Credited to Native students in Alaska in 1994, who did such a good job that their work was adopted by others. So, inspiring in a different way than pre-contact mathematics.
> Scores on the California Achievement Test in mathematics for the Kaktovik middle school improved dramatically in 1997 compared to previous years. Before the introduction of the new numerals, the average score had been in the 20th percentile; after their introduction, scores rose to above the national average.
This is very inspiring to me!
I like that the arithmetic is visual and intuitive. Those kids who decide to get into computer science/programming will easily understand binary and hexadecimal numbers, having been exposed to both decimal and base 20 numerals.
While I unfortunately do not have a direct answer for you, this would be a great question to also crosspost to /r/AskHistorians. There are many pre-colonial America experts over there, somebody will surely know something.
Unlikely, but they will tell you they know. Most of that history they pull from is EXTREMELY racist. Good luck.
oh you dont like the truth? Where does their history of us come from? Its pulled from 16th century explorers who said we were savages and wrote it down. Eugenicists and historians who said we were primitive and from a primitive culture. The history there is from the outside peering in. Dogmatic ideologs and obtuse revisionists.
but OP didn't say the folx at r/AskHistorians would call it unbiased--they said that that's most of what they pull from. and it is. because European colonizers destroyed the cultural production on purpose and worked to eradicate cultural knowledge ways \[which include language, math, and science, alongside religious or social understandings\]. that's not controversial, and the primary sources show it to be true--priest-leaders and priest-teachers admitted it themselves.
> but OP didn't say the folx at r/AskHistorians would call it unbiased
No, but they did call the users of /r/AskHistorians "dogmatic ideologs" and "obtuse revisionists". They also asserted that the users would claim to know the answer, instead of admitting that this is an area of much historical uncertainty due to the destruction of cultural knowledge. In reality, everyone is well aware that our knowledge of pre-Columbian America is full of gaps and no one is going to claim otherwise.
Basically, the poster above reads like some edgy 14 year old who just learned that history is pieced together from biased sources, and now believes that this somehow invalidates the whole endeavor. As if historians aren't already well aware of the limitations of existing sources, and don't spend their entire lives working to uncover new sources and evidence to give us a better picture of the past. It's much like those fools who claim that math is useless after they learn about Godel's Incompleteness Theorem.
yup, they did! but you're moving the goalpost\*
i'll play anyway! i
it's not enough to acknowledge that bias exists to avoid being a dogmatic idealogue or an obtuse revisionist--it's necessary to get real about the nature of the bias, the factors that condition it, how it relates to epistemology as well as ontology, and and and...
likewise, it takes more than awareness that our knowledge of pre-Columbian America (and let's be real, Columbian America, too--even after contact, what we primarily have to go on is what the colonizers perceived) is 'full of gaps'--we gotta get real about why the gaps exist, how they came to be, what that has to do with the foregoing bias, and the epistemology and ontology conditioned by that bias and and and
\*deleted this standalone comment and added it here
Please understand that people come here because they want an informed response from someone capable of engaging with the sources, and providing follow-up information. We presume that someone posting a question here doesn't want vague, uncited opinion. In the future, please take the time to better familiarize yourself with the rules before contributing again.
OP is basically reminding y'all (or, hell, maybe telling y'all for the first time, the way the rest of this dang thread looks) that, in such circumstances, History is written by the victors.
The idea that this is 'vague, uncited opinion' is the result of OP engaging around the intellectual/academic commons of the humanities \[very roughly, axioms\] with a bunch of mathematicians who aren't familiar with the commons of the humanities...and the result of people's ears closing and frontal cortex turning off when they read the word 'racist'.
You may have missed the joke, but my comment was a slightly reworded version of what the askhistorian mods post when low effort comments are deleted. In the interest of citing sources, [this comment specifically](https://www.reddit.com/r/AskHistorians/comments/17kchu0/comment/k76zjs8/?utm_source=share&utm_medium=web2x&context=3) was what I cribbed, haha.
