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When I talk about the western mathematical tradition, I think roughly of the mathematics done by the Greeks, and then take up by European countries and then countries with primarily European descendants. Also, I'm not thinking of the mathematicians in particular, but the methods they used, and the work they built upon and added to.

What non-western and non-modern mathematical traditions contributed to modern mathematics? The biggest example I know of is Algebra. I also know some small things, like the Chinese remainder theorem, and I remember hearing that African mathematicians independently discovered Fibonacci numbers at some point.

Note that I would say that Ramanujan's work counts as western or modern mathematics, even though he himself is Indian, since he primarily worked with mathematicians using western techniques, and built upon that tradition.

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This is a meaningless question. First, you count medieval "Arab" contribution as "non-western", while it is well-known that Arab mathematics comes from the Greek and Mesopotamian sources. Arabs did not invent mathematics themselves but developed Greek and Mesopotamian mathematics. Is Mesopotamian mathematics "western"?

Really, mathematics cannot be separated into "western" and "non-western" until 17 century. Greek mathematics was influenced by Mesopotamian tradition. Muslim peoples inherited Greek and Mesopotamian mathematics. Indians too, to some extent. Then Europeans of 16 century learned a lot from Muslim (and Indian) mathematics, including many achievements of the Greeks. Much of trigonometry, and positional number system is of non-European origin.

And why is Greek mathematics WESTERN? Greece is in Eastern Europe, by the way, and most significant Greek mathematicians worked in Africa, namely in Alexandria. And Western European civilization is very different from the Greek one. Yes, we inherited Greek mathematics but so also the "Arabs" did. (I use quotation marks because many medieval mathematicians were not Arabs but Iranians and people from Middle Asia).

It is indeed very hard to tell which tradition never experienced Greek influence. (Perhaps American-Indian is the only clear case).

Second, you mention Chinese reminder theorem and African independent discovery of Fibonacci numbers. Can we really say that these independent discoveries CONTRIBUTED anything to Western Mathematics? Did Western mathematicians really learn them from Africans and Chinese? No. There is a difference between "discover independently" and "contribute".

Let me give another clear example which is well documented. In 18th century Japanese mathematician Seki Takakazu developed the theory of determinants. Few decades earlier than his European counterparts. But did European mathematicians learn anything about determinants from him? Did Japanese mathematicians of 20th century study determinants from his work? No, they studied them from "western" sources.

After the Meiji revolution, the door for cultural exchange was wide open, Japanese students learned "western mathematics" very quickly and started to contribute to it on the highest level, since approximately 1900. But they learned it from European sources, not from native Japanese sources.

Did European mathematicians really learn anything new from pre-19 century Japanese mathematics? I strongly doubt it. There was a high quality mathematics in Japan before 19th century, but it did not contribute to modern mathematics.

I dare to claim that none of the famous mathematicians who made a substantial contribution to modern mathematics did this by developing any "non-western" tradition, no mater where this mathematician comes from. Even when some of them claim the contrary. One cannot contribute anything essential to modern mathematics without being educated in this "western tradition".

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    $\begingroup$ +1. Also, Cordoba/Seville in Spain was a important center of "arab" science. Was it "western" because it was located what is now Western Europe? Was it "eastern" because those were Muslim (and Jewish, but not Christian) scholars? $\endgroup$ – Peter M. - stands for Monica Nov 7 '17 at 21:03
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The question as formulated 'What non-western and non-modern mathematical traditions contributed to modern mathematics?' does not leave much place for anything except a strictly negative answer. The nature of mathematics being what it is, it is possible to (re)discover results independently of preceding authors and traditions, e.g. we know that Leibniz discovered the expansion of pi later than Madhava and naming the series with the hyphenated Madhava-Leibniz acknowledges priority, not influence or contribution.

The self reliance of mathematics exists along with an historical deficience of communications, so the West did not know anything of Indian mathematics which developped independently to an impressive degree. It is an historical fact that Christianity deleted pratically all of the Greek mathematics and the Scholastic revival started with translation from Arab copies into Latin e.g. Adelard of Bath translation of Euclid.

Recent research has uncovered a remarkably high level of mathematics (e.g. Plimpton 322 and YBC 7289) during the Old Babylonian epoch, a millenium before the Greeks, but its later transmission remains unclear. Some of it went to Egypt but mostly it was reconstructed by the Greeks as Joran Friberg demonstrated in Amazing Traces of a Babylonian Origin in Greek Mathematics, 2007, World Scientific.

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I was taught that the standard notation for representing integers was invented in India and transmitted to Europe by Arabs. I suppose this is part of mathematics.

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Counting is non-western in origin:

Around 35,000 BC, somebody in South Africa made the earliest known counting stick, or tally stick, and left it in Lebombo Cave. Somebody cut 29 notches into the stick.

The next advance in counting wasn't made until the early 20th C by Cantor in his cardinal arithmetic.

As 29 is roughly the number of days between one full moon and another, we could also say that this is also the earliest evidence we have of astronomical observation and so one of the earliest hints of science.

Also mensuration or the measurement of length, area and volume in ancient Sumeria, Babylonia and the Harrapan civilisation.

The first unification result in mathematics is likely to have been this - not only can we count things, we can count length, area and volume and then it developed into a distinct discipline - geometry - which as the name describes, is the measurement of the earth.

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    $\begingroup$ To be fair, the distinction between what is western and non-western is a bit of a moot point, when you go back to a time when most of Europe was still covered by ice and permafrost and America won't even settled by any human inhabitants for another 20,000 years. $\endgroup$ – mlk Nov 9 '17 at 15:32
  • $\begingroup$ I can see what you're driving at; however, it might have been the case that mathematics as such wasn't discovered until much later; this shows that mathematics (and observation) goes back to very early times. $\endgroup$ – Mozibur Ullah Nov 15 '17 at 12:58
  • $\begingroup$ To claim that there was no "advance in counting" between 35,000 BC and the early 20th century is preposterous humbug. $\endgroup$ – fdb Nov 16 '17 at 15:58
  • $\begingroup$ @fdb: ok, since you know so much about this topic, you tell me what advance in counting was there? And I'm not talking about arithmetic. $\endgroup$ – Mozibur Ullah Nov 16 '17 at 17:18
  • $\begingroup$ @fdb: the point I'm making is that counting is the simplest mathematical operation that it's hard to see how one can extend it; this was discovered by Cantor in the context of Set Theory when he defined the cardinality of a set; this is the most straight-forward extension of the notion of counting. $\endgroup$ – Mozibur Ullah Nov 16 '17 at 18:36

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