A recent paper about the Antikythera Mechanism, argues that one of the dials on the device used to predict eclipses pushes the construction date of the device back at least fifty to one hundred years to at least before 200 B.C.
In doing so, it changes the possibilities for the mathematics used to develop the device away from a trigonometric system as the Greeks had not fully developed it as a system.
This led me to ponder how math has been developed over the centuries. What developments and needs led to it's creation and development?
Also it made me ponder how systems of knowledge were conveyed in the Classical World. How did Babylonian ideas become exchanged with the Greeks and vice versa?
This led me to ponder how math has been developed over the centuries. What developments and needs led to it's creation and development?
Can I ask you to please specify a time period, as that is a very large and vague question. Set Theory for example in the 1800's would have been created for very different reasons to types of Geometry. Anything specific, I assume you mean in the Classical world, but what exactly and who? (It would be different for the Greeks and the Babylonians)
Also it made me ponder how systems of knowledge were conveyed in the Classical World. How did Babylonian ideas become exchanged with the Greeks and vice versa?
Well I can give you an answer for your example (though it may be different for different examples). The spread from Babylon to Greece happened shortly after the conquests of Alexander the Great, so much so that the deaths of Alexander and Aristotle around the same time is seen as a convenient dividing line in Greek Mathematics, after which comes what is usually seen as the "Golden Age" of Greek Mathematics.
Anyway, according to Simplicius of Cilicia (which is a very late source, coming from the 6th Century AD) Alexander ordered the translation of the historical astronomical records under supervision of his chronicler Callisthenes of Olynthus, who sent it to Aristotle. This is just one of the ways ideas would have spread after the conquest.
At the same time though, this conquest spread Greek culture with it, where it partially mixed with the culture in places like Babylon (though there was always a gap between them) and transplanted there own mathematics, with there new ideas, into these new surroundings. And from that a mix of the two was formed.
It is however strange that most of these new mathematical discoveries occurred in Ptolemaic Egypt, and not in fact Mesopotamia. A reason for this could be because Egypt was much more central, with Alexandria being a port that basically became an intellectual capital. In comparison Babylon was much more remote and was soon replaced by Seleucia capital of the Seleucids.
It would be worth pointing out native Babylonian astronomy did continue developing, and reached its height under the Seleucids, as well as there more coastal cities closer to the Mediterranean did have some important Greek Schools, it wasn't a backwater of development.