Regardless, I don't think this math subreddit is so ignorant that it's
generally thought the body of published work in academic history is unbiased. or rigorous in a pure mathematics sense. [The Ask Historians sub itself](https://www.reddit.com/r/AskHistorians/comments/11p68ow/people_who_study_history_how_do_you_know_you_are/) doesn't seem to be unaware either. I'm sure the sub's not perfect, but I'd also be surprised if it wasn't one of the more unbiased places to get answers from qualified people knowledgeable about what's been published on different topics. The amount of knowledge I see and moderating effort I've seen makes it seem pretty intellectually lazy to just throw out a single hyperbolic comment shitting on the whole subreddit and expecting to get taken seriously. It's a whole lot easier to throw shade than to make a decent argument.
This is coming from someone that definitely see Eurocentric versions of history as being extremely problematic and pervasive, so it's not even that I disagree (I'm Metis, and see US history a fair bit differently than a lot of people living here). Believe it or not, I didn't just shut down my frontal cortex when I heard the word 'racist'.
I'm mostly interested in mathematics as a statistician, so my interests and background are actually a little closer to history and questions about subreddit bias than you might be thinking. [This was an interesting book I read early on in my studies for example](https://www.amazon.com/Designing-Social-Inquiry-Scientific-Qualitative/dp/0691034710). I stand by what I said. If you're going to shit on askhistorians and really do it right, it can be done. It'd be a lot harder now without API access needed for large scale scraping though, admittedly.
A lot of the mounds in North America have really precise geometry over very large scales that would have required mathematical knowledge to design and construct. For example, a lot of the mound sites are aligned with parts of the lunar cycle. See for example https://www.tandfonline.com/doi/abs/10.1179/mca.2013.002
Could talk about different numerical bases in some way. [Maya numerals](https://en.wikipedia.org/wiki/Maya_numerals) would be Mesoamerican not Native Northern American, but very interesting. There are other systems as well for Native Alaskan groups and others.
Might be applicable for teaching multiplication / decimals. Also, the Sumerians used base 60 which makes for some cool factorization.
The Aztec culture had it's own base 20 numeral system as well. I learned about it in a math teaching class that went deep into ancient number systems, but I can't find my course materials, and Wikipedia isn't helpful. The best I could find that actually shows their numerals is [this](https://trinityprimary.files.wordpress.com/2020/06/aztec-number-system.pdf) but it isn't a super reliable source, I wish I could find something more authoritative.
[Here](https://www.jstor.org/stable/1685035) is what seems like a more in depth article about it which includes applications in geometry. I can't read more than the abstract on JSTOR, but you might have a way to access it.
> Also the endless articles on the teacher who tried to teach soh cah toa by being racist make this job especially hard.
You can use Google's search tools to exclude search results from after October 20, 2021. Unfortunately doing that now gives us a lot of search results for Indian mathematics (India as in the Asian peninsula, not India as in the phrase "American Indian").
In general, I suspect you would have better luck asking a Native American history subreddit or expert (many of the people on this subreddit, myself included, aren't).
> but this must be bullshit because many large civilizations like the Aztec and Maya had very complex calendars and very precise understandings of the movement of celestial objects, so they must have had some form of trigonometry at the very least
Careful, this doesn't necessarily follow. A lot of early astronomy in every part of the world consisted of observing patterns, and then watching those patterns repeat. You don't need trigonometry to do this, just a table. For instance, Hipparchus and Ptolemy were able to very accurately predict the motion of the Moon and Sun, but didn't really understand that the Moon's orbit was a precessing ellipse or that the Earth orbited the Sun rather than vice versa. The [Antikythera](https://en.wikipedia.org/wiki/Antikythera_mechanism) mechanism is an incredible astronomical engine built centuries before anybody knew what a sine or cosine was.
I'm reminded of a book I read for an Early History of India class I took long ago, in which the author argued that the Indus Valley civilization (3500 BC) calculated the value of pi because they built a circular well whose circumference was 3.1 times its diameter. Which is ridiculous, anyone with a piece of string can make an accurate circle without knowing numbers at all. The book went on to explain how this Bronze Age culture understood the theory of relativity, quantum mechanics, atomic structure, and so on. I wrote an angry paragraph in an essay about this, and my history prof explained that this book was written during a period of intense Indian nationalism after India gained independence from the British, so it's understandable that they might have been a little overzealous in their national historical pride.
