than what the ancients could have imagined, and it’s what modern scientists believe. Einstein drew a picture of the universe that might be infinite to an extent; perhaps flat, perhaps curved, but space and time are woven into this one fabric that extends in all directions. Some scientists even speculate that there are worlds and universes pretty much like our own, unobservable, either very distant or very close in a different dimension. All of this is mind-boggling, but scientists with their instruments and their telescopes are answering some questions that could never have been posed years ago, in a world without infinity and a world without zero. The biggest question perhaps in cosmology today is: what is pushing the universe apart? In the deepest vacuums, in the regions where there is nothing around there is a mysterious force pushing galaxies apart. And scientists now are trying to figure it out, and they’re actually getting somewhere, they’re making progress, they’re observing the Big Bang, the afterglow of the Big Bang, when the universe was only a couple of hundred thousand years old. It glowed with light. And by observing that light, they can get an idea of why our universe is not nothing.

Perhaps on some level, zero would exist even if humans weren’t there to observe it. Only when scientists could write down zero could they understand the nothingness in the universe, understand how zero relates to space, to black holes, to quantum mechanics, the smallest and the largest things in the universe. The wonderful telescopes that scientists use to understand the universe are all based upon digital technology. Ones and zeros stored in memory banks that represent pictures. The famed picture of the Eagle Nebula with the pillars of God, green- glowing, are all just ones and zeros stored somewhere on a hard drive.

When we were counting on our digital digits to the time when we were counting on binary digits there were thirty thousand years. But in the few years since we moved to binary digital processing science has exploded. Computers have given scientists a whole new way of looking at the universe; a way of understanding billions of sums at once. But without zero, none of this would have been possible. Zero was central to science, the idea of zero is at the center of the number line and at the center of science. Zero is fundamental to our understanding of the way the universe works. Without zero the number line doesn’t make sense and science doesn’t hold together.