Integers, rational numbers, irrational numbers, transcendental numbers … There are many types of numbers, but among them zero is truly exceptional. Zero is a form of darkness that does not exist on the number line. Walking along it, we could slip into a basin-like hollow, find the surroundings altered, and reappear suddenly in another dimension. The concept of zero is exactly like a black hole in astrophysics.
When we expand the domain of numbers to include complex numbers, a second dimension opens up around the line, a plane on whose surface bristles an infinite amount of imaginary numbers. We can solve quadratic or second-degree equations without postulating their existence, but not cubic or third-degree equations.
On either side of the road I am traveling expands a vast grassland, home to innumerable types of plant life. Some grasses roll along the ground, blown by the wind, their stalks trembling uncertainly. Other types of grass solely plant strong, deep roots in the earth. One can’t help but wonder just how many types of fauna are concealed in the flora.
Imaginary numbers are like spirits wandering between being and nothingness, and are again much more numerous than real numbers. Unlike real numbers which are expressed as a line, in one dimension, complex numbers extend their realm onto a plane, in two dimensions. Without the help of these phantoms, we are unable to describe the physical world using the language of mathematics. What does this mean in reality?
A shock from below jerks my hands on the steering wheel — the jeep veers off to the side. My drowsiness suddenly dissipates, and I grab the wheel and correct course. It takes me a moment to work out what happened. They say that drowsiness is catching — I must have dozed off.
I pull up to the side of the road thinking it a good idea to get some air and stretch my legs. The sun beats down strongly, the air dry. I stretch my arms and look back at the road the jeep had come down. About twenty meters from me I see a round hole in the asphalt. The road is in pretty bad shape overall, with little pockmarks here and there, and I must have driven straight into one of the larger holes.
I kick at some nearby pebbles and voice an idea: “Our world isn’t built as sturdily as everyone thinks.”
It’s as if we’ve been walking along a bridge that, from good luck or chance, simply hasn’t crumbled yet. Modern technology cannot be maintained without resort to imaginary numbers, which cannot exist in reality. This begs the question: what if a mathematical genius denied that such numbers exist and offered a flawless proof that they don’t? In real-world terms that would be the same as discovering, only after a bridge has been completed and walked across, that its legs contained no bolts whatsoever to enforce them. With that would be born the realization that the bridge could collapse at any minute.
If the world as we know it ever begins to collapse, then our first signal would be a small shift in mathematics. Such a shift would be evidence that we have misinterpreted the world and engaged in negligent building practices.
The number zero poses an even bigger problem. In calculations involving the physical constants of the universe, the moment zero appears in the denominator, it gives rise to infinity and botches all attempt at quantification. Zero has the ability to blow it all up. That is why mathematicians have devised means to tame and paper over zero. It’s almost as though they’ve been telling a string of lies that would be discovered eventually, and I wonder what payback the universe has in store for us when the deceit becomes unmanageable. I shudder at the thought of it. The appearance of zero where it shouldn’t be is a harbinger that the structure of the universe is on the verge of collapse, a sign that mockingly admonishes, “Pardon me, but it is too late to restore the status quo ante.”
I hold my hands up to shield my eyes, and across the road I can make out a greenish sign indicating the distance left to the Tiwanaku ruins: nine miles. I’m comforted that it’s an integer. I also find it in poor taste that it’s almost a round figure but isn’t.
It is my first time to visit the Tiwanaku site. Taking in the view from where I parked my jeep the ruins seem to blend into the nondescript, endlessly vast brown earth. The site is about a kilometer long and five hundred meters wide, and it would probably take about an hour to walk around its circumference. My usual routine when visiting sites is to take a walk around the area to get a sense of the whole before moving on to examining the various parts. Today, I decide the walk would be too much and follow the arrow sign at the entrance, proceeding through the Kantatallita Temple toward the Puerta del Sol — Gateway of the Sun — that stands in the northwest corner of the Kalasaya platform.
In two days I will visit Peru’s “city in the sky,” Machu Picchu, located 2,400 meters above sea level on a sheer mountain ridge. Tiwanaku itself is 3,700 meters above sea level but finds itself on a barren plain. What they have in common are massive stone buildings.
In both cases it is unknown how stones weighing hundreds of tons were carried up to be piled at such high elevations. Why did the Mayans have to brave pain and suffering to build such enormous stone structures? The scale of the endeavor is mind-boggling, and yet one day, they completely abandoned the city of stone, the fruit of so much toil, and disappeared somewhere, their reasons again a mystery.
Visiting the world’s ancient ruins, I often wonder if the stones’ placement, aligning as they do with the movements of celestial bodies, expresses some meaning. This was particularly the case when I visited Stonehenge when living as a student in England. One of the more resilient theories regarding the 5,000-year-old circular structure is that it is a calendar. My visit did not help me to ascertain whether or not this is the case, but if people living 5,000 years ago had knowledge of solar years and the cycle of the moon, then our understanding of the history of civilization is thrown into confusion.
Generally accepted history tells us that in 1543 Copernicus wrote his treatise on the motion of celestial bodies and brought about the paradigm shift from a geocentric to a heliocentric view. However, there are various signs that suggest ancient cultures knew not just the orbital period of the Earth but even about precessional movement. Could they really have built these structures to represent the movements of celestial bodies? It indeed seems hard to fathom that the ancients would have gone to so much effort without a clear purpose. Calendar or not, the stones’ arrangement must have some meaning.
Standing to the northwest of Kalasaya with two feet braced on dry earth is the Gateway of the Sun, a large structure carved from a single block of Andesite stone. Apart from the obvious differences in size and surroundings, it looks like a smaller version of the Arc de Triomphe in Paris. As I’ve already seen in photographs, the eastern face of the gate is covered in exquisite patterns said to represent a language undeciphered to this day. Various theories abound as to the content — that they detail the scientific knowledge of the time, that they contain data on the movement of the heavens …
Indeed, whether it is the Pyramids or Stonehenge, the structures’ connection to the movement of celestial bodies, beginning with the Sun, has always been the issue. Ancient civilizations painstakingly observed the skies to measure time. A day was measured in the 24 hours it takes the Earth to rotate on its axis, and a year was measured in the 365 days, 5 hours, 48 minutes, and 45 seconds it takes the Earth to make a complete orbit around the sun. Celestial motion defined time and resulted in a calendar. For mainly agricultural societies, a calendar was necessary to accurately track the seasons. Yet, although grasping the lengths of a day and a year and the changing seasons should have been sufficient for the purposes of farming, a calendar extending 1,200 years into the future was created in this land.