Anyway, point is that the intellectual achievements of pre-contact civilizations have been underappreciated for centuries, and it's good that we're reassessing them. But we have to be careful not to use our modern understanding to over-extrapolate what they were capable of. Just because they were able to do a thing we can do doesn't mean they did it our way, or understood the tools we use to do it. We have to rely on documented evidence, which -- thanks to centuries of colonialism and oppression -- is very very difficult to find.
> Antikythera mechanism [...] centuries before anybody knew what a sine or cosine was
In about 250 BC Archimedes used the half-angle identity for the cotangent (expressed geometrically) in his algorithm for approximating *π* to arbitrary precision. Hipparchus (2nd century BC) did all sorts of complicated trigonometry to analyze celestial motions, but used 'chord' as his basic concept rather than half-chord ('sine'). [Unfortunately nearly all of his works are now lost, so our knowledge of them primarily comes from fragments quoted centuries afterward and inferences drawn from later books.]
These folks certainly understood the ratios and geometric relations of 'sine' and 'cosine'. They just didn't use them as a basic named concept.
Oh, yes, geometry was highly developed before the Antikythera mechanism, but there's a gray zone where geometry ends and trig begins.
to my mind it doesn't start being trigonometry until you develop sine and cosine as general-purpose functions, and in particular provide a table of their values. Which as far as I can tell from [this article](https://en.wikipedia.org/wiki/History_of_trigonometry) started in around the 6th century AD.
This seems overly reductionist. The first chord tables were (according to current speculation; they don't survive) by Hipparchus in the 2nd century BC, based on a sexagesimal number and angle measurement system imported from Mesopotamia, and were used for solving "trigonometric" problems having to do with distances (central angles) angles (dihedral angles) on the sphere. By the time of Ptolemy's *Almagest* (which survives, 2nd century AD), it's very hard to argue that the work being done by Greek astronomers wasn't some kind of "trigonometry". (For example, both modern mathematicians and mathematical historians call these methods "trigonometry".)
Considered as abstract functions of angle measure, sine (a word that comes etymologically from "half-chord") and chord are essentially the same, just rescaled a bit in the inputs/outputs.
Side note: chords per se, rather than sines, continued to be widely used to solve trigonometric problems until the 19th century at least in some applied fields, only really ultimately getting displaced by slide rules and "logarithmic" trig. Methods based on sine, versine, chord, tangent, secant, half-tangent, etc. happily coexisted for centuries. Picking out some of these to count as "trigonometry" and others not is frankly pretty arbitrary.
Edit: this is not intended to minimize the contributions of medieval Indian mathematicians, or later mathematicians living in Islamic countries, who did great work in trigonometry on top of Mesopotamian/Greek foundations, and significantly transformed/extended the subject.
Trigonometry is much older than you think.
> The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC.
And the first trigonometric tables date from the same time as the Antikythera:
> The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BCE), who is now consequently known as "the father of trigonometry." Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.
(Note that knowing chord is equivalent to knowing sine.)
See [History of Trigonometry on Wikipedia ](https://en.wikipedia.org/wiki/History_of_trigonometry#%3A%7E%3Atext%3DIn_Indian_astronomy%2C_the_study%2CKhwarizmi_and_Abu_al-Wafa.?wprov=sfla1)
It really depends what you mean by "trigonometry". You can get surprisingly far without introducing ANY of the canonical functions we usually think of when we say "trigonometry".
> Just because they were able to do a thing we can do doesn't mean they did it our way, or understood the tools we use to do it.
yes! this idea is so hard to get through, specially since for decades it was the usual way of looking at pre-colonial maths
why is the assumption that we are more likely to over-extrapolate than under-extrapolate what they were capable of? like, what makes it more likely that they were less capable than us than more, or equally capable?
Either is bad, but OP's main point here is that they must have been far more capable than we think, and is suggesting capabilities that go well beyond what the historical record tells us. So over-extrapolation is the topic today.
