I replied to the other thread, but since you moved the question...
Invention means that the laws of nature are nothing but an outgrowth of human activities; other thinking beings at other places or times may invent completely different systems fitting for their peculiar needs. At the best, we may come up with some approximation to something intrinsically intangible, because there are no absolute truths. This statement, of course, must be an absolute truth, which opens a different can of worms labeled Godels theory.
Discovery means that the laws of nature exist in a defined form, totally independent of humans or anybody else below the level of an almighty being, and that there is a possibility to discover them in total (if there is a finite number of natural laws) or at least in parts and to describe them in some language (including the language of mathematics). Maybe we find only parts, or we see the laws coarse-grained (i.e. in some approximations), but it is out there to be discovered.
I believe that our ancestors discovered mathematics due to particular needs that warranted defining. One case in point:
A 35,000 year old, fossilized baboon bone found in Zaire, the Ishango Bone, is covered with a series of notches or tally marks, which makes it the oldest mathematical object in the world, and the world's earliest number system. The bone is also a lunar phase counter, which suggest that African women were the first mathematicians, since keeping track of menstrual cycles requires a lunar calendar.
As needs are met throughout the history of mankind, curiosity takes over. Mathematics and Language become the end all of the modern thinkers of the day. As new concepts are discovered, so are new needs. When certain civilizations discover a new, bold way of keeping track of seasonal shifts, then a way to make more out of their harvesting becomes clearer, and the drive to make an even more accurate calendar arises. Another case in point:
Mathematics in Africa started much earlier from the first written numerals of ancient Egypt around 3100 BC. Ancient African calendars made use of numbers and calculation at an early stage. Ancient Africans also discovered and use the concept of zero, and wrote several texts on math and other subjects.
Where did zero come from--and what, exactly, does it mean? The Nothing That Is begins as a mystery story, tracing back to ancient times the way this symbol for nothing developed, constantly changing shape, even going underground at times. (The ancient Greeks, mathematically brilliant as they were, didn't have zero--or did they?) The trail leads from Babylon through Athens, to India, then to Europe in the Middle Ages. Brought to the West by Arab traders, zero was called "dangerous Saracen magic" at first, but quickly made itself indispensable. With the invention (discovery?) of calculus in the seventeenth century, zero became a linchpin of the Scientific Revolution. And in our own time, even deeper layers of this thing that is nothing are coming to light: our computers speak only in zeros and ones, and modern mathematics and physics have shown that "nothing" can be the source of everything.
Was zero invented? I think not. As we progress in our own evolutionary way, and our minds are capable of grasping newer and fresher concepts, the more readily we will be able to find the ways to discovery.
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