Why were the Mayans so advanced?
It really depends on what Old World Civilization we are supposed to compare them to as a technological benchmark. The short answer is that they were more advanced than the Europeans of the time on a some areas but less advanced in other areas that the Europeans were already much better at.
So, I will chose three Old World advanced civilizations and compare them with the 3 main Native American civilizations while giving examples of what the Native Americans actually managed to do better than their Old World counterpart. But before we start let me reiterate, just as the Mayas, Aztecs and Incas were better than the civilizations of the Old World at some things, so were the Old World civilizations better than the Maya, Aztec, and Inca in other areas.
Comparison number 1:
The Roman Empire vs The Inca Empire
There are 2 main things I believe the Inca developed better than the Romans. I imagine there could be several more but while I am a World History buff and have traveled to Peru, I don't have a degree on the history of the Inca so is quite possible I may be missing something. Still, there are 2 main things I believe the Inca developed to a much better degree than the Romans.
The first was Ancient Inca stone joinery. It is still not known exactly how they achieved the close fit of their masonry but even nowdays these original Inca structures, made without cement of any kind, are so well put together that you cannot successfuly fit a credit card in between the blocks of stone. Extremely remarkable for massive constructions that lacked cement in its entirety.
Some Conquistadors reported that they had been told a plant of some sort was known to the ancients that could soften stone, but this is not taken seriously. (One reportedly claimed to have walked through some kind of plants that melted the spurs off his boots - again not taken seriously.)
Bottom line regarding this stone joinery techniques is that the level of quality of the buildings using this techniques are a lost specialized craft and that these original stone structures that can be seen in Peru to this day have lasted 5 times longer than the Empire that built them (The Inca Civilization was the Golden Age of Peru as it was also the culmination of many advances from prior civilizations blended together with efficient political administration but it lasted barely more than a century because of European colonization)
The second main thing is that the Inca learned how to dehidrate their potatoes to the point they could be stored for over ten years without rotting. That, along with their mass production agricultural techniques allowed the Inca to create a massive surplus of food that made countless villages from the Pacific coast of Ecuador to the northern borders of nowdays Argentina and Bolivia join the Inca Empire and voluntarily submit to it's rule. To this day, the Inca Empire is one of the very few civilizations in all of human history that essentially managed to erradicate hunger because every household in every settlement of every province had enough food for all the members of their respective families to eat 3 times a day. If that isn't a remarkable achievement for the subjects of an empire I don't know what it is.
Now, let's talk about the Romans in these 2 areas-stone-joinery and agriculture- so we can compare.
I don't mean to disregard the remarkable construction achievements of the Roman Empire in this answer. The Romans certainly had the most brilliant engineers of their day and their massive wonders of constructions, like the Colisseum and the Pantheon, are awe-inspiring even in this modern age. The Pantheon in particular continues to baffle modern architects and engineers in many ways, for example, especially because of our current inability to successfully make the accurate plans for the construction of a 360 degree dome that colossal, much less build it, without the use of a computer, a tool the Romans most definitely lacked.
However, when it comes to stonework masterpieces, I personally have to give the prize to the Inca and here is why: for all their magnificent structures, the Romans never quite learned to shape stone to their will the way the Inca did and I believe the special cement they developed with the use of volcanic ash was the main reason. When you know how to make a cement of a far superior quality to the kind your neighbors can produce; a cement that even works well with underwater structures made over 15 centures ago, the people probably won't see the need to develop such precision when crafting stones when they can just paste the pieces together much faster and easier that way.
On the second topic, massive production of food through agriculture and capacity to store it for long periods of time, here is what I got to say: the Romans advanced road system and the fact the Mediterranean was once their own private lake for all intents and purposes allowed the citizens of the Empire to transport goods far and wide. That ensured major urban centers, such as the city of Rome, would always have a constant supply of food coming from all corners of the Empire. However, the capital of the Empire depended heavily on imported food, especially from Egypt, to be able to sustain their massive population. Take Egypt out of the equation and Rome would get cancer in the stomach until the population levels drop to a more managable level.
Cusco didn't have that overwhelming dependency problem. Yes, it was easier to transport goods on their own road system over long distances than it was at, say, the Aztec Empire but their mastery over agricultural production has reached such levels of surplus only highly disruptive nation-wide warfare and/or an extremely spread-out plague could truly make large groups of people start to starve and even then, only after their storages of dehidrated potatoes were gone. So, when it comes to working hard to keep most of their citizens from starving, I personally concede the victory to the Inca Empire as well.
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