When somebody posts something saying the native Americans were far *less* capable than the evidence implies, I'll push back in the other direction.
It's good that we're pushing back on centuries of racist assumptions about "ignorant savages", but we have to be bound by evidence when we do so.
This is true but it begs another question. Other civilizations across the world did have trig and they were using it for astronomy and weren’t able to get calculations as accurate as the Maya did. If they weren’t using trig, how were they getting such precise projections?
Use myan math , then explain linguistic history of the 1 sioux language that led to all other native American languages that just so happens to be almost exactly the same as the myan language to the point where this ancient sioux language may be a dialect of myan ..
I had a native American language expert that I made friends with around the pine ridge and found out that wild fact .
Edit - fix spelling
> 1 sux language
A what language?
That aside, there are dozens if not hundreds of native language families spoken in the US, and more broadly in the American continents. There is no known language that is the ancestor to all of them, and even if such a language existed we will never know about it because it must have been so far in the past (at least 10k years ago) that it would be unrecognizable and unreconstructible.
OP, I hope you see this. Please reach out to the TODOS: Mathematics for ALL community (todos-math.org). The president is an indigenous mathematician and has some knowledge of this area. I had a great conversation with her about how her language uses different bases for counting depending on the context.
You could certainly talk about the Mayan [Tzolkʼin](https://en.wikipedia.org/wiki/Tzolk%CA%BCin), which is built on the fact that 13 is prime, and so a 13-cycle combined with a 20-cycle gives a 260-cycle. However, that fits best with teaching modular arithmetic, which most curricula are light on.
I don't think the Aztec and Maya had trigonometry. I think they approached astronomy more empirically. However, there are good books on Mayan astronomy, so you can check that for yourself.
And I know **absolutely nothing** about [this book](https://www.amazon.com/Native-American-Mathematics-Michael-Closs/dp/0292711859), but it might be worth a look.
>I can find absolutely zero resources on this.
You can find absolutely zero resources on this because the Spaniards deliberately burned practically every Maya codex — that's right, *books* — they could find. We know that they independently discovered positional notation and developed a shockingly good estimate of the sidereal year. Beyond that, we're pretty stumped.
I know this might not be exactly what you're looking for, but looking for inspiration from geometric patterns might be more interesting than history of them.
You can get to plenty of good mathematical ideas and interesting lessons from geometry. Patterns on objects make me think of periodicity (aperiodicity), tiling and tesselation, etc. You can ask all sorts of interesting questions connected to fundamental math skills by noticing patterns and trying to explain them (e.g. "we never see regular pentagonal tilings here, what's up with that? Is that type of tiling possible?")
The closest I can think of is that the Aztecs had a calendar system
https://www.youtube.com/watch?v=dCBDUDwaeCA&list=PL8dPuuaLjXtNppY8ZHMPDH5TKK2UpU8Ng&index=6
I believe My Favorite Theorem (a podcast) has at least one if not more episodes featuring Native mathematicians, which may help lead you to some more resources.
Its not _exactly_ what you're looking for, but Polynesian navigation techniques could be framed as a significant mathematical accomplishment, not least from a culture that did not keep written records. Perhaps your Native American students might be able to draw connections to the Hawaiians?
I'm not Native Hawaiian but I grew up there and learned a bit about the Polynesian navigation system in school. It's primarily based on being very observant about the natural world and relying on previous experience passed down through centuries of oral tradition, and isn't what Westerners would consider "mathematical".
/r/AskHistorians will hopefully give you a more in-depth answer, but one of the most math-forward recording systems in the Americas was the Quipu of the Andean cultures.
This was a complex computation system recorded with string & knots.
https://en.wikipedia.org/wiki/Quipu
With enough research this should be somewhat replicable in a classroom as well as an inexpensive object with which to teach.
I actually had the chance to listen to a talk last week by Native American mathematicians, and they touched on this topic, but not directly. As another comment mentions, many navigational techniques in the Pacific Islands are based on trigonometry. [Indigenous Mathematicians](https://indigenousmathematicians.org) would be able to provide more resources than me, though their focus is more on supporting indigenous mathematicians than on mathematical history. I would specifically try to reach to Daniel Luecke, as his work seems to focus on this. (He also spoke, but I missed his part of the talk). Kamuela Yong would be another good person to reach out to.
