The Vedas are ancient Hindu texts dating back as far as 1500 BCE. I first became aware of them when in discussion in the online portion of a Calculus Analysis class. I was leading a discussion about the nature of Infinity on the back of reading Gödel's Incompleteness Theorem. The discussion led to my reading about Cantor's proof of multiple infinities, and I came across information about how Infinity was treated in the Vedas.
The Vedas support Cantor's hard won conclusion, taking it several steps further by describing never ending finite sets, such as the set of all Integers; infinite sets, such as the set of Real Numbers, and infinite sets of infinite sets, such as the set of all Imaginary numbers. Cantor proved that there were infinite sets that were larger or smaller than other infinite sets. He used a proof that was actually very similar to Gödel's Incompleteness Theorem. Both Cantor's and Gödel's concepts are reportedly found in these ancient Indian scrolls. I'm not so sure Gödel's incompleteness is there, but the idea of constructs containing models of itself are.
The sophistication of mathematics from these early times makes me wonder if Newton and Leibniz were the first inventors of Calculus. In the Vedas, from what I can remember from my very casual research, actually describe three basic types of infinities, as I touched on above. Each of these three basic categories of Infinity were further divided into the sub-categories. That's about as far as my reading had taken me. I found this in some online description of early Hindu mathematics.
I did not read the Vedas or a translation of them, just a summery of these very specific concepts. It should be understood that the Vedas are a series of texts that are primarily religious, so I can not even vouch for the accuracy of any translation. These mathematical ideas might even have been read as metaphors that some scholar took as proof of rigorous treatments of math when they may not have been. However, I see no reason to believe such scientific thinking was not the case. Certainly, if you read Bertrand Russell's A History of Western Philosophy, ancient India can be credited with significant contributions to modern maths and science.
-Will