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I know that Mathematics is nearly as old if not AS old as humanity itself, but which empire/nation can be credited as the cornerstone of what we know today as Mathematics?

My initial guess is Mesopotamia as it contains some of the earliest historical evidence of what we called Mathematics today. If anyone has some hard primary source information, I'd love to hear it.

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    $\begingroup$ What do you mean by "mathematics"? Calculations? Proofs? Abstraction? $\endgroup$ – Rory Daulton Jul 22 '15 at 13:27
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    $\begingroup$ It really isn't that difficult of a question to me but I may not have the terminology you're looking for. Who was the first major contributing nation/empire to the field of mathematics. Who established the first fundamental theories and simplest concepts of math. Where was its earliest origins, if known. $\endgroup$ – Adam Jul 22 '15 at 13:43
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The word "mathematics" can have several different meanings. If you include things like counting (all sort of things, say sheep or beer or days of the year, stars in the sky or solstices and other astronomical events) or making the corners of a building right and the walls straight, this kind of mathematics is as old as mankind.

But if you are asking of mathematics "as it is understood now", in the narrow sense, that is a system of knowledge based on rigorous proofs, this was invented in one country, probably in one place approximately in 6-th century BC. We do not now exactly, but according to the tradition this invention was made by Thales (624-546 BC, dates from Wikipedia but they are only guesses). He lived in Miletus, on the territory of modern Turkey, but was Greek by language. He is credited with the crucial discovery: that some true things can be PROVED. (Not just verified by experience). Some simple things he proved are still taught in schools, essentially with the same proofs.

This is a semi-legendary person, his writings do not survive, and we know about them only from much later sources. So one cannot be sure about the details. But the fact is that mathematics in the modern sense was invented by the ancient Greeks approximately 6 centuries BC. There is no record in history of an independent invention anywhere else.

EDIT. We cannot be certain, of course that Thales really proved any theorems, this statement is based on ancient histories written many centuries after him. But we can be reasonably certain that mathematics was invented by some Greek, and probably this happened in 6-th century BC.

The earliest surviving complete mathematical works, with proofs, come, as I understand, from the 3-d century BC. (Correct me if I missed some earlier work). But this is already fully developed, essentially modern mathematics.

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    $\begingroup$ The latter was what I was asking for. I appreciate it. $\endgroup$ – Adam Jul 22 '15 at 14:13
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Alexandre Eremenko's answer is all good, but here are some extra things :

Mathematics did sort of develop independently several times in history, I believe basically three times during the Iron Age : Greece (as mentioned above), India and China. Before then, people didn't care all that much about proofs and mostly had ad-hoc formulas and big tables of values (Sumerian math is famous for giant tables).

China had such works as "The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven" during the Zhou dynasty (1122–256 BCE), which contains a proof of the Pythagorean theorem, and the Mo Jing in 330 BCE.

Edit : Some more things, I guess :

I cannot find much in the ways of sources indicating foreign origins of chinese mathematics, except perhaps babylonian (Chinese mathematics was pretty tied to astronomy). For dates, a comment from Mathematics across cultures :

"On the whole, if Greek and Chinese mathematical astronomy are compared from a merely technical and non-historical point of view, as an abstract catalogue of isolated results, it clearly appears that Chinese developments, such as those connected with the discovery of the precession, or the solar and lunar inequality, occur much later than the former, usually five or more century"

As for Indian mathematics, the oldest geometry texts are the Sulbasūtras, parts of religious texts used for altar construction. The seven big ones are the Bodhāyana, Āpasthamba, Kātyāyana, Mānava, Maitrāyana, Varāha and Vidhūla, written between 800 and 500 BCE. They include the proofs of such theorems as :

  • The diagonal of a rectangle divides it into equal parts
  • The diagonals of a rectangle bisecting each other and opposite areas are equal
  • The perpendicular through the vertex of an isosceles triangle on the base divides the triangle into two equal halves
  • A rectangle and a paralleloram on the same base and between the parallels are equal in area
  • The diagonals of a rhombus bisect each otbher at right angles
  • The pythagorean theorem
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  • $\begingroup$ Is there an English translation of the Chinese work you refer to? Unfortunately I do not read Chinese. No English source on Chinese mathematics that I know hints on independent origin of mathematics in China. $\endgroup$ – Alexandre Eremenko Jul 23 '15 at 15:17
  • $\begingroup$ What is the reference (if available in English) on the independent origin of mathematics in India? $\endgroup$ – Alexandre Eremenko Jul 23 '15 at 15:17
  • $\begingroup$ Do you have some references for the second paragraph? $\endgroup$ – HDE 226868 Jul 23 '15 at 20:43
  • $\begingroup$ Added some things. Overall btw my references on non-western mathematics are mostly from "Mathematics across cultures" and "A history of mathematical notations", which contain both a good overview and a good bibliography on the topic. $\endgroup$ – Slereah Jul 24 '15 at 8:27
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If we understand by mathematics simple counting and addition, then the oldest known relicts are the Ishango bone (20000 years) and the even older Lebombo bone (44000 years). It is even speculated that the duodecimal system and the sexagesimalsystem have their roots there. The "nations" producing it are today named Kongo and South Africa.

If we understand by mathematics some more sophisticated calculating like geometric series and drawing rectangular triangles, then Egypt, and Mesopotamia usually come to mind.

When we include astronomical observation and calculation, then not only China should be mentioned (execution of the astronomers Hsi and Ho 2000 BC) but also the sky disc of Nebra, recently found in Germany, stemming also from about 2000 BC.

For the modern meaning of proof-based mathematics, see the answers already given: Greek, India, China, all several centuries BC.

Finally it should be noted that the word μάθημα máthēma has been created by the Pythagoreans.

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