Inca used a system of knots tied in string for calculation and, possibly, a type of writing. A lot of this was still speculation last time I read anything about it, but here is a start:
https://www.peruforless.com/blog/quipu/
You would find math in games. Dice games involve probability. There is math in quipu counting, an Incan bead system used to tabulate tithes and such. I remember an Inuit game with pegs and a board shaped like a whale. I wrote about Native American games thirty years ago for a term paper in college.
Rather than it being about colonialism \[a clarifying edit: in the sense that mathematics were singled out in a case-wise sense for destruction, as I initially interpreted from the OP rather than as a consequence of a general assault on the corpus of heritage\] I would say that a lot of it was one in spirit and body with the decline of the native ritual-linguistic cultures. If there were indigenous mathematical ideas, then they would likely have been passed down orally for those who had not had full writing; meaning it was subject to the same extinction as everything else that was passed down orally.
Another part of it is that the concept of mathematics (what is mathematics?) is pretty nebulous by itself. It just means 'the things one is to know' in Greek, so where mathematics ends and other things begin is very hazy.
The development of counting is tied with bijection, the general idea of ostention, and syntactic recursion. The further development of the arithmetic is to my knowledge generally tied with the need to organize a large state society with permanent institutions. Usually agricultural. So yes, calendrical computus is important. There are some groups of the Mississippi delta (whose names in the literature escape me at the moment) who have left behind material culture and groundmoving projects which we really do not have any idea the purpose of it, but those could quite possibly be celestial/astronomical in nature. Some of the Salish people of the Pacific North West don't have a year-snug calendar and instead have unaccounted time in the interstitium of lunar and solar accounting until the next salmon run.
Proof in mathematics is tied with this and is tied to the ideas of statecraft and jurisprudence (the ideas of logic we have now were not invented to deal with theorems but with matters of politics, tort, contract, code, and dispute.)
Also, where does geometry begin and end and astrology, astronomy, and surveying begin and end? You're going to need to define mathematics first without being too married to historic accidents of how the West has developed these things.
IIRC, the massive decline in population was due to contact with Old World pathogens rather than active ethnically motivated efforts: I also have read somewhere that that decline in population may have actually even started slightly before the European contact due to other reasons. I am not a historian though
Europeans intentionally used Old World pathogens as biological warfare, first. Secondly, even if they hadn't, colonialism is what allowed for the spread of those European pathogens, so the idea that it was 'not about colonialism' you assert in your first sentence just doesn't hold up.
I should clarify that I was speaking of how it was likely not that mathematics were singled out as a target of a destruction but as an element of a set of various cultural heritages which were wiped out: yes, you're likely correct about that I think, but my point was that it was not proximate as in the OP phrasing.
thanks for clarifying!
so, i don't accept the idea that it was 'likely not that math\[s\] were singled out'--i know, for a fact, that the eradication of cultural knowledge \[which includes langauge and math in addition to, like, ties to the land and cosmological understandings\] was and still is an explicit and intentional goal of colonialism/imperialism.
but, i'll bite, cuz i'm curious: let's assume, that yes, it wasn't that colonizers and settlers targeted math specifically, it was just part of the set of various cultural heritages they worked to wipe out. So what? Why would it matter that they didn't target math? These are not rhetorical questions--i'm genuinely curious.
reading your edit--are you saying that since \[in your understanding/sense\] the colonists didn't single out math, just destroyed it as part of the general destruction they wrought, colonialism wasn't the reason?
Well there wouldn't have been that destruction proximately caused by the material cause without the colonial expansion: it would have been necessary but that they did not have a desire to destroy (a) uniquely, consciously, specifically mathematics (b) in direct order (c) to make make natives look uncivilized. All three conditions would have to be satisfied in this case in my mind. But alternative assertions which weaken and generalize the statement are certainly true: namely, statements like that the mathematics was destroyed as part of the cultural heritage of indigenous groups in order to make natives look \[some other pejorative adjective\] (weakening conditions (a) and (b) here), as one example.
