A comment on another site brought the article How Does Race Affect a Student's Math Education? to my attention. In the article, the author observes (emphasis mine),

But she’s also constrained by the institutional aspects of whiteness in her classroom that exist outside her teaching methods—not simply the how of teaching, but what the state standards value. She and her students share a culture that isn’t reflected in the way she’s expected to teach math. Required to rely on what she calls a “western white lens,” other sources of math knowledge that would be relevant to her students remain untapped. “What are the theorems that we have known here in America before colonization? What indigenous mathematicians have we had? We’re not a written society, so we don't have these books that say, ‘Here’s this Ojibwe person’s knowledge.’..."

Do we know anything about what mathematical theorems were known in the Americas prior to European contact and settlement? It's become somewhat of a trend to point out that the Maya civilization of present-day Mexico had the concept of zero, but little else seems to be said about the theoretical mathematical knowledge of native civilizations of the Americas - that is, whether they conceived of math simply as a set of basic operations (+, -, etc.), with more advanced math coming only through European educators, or whether or not they had developed an advanced conceptual knowledge of numbers, geometry, etc. in a way analogous to how Europeans at the beginning of the Age of Sail and colonization were studying and developing theorems at universities.

To be clear, the article mentions that native peoples of the Americas may not have expressed mathematical truths in the same way that Europeans do in terms of peer-reviewed mathematical journals, textbooks, specific written notations developed by European mathematicians, etc., so I'm not expecting to see that. This doesn't mean that they couldn't have known about such truths - maybe they had an oral tradition/story/ritual that clearly demonstrates knowledge that the square root of two is irrational, or maybe some archaeologist found a twelfth-century totem pole somewhere in Oregon that appears to demonstrate knowledge of how the Pythagorean Theorem works.

In response to comments, I'm not looking for a specific type of pre-Columbian mathematical paper, publication, journal, dissertation, or presentation that would meet modern Eurocentric standards of academic scholarship. I'm looking for evidence of a pre-contact understanding of mathematics - that is, whether natives of the Americas conceived of math as a theoretical discipline that could be studied intentionally or whether it was only ever a practical endeavor (e.g. "I had three tents yesterday. I built another one, so now I have four tents." or "I had five children, one died, so now I have, uhh, one, two, three, four children.").

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    $\begingroup$ As far as we know, the idea of mathematical proof was invented only once in history: in Greece, in 6th century BC. It spread to the rest of the world from there. So there are no mathematical proofs in Americas before the contact. $\endgroup$ Commented Jun 24, 2020 at 12:59
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    $\begingroup$ The issue is not that "may not have expressed mathematical truths in the same way that Europeans do in terms of peer-reviewed mathematical journals, textbooks.." America was "discovered" in 1492 and at that time there were no "peer-reviewed mathematical journals" in Europe. The issue is that we have very very few records of "mathematical activities" of native American and I think we have very very few chances to discover some new one. $\endgroup$ Commented Jun 24, 2020 at 14:22
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    $\begingroup$ For what I know, only some native American people have writing $\endgroup$ Commented Jun 24, 2020 at 14:26
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    $\begingroup$ For an overview see Native American Mathematics volume edited by Closs. However, phrasing it in terms of "theorems" is part of "the way that Europeans do", this sort of abstraction is alien to native cultures and is best not forced on them. $\endgroup$
    – Conifold
    Commented Jun 24, 2020 at 21:52
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    $\begingroup$ You folks should read "1491." There is reasonable evidence that some South American civilisations had reasonably advanced writing - not just cuneiform but also the "woven strip messages." We can back-deduce mathematical knowledge, if not formal theorems, from the type and quality of architecture, for example. $\endgroup$ Commented Jun 25, 2020 at 11:36

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The Greek word "theorem" has a precise meaning "a statement for which a mathematical proof exists". As far as we know these notions were invented only once: in ancient Greece in 6 century BC. More precisely in the Greek city of Miletus on the territory of modern Turkey. This is what the Greek tradition says. From what we know, this discovery was never made independently in any other place and time. This does not mean that other civilizations had no mathematical knowledge. For example, Babylonian and Chinese civilizations accumulated a lot of sophisticated mathematical knowledge, but they had no "theorems".

The knowledge that (for example) a triangle with sides 3,4,5 has a right angle is DIFFERENT from the Pythagorean theorem, because Pythagorean theorem is proved. One of the theorems credited to Thales says that "vertical angles obtained by intersection of two straight lines are equal". Or that "all right angles are equal". They say that Thales was ridiculed for such trivial statements. Of course, everyone understands that this is so.

The unique contribution of the Greeks is that such things can be PROVED, by a very specific procedure which is called a "mathematical proof". Many civilizations knew the empirical fact that there are 5 regular polytopes. But the theorem saying that a) these 5 really exist (=can be constructed by a very specific procedure using a compass and a ruler), and b) there are only 5 of them and no more, is a Greek achievement, and as far as we know, nobody even tried to give a precise definition of a regular polytope, and to prove these exact statements.

There is no evidence that someone somewhere else at some other time proposed this idea independently.

  • $\begingroup$ Didn't the Chinese prove the Pythagorean theorem (under the name of Gougi's theorem) way before the Greek? $\endgroup$
    – Taladris
    Commented Jul 8, 2020 at 22:54
  • $\begingroup$ @Taladris: they did not. To be sure, I do not read Chinese. But all I read on Chinese mathematics in the languages that I do understand confirms that there was no notion of proof in China before it was brought there by Europeans. $\endgroup$ Commented Jul 9, 2020 at 1:58

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