- Education and Science
Math is crowned king
The one thing that we have discovered in recent years that has opened up science and our understanding of the world is that mathematics is not just an invention of mankind. It is not just a construct of convenience as so many philosophers have suggested. Mathematics is built into every aspect of existence. One could say that the patterns of existence are decidedly mathematical.
Euclid was one of the great Greek mathematicians who gave us modern geometry. Pythagoras expanded on Euclid with his theorem and then went on to build a religion formed partly around math. To him, his science and his religion were inseparable. Unfortunately not much is known for sure about the particulars of his religion as it was an ultra secret cult. However, he was not the first or the last mathematician to see something in mathematics that goes beyond a simple human construct for convenience.
It has long been known that music is mathematical in nature. Bach was a master at creating what critics have called mathematically perfect music. But one could argue that math is just a way for us to define certain patterns that could be defined in other ways just as effectively. We could use a different base than ten, for example.
But all that is irrelevant. Whatever we call it, whatever symbols we use, math describes real patterns. The way it does this is by giving us formulas that describe the pattern. The patterns are real and they are all around us. We use those patterns in our daily lives.
Geometry is all around us. Nature loves to produce spheres like the sun and the moon and the planets. If you go to visit salt mines you notice that salt crystals are amazing cubes. They mimic their atomic structure.
If you blow a bubble you notice how almost perfectly spherical it is. Bees create amazing honey combs with almost perfect hexagonal design.
If you watch expert bubble blowers, and yes there are experts, when you join enough bubbles together they naturally form that hexagonal pattern. Join bubbles together in specific ways and you can even form cubes. It’s all a matter of which shape becomes the default under specific conditions. One of those conditions, of course, is pressure. Spheres can’t sit side by side without leaving spaces. Pressure forces these spaces closed forming new geometric shapes.
With this tool called mathematics we can create formulas that perfectly describe and predict how and why nature naturally creates these geometrically perfect shapes. And it turns out that the most complex structures are due to very simple rules.
But nature is not all about perfect geometric shapes. Benoît Mandelbrot, who figures prominently in this discussion, said that the one thing that always bothered him was the roughness of the world around us. By roughness he means the apparent random nature of nature.
Look at a tree, for instance. Nothing very geometric about that. Mandelbrot has spent his life looking for answers to that question, and he found them. Nature is not random. It is chaotic. And there is a big difference.
Randomness is action that has no pattern. What science has discovered is that randomness does not exist in a cause and effect world. When we first started trying to generate random numbers we thought it was going to be easy. After all, we all know about playing dice, flipping a coin, bingo balls and a host of other ways we have mimicked chance since ancient times.
So it was a surprise when we found out how really hard it is to produce true randomness. In fact, it is virtually impossible. We can produce apparent randomness with great success now, but true randomness would have to come from actions without history or cause. If there is cause then there is a pattern, even if that pattern is seemingly impossible for us to predict or find.
Chaos, on the other hand, follows very specific patterns. Chaos breeds order. That is a rather counter intuitive statement in light of the way we have been taught to view chaos. But it is remarkably true.
Rather than being a mass of disorder as so many dictionaries would have us believe, chaos is complexity. It is what makes the world what it is, and again, it follows very simple rules to produce almost infinite complexity, some of which looks like true randomness.
This is an amazing discovery. The butterfly effect is real. Small events can snowball into large complex events very quickly or they can accumulate over a long period. The air that a butterfly moves by flapping its wings half way around the world can affect the weather in some far off place.
Edward Lorenz coined the phrase “the butterfly effect” in 1969. Lorenz discovered that weather could not be predicted accurately for more than a few days at best due to what became known as “the Lorenz attractor.”
This was a surprise to many who thought we would be able to predict local weather perfectly if we knew all the particulars of the weather patterns. It turns out that many aspects of the weather are predictable. But it is impossible to know the starting state of many of the variables in those weather patterns, such as the flapping of the wings of one or a million butterflies. Lorenz discovered that very small variations in these variables resulted in major variations in the probability of specific weather patterns over time.
One could describe this problem by saying that if you put a ball on top of a hill, depending on exactly where you put it it could roll down the hill in any of dozens of slightly different directions, making where it actually lands completely dependent on where it starts from. The game of Plinko comes to mind as another perfect example.
Since we can’t know all the variables of what causes a weather pattern or what affects it millisecond to millisecond, we cannot predict the weather accurately for more than a week at a time at best. The longer we look into the future the more changes and influences sneak in, making it anyone’s guess.
It all comes down to whether or not that butterfly flapped his wings at a specific time in a specific place. And of course, there are millions of butterflies in the world. It’s a wonder we can predict anything at all. But because of the rules behind this chaos, we can predict quit a bit about most things regardless of non-local variables.
Non local cause is something another branch of science had to grapple with in the early part of the twentieth century. Physics always assumed that local cause and effect were isolated. Sure, there were lingering effects that spread out from any event, but they diminished in their influence over time. A man riding a snail on mount Fuji knocking pebbles down the mountain could not be responsible for the outcome of an election in Denmark.
But what Lorenz told us was that indeed such an event could play a major part in a Danish election; and that cause can be non-local. The problem again is that it seems that all the events of the world affect each other on an ongoing basis moment to moment. The weather doesn’t have a starting point unless we go back to the formation of the earth 4.5 billion years ago, and neither do the variables involved in it.
Choosing an arbitrary starting point is fine, but knowing all the variables which have created that point in time is next to impossible.
Yet the idea that we had that we could eventually predict the weather was not farfetched. We had already learned that some things like the movement of heavenly bodies are completely predictable. We have been able to predict what the sky would look like thousands of years in to the future with almost perfect accuracy for hundreds if not a few thousand years. Similarly we know what the sky would have looked like thousands of years in to the past. We fully expected that if we knew the formulas, all events would become predictable in the same way. We would eventually be able to predict the future of everything with pin point accuracy
So Lorenz’s discovery was radical. So radical that it spawned a new field of study called chaos theory.
Once the genie was out of the bottle, so to speak, it was just a matter of time before others started to see the same patterns and make their own related discoveries about chaos and its connection with the order of the universe.
We had long known about the strange properties of opposites in math. They tend to cancel out. But all through the twentieth century we started seeing that phenomenon in reality. It isn’t just an artefact of math.
Everyone knows what a car engine sounds like without a muffler. But if you generate an opposite and equal noise, (frequency) the two cancel out and all you have is absolute silence. The sounds are not gone. They have not destroyed each other. They have locked each other in, creating a new order from the chaos of the competing frequencies.
Then there is spin glass. In this process different fibre molecules with wildly different “spins” at high energy are thrown together. The opposite spins cancel out leaving a very strong sheet of glass as the new order of the situation.
In this case, chaos is in the form of conflict. Conflict always demands resolution. A perfect example of this is war. Wars always end because someone runs out of resources and the ability to fight. A war might be lost because someone overpowers their opponent. It might end in stale mate. There are hundreds of reasons that war ends, but it always ends due to one side or the other literally running out of energy.
War is a high energy endeavour, and high energy can never be maintained indefinitely. The result is always a new order and a lower overall energy output.
All of this is, of course, part of physics concerning the conservation of energy and thermodynamics; which are some of those rules of chaos I was talking about. We will be getting to all that a little later on. But first, let’s look at a few other aspects of chaos in part two.