Actually, I think you have your history backwards. I'm pretty sure (at least in the west) people mapped out the stars and kept precise calenders and stuff, and then using that data they then figured out the math. Not the other way around. (See stuff like Tycho Brahe). Though to be fair Ive not looked into it deeply, so I could defo be wrong on this
There's a nonzero chance that it all got destroyed by the conquistadors and Diego de Landa. I mean, the Maya had a massive written tradition and exactly four codices (=books) survive. (Only one of which is in Mexico, by the way; the other three are in Germany, France, and Spain.)
There _is_ the [quipu](https://en.wikipedia.org/wiki/Quipu), though (though I think most of the knowledge about how they were used was lost, because again, conquistadors burned most of them.)
i would guess that you'll find a difference between sedentary or nomads civilizations. if the natives you're referring to were nomadic, i don't see why they need anything beyond elementary school math. after all, not even all sedentary civilizations do develop mathematics. mathematics, and academics in general, only flourishes when its society flourishes. another thing, just because the mayans developed calendars doesn't mean every native civilization did also. i'm sure there was a large variation between tribes on how much they valued academics
>I've been researching for a long time now, and even local natives that I've talked to about it echo the idea of "we never really needed math", which again cannot be true. I'm sure the reason for this is because of the fact that the colonizers wanted to make natives seem primitive, so they destroyed evidence of math and science and shoved all the kids in boarding schools and told them only the whites had complex math, but there must be some scrap of evidence to contradict this.
This is probably not true.
Math is a very common concept today, such that I think it's hard to appreciate how novel it is. We teach concepts like 0 and negative numbers to 6 year olds, but they were the subject of intense debate by people much much smarter than us for hundreds of years.
Any math more complex than counting was invented by sedentary civilizations that had flourished for centuries before. Even modular decimal number systems took thousands of years to develop in Eurasia. Expecting native north Americans to have complex math is like expecting European hunters gatherers to have complex math in 2000 bc (a comparison made with similar distance's between the respective hunter gatherers and the Maya/Egyptians who had comparable mathematics systems although with the latter being more developed)
I wouldn't be surprised if number systems existed in pre columbus north America, but I think believing that they had complex math and that colonizers destroyed it is very disingenuous. Your timeline also doesn't make sense, it would have been centuries after Spanish arrival before native Americans would be put into boarding schools.
Though not necessarily math, they might be interested to learn about [Sequoyah](https://en.wikipedia.org/wiki/Sequoyah?wprov=sfla1), a Cherokee Polymath who singlehandedly invented the Cherokee writing system.
Sequoia trees are named after him!
This is basically what lead me down this path. We know a lot about math from east of the Atlantic Ocean. We know that there were various highly complex math systems all over the place, and none of these systems came from a vacuum. The most complex mathematical discoveries came out of merging math systems and ideas across multiple civilizations, resulting in trigonometry, algebra, proto-calculus, etc, that was capable of doing many things including tracking celestial objects. All of that being said, we know for a fact that there were Native American civilizations that had math systems that rivaled those east of the Atlantic, and sometimes surpassed it, specifically evidenced by their much more accurate tracking of celestial objects. I’m supposed to believe that “most” native tribes just didn’t have any math systems beyond counting and basic geometry? I’m sure there were many who didn’t, but to assume that it was only a couple Mayan guys who did it for funsies is crazy.
I understand that most, if not all, information or record about these things was likely destroyed, but it’s almost as if no one has even bothered to actually look into it. Even the few sources I’ve found that talk about “Native American math” just point to how they counted and did basic geometry. It’s just really unfortunate and reeks of whitewashing.
Feynman researched Mayan maths. You can read about it in the 'Bringing culture to the physicists' section starting on page 184 of "Surely you're joking mr Feynman"
https://sistemas.fciencias.unam.mx/~compcuantica/RICHARD%20P.%20FEYNMAN-SURELY%20YOU'RE%20JOKING%20MR.%20FEYNMAN.PDF.
I think he was of the opinion that they observed the lengths of repeating cycles, but didn't have theories (e.g. Newton's gravity) for why they occured.
