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The Greeks adapt positional math notation and the number zero.
-- This is just the early part of a work in progress, comments and suggestions are welcomed in the "talk" section --
In our timeline, the Greek system for notating numbers was quite awkward This made doing math very difficult and ultimately limited their progress in the sciences. The Babylonians had developed positional notation by around 2500 b.c. Sometime between 700 and 300 bc they started using a symbol of one dot over another to represent an empty position : This was the number zero.
Greek mathematicians learned much of their craft from Egypt, but they also learned from the Babylonians. For example, some historians believe that Pythagoras brought his famous theorem from the east and Diophantus brought some of the rudiments of algebra from the same region. What if the concept of position based math notation is picked up by the Greeks around 350 b.c? And what if the Babylonian zero becomes used by the Greeks, not just in between numbers, but also at the end of numbers? What if the Greeks develop a much stronger math than in OTL, which is also much easier to learn?
The Babylonians had several different numbering systems including a base 60 and a base 12 system. Plato and his followers held the number 12 in mystic awe, so let's say that the Greeks adopt the base twelve system. They continue to use the Greek letters of the alphabet to represent the numbers one through twelve but use the " : " to denote zero.
POD: Anaxarchus injured at the battle of Gaugamela
Pyrrho and Anaxarchus
At the Battle of Gaugamela elements of Darius' Royal Guard, along with some Greek mercenaries, broke through Alexander The Great's lines and attacked the Macedonian camp. Alexander went on to win this pivotal battle but, as in any battle, there were casualties and injuries on all sides.
Amongst the injured was the philosopher Anaxarchus, who was in the Macedonian camp at the time. Anaxarchus was able to stay with Alexander's army through their takeover of Babylon, but then he decided he should take some time to recover. He stayed in Babylon for almost three years along with his student, Pyrrho of Elis. Together the two philosophers learned much of Babylonian science, astronomy, mysticism, and mathematics -- including their method of positional notation for numbers and of their use of ":" as a placeholder for numbers which are not there (the zero). In 328 they traveled to the recently established city of Alexandria and continued to learn and teach -- they brought the zero with them.
Pyrrho regretted not continuing with Alexander to India because he wanted to converse with the famous "naked philosophers". But he was not convinced that the gumnosophistai truly exist or that, if they did exist, that they would have anything to contribute to his understanding of the world (Pyrrho was a skeptic and cynic of the first order).
In Alexandria, Pyrrho became a teacher of great reputation. His teaching refined some existing Greek ideas: mainly that the world of perception cannot be relied upon. He thought that the world of intellect was also unreliable but was significantly more trustworthy than the physical world. As part of his turning toward the intellect, he held both logic and mathematics in high esteem: he considered them to be the least unreliable forms of knowledge. Other elite Greek philosophers were greatly influenced by Pyrrho (through the writings of his student Timon) and along the way they also adopted position notation of math -- the zero sneaks along as a stowaway.
Pyrrho was one of the founders of the Skeptics and was in opposition to the Dogmatic philosophers of the time. Part of the Skeptics' belief was that all knowledge is based on uncertainty. As such, he was able to use the ":" in his calculations without needing it to be fully defined or understood.
Prevailing Greek philosophy taught that perceptions were not to be relied upon. Archimedes, in part, rejected this notion and chose to live in the real world. He carefully observed the world and created devices which were useful in it. Archimedes' interest in math caused him to read the works of Pyrrho and other Skeptics. He used positional notation and the ":" in his calculations which made it much easier for him to advance his own studies. It also made it much easier for other people to be able to read and understand Archimedes' notes.
Archimedes found irrational numbers to be very unsettling. He could not avoid the realization that the square root of two and his own pi are irrational numbers. Archimedes suffered a brief crisis but found comfort in the Skeptic philosophy - much of the world may be beyond our ability to understand it, but we can still progress our understanding by making use of concepts we do not fully grasp.
Archimedes went on to develop infinitesimal math and then calculus. His notes on the subject were easily understandable to the serious student. He also ushered in the concept of zero - no longer a second class citizen or an artifact of positional notation, zero became a mathematical entity in its own right.
Back in Egypt, where Pyrrho taught, the zero was associated with the God Niu.
200 B. C. - Apollonius develops geometry
250 a.d. - Diophantus develops algebra
The new numbering system is much easier to learn, work with, and teach. More great mathematicians emerge and math is more widely used throughout society. Roman merchants become well versed in arithmetic, and military engineers become capable of advanced math. In OTL there was no taboo against teaching math to slaves and having them use the skill in their work. The same would be true in ATL. In short: the new math becomes used at all levels of society - from the very top to the very bottom.