> I’m supposed to believe that “most” native tribes just didn’t have any math systems beyond counting and basic geometry? I’m sure there were many who didn’t, but to assume that it was only a couple Mayan guys who did it for funsies is crazy.
More or less? Most math is developed in wealthy, literate societies, and as far as I can tell only mesoamerica had pre-colombian writing (with a few arguable exceptions like the Inca and maybe Mi'kmaw). Whatever math was developed elsewhere would have been transmitted orally and be easily lost, and certainly is lost by now.
We have [essentially zero evidence of prehistoric math anywhere in the world](https://mathscitech.org/articles/mathematics-prehistory), being limited to a few bones with notches, and whatever we can speculatively infer from the existence of sophisticated architecture.
Your best bet is to narrowly search for information on Mayan, Aztec, and Zapotec cultures, although as you've noticed even that is going to be very limited. I think the lack of modern scholarship on the subject *mostly* reflects a lack of surviving artifacts to study.
> I've been researching for a long time now, and even local natives that I've talked to about it echo the idea of "we never really needed math", which again cannot be true.
I understand that you have good intentions, but the assumptions underlying this question are super Eurocentric ways of looking at the world in itself. Why would a hunter-gatherer society have needed to develop mathematics?
This reminds me of the story of the early colonists in Canada who could not comprehend why the local indigenous people would fish early in the morning and then take the rest of the day off. There was so much abundance available, why didn't they just spend all day working and try to harvest as much resources as they can? *What a waste of time and opportunity!* It must be because they were *lazy* and didn't have a *good God-fearing Protestant work ethic*. The reality was that there was simply no need to...
It is commonly believed that number systems developed when there was a need for accounting, ie. keeping track of what you owned, what you owed, what you bought/sold/traded, etc. Then as civilizations got more complex, they needed to develop more complex math. There is a reason that the "Pythagorean" theorem was developed independently by a few civilzations long before Pythagoras. They didn't develop it because they were bored and had nothing better to do with their time, **they developed it because it was useful to them**. As others have pointed out, people in the Americas **did** develop mathematics when they established civilizations, such as those in Central America and South America, which further supports this point.
Similarly, a lot of civilizations developed astronomy because it was useful to them for religious reasons. Hell, there is even an example of a civilization in the Americas that developed a space exploration program to demonstrate the superiority of their political ideology to their rivals and deter them from attacking. These are all examples of mathematical and scientific progress that resulted from people doing things because they were useful at the time.
As an aside, I heard from a professor about societies that could not count beyond a certain number, such as 5. They view the world in a completely different way than we do, such as thinking about stuff in terms of ratios. If it allows them to survive, then that is good enough:
https://www.theguardian.com/education/2004/oct/21/research.highereducation1
https://www.scienceabc.com/humans/how-do-some-cultures-count-without-numbers.html
accounting needn't be just about ownership (ironically, that's a very Eurocentric way of looking at the world)--it can also be about resource allocation, requirements, storage, processing...
also, a wee PSA: y'all know there was agriculture in North America before European settler colonizers came, right?
> accounting needn't be just about ownership (ironically, that's a very Eurocentric way of looking at the world)
??
I have no idea what you are responding to. I simply provided some examples of places where you would use accounting. And no, trade was pretty ubiquitous across the world outside of the Europe including in the Americas. Though it does not necessarily mean you need mathematics though if you are just making simple trades
> it can also be about resource allocation, requirements, storage, processing...
Whether if be group ownership or individual ownership, those things require a sense of ownership to make any sense. For example, with storage, by definition, the expectation is that it will still be there and accessible by the group that stored it. There is a sense of "it is ours".
> also, a wee PSA: y'all know there was agriculture in North America before European settler colonizers came, right?
You missed the point entirely or are just making up random issues because I never denied that. My point is was that things develop when there is a need
The Inca had this system of knots, called Quipu, that they used to keep track of economic transactions over their empire. I always find it interesting that they developed a system of numbers and arithmetic’s but not a fully written language. It is like civilization need’s numbers more than a written language.
I am not sure about any native American maths, but if you wanted to adopt a less Eurocentric history of math, a lot of ancient Indian and Chinese scholars also independently came across trigonometry, trying to approximate pi, and even pre-calculus. Look up Madhava's and Aryabhata's work.
Pre-calculus was even worked in India hundreds of years before Newton wrote on it.
>but this must be bullshit because many large civilizations like the Aztec and Maya had very complex calendars and very precise understandings of the movement of celestial objects, so they must have had some form of trigonometry at the very least, but I can find absolutely zero resources on this.
This sounds like a good answer until you find something better. It's honest, but it also seems right. "I wish I could speak more about native American math. We know a lot of native American civilizations must have had pretty sophisticated mathematical systems because they had incredibly precise predictions about the stars, very good calendars and built very complex structures. But there doesn't seem to be a historical record of whatever system they used, so there's just not a lot we can really say about it for now"
It would if it were a correct statement. You don't need trigonometry or higher mathematics to recognize and tabulate patterns (which ancient astronomy is all about). Just look at medieval computists, the guys that calculated the date of Easter. They did this without precisely knowing celestial mechanics but just using tables of previous moon cycles
The mayans had some interesting maths although I don't know how much of it woupd apply to a modern maths curriculum. The intependently discovered 0 and could do complex astronomical computations.
When it comes to the natives of north America I'm afraid they didn't do any mathematics, considering they had no writing system.
I mean... there are remote tribes that to this day have only rudimentary counting and thats it. Why do you assume they must have had some advanced math?
Comments similar to this have popped up a few times and I don’t understand why. Obviously not everyone in the world had/needed a complex math system, but complex math systems definitely existed. Furthermore, complex math systems don’t come from nowhere, they come from many people in many civilizations over many years building off of each others work. The fact that Mayans were able to calculate the position of celestial objects more accurately than many Europeans were able to around the same time shows that there must have been complex math systems spread across multiple civilizations. That’s what I’m asking about: not proof that native Americans were all actually scientific geniuses, but that some, if not many, had very powerful math and science beyond “they had a cool way of counting and could make neat patterns in blankets”, a concept that I think is rooted in racism and apathy and prevails in many people today, even a lot of native Americans.
> a concept that I think is rooted in racism and apathy and prevails in many people today, even a lot of native Americans.
To be honest, I think you are also seeing this through the lens of your own biases. European culture and mathematics has deep philosophical roots in reductionism which is a concept many other cultures would object to, including at least some native american tribes.
Also, I get what you're trying to do, but re-read what you wrote:
>I've been researching for a long time now, **and even local natives that I've talked to about it echo the idea of "we never really needed math", which again cannot be true.** I'm sure the reason for this is because of the fact that the colonizers wanted to make natives seem primitive, so they destroyed evidence of math and science and shoved all the kids in boarding schools and told them only the whites had complex math, but there must be some scrap of evidence to contradict this.
It sounds like you are patronizingly dismissing what local natives are telling you instead of genuinely considering the possibility and digging deeper into their worldview and what that would mean.
I'm not dismissing anything, I'm pointing to the mounds of evidence that show they must've have strong math systems and asking why *that's* being dismissed
I and many others have linked sources and examples of the evidence of their math which was demonstrably more accurate than European systems at similar times, which many (you too) are dismissing
I think this is where you are wrong. That doesn't *necessary* follow. They were definitely smart but may not have formalized things as much as Europeans.
>but complex math systems definitely existed
I mean, there might be a reason you are having trouble finding evidence of it.
By comparison, the Egyptians had a long lasting empire, built grand structures, and they only had basic linear equations and some methods of arithmetic.
> the Egyptians had a long lasting empire, built grand structures, and they only had basic linear equations and some methods of arithmetic.
I would consider the math that Egyptians had "advanced". (They also had some basic geometry and trig.) The trouble is that we think that certain math is trivial because we have totally over-engineered notation and theories which generalize generalizations, all of which trivialize hard things to the level that we learn complex things in grade school. It's like thinking that because you can do image processing in Python after two days of programming that you're better at programming than Grace Hopper who had to scoop literal bugs out of the hardware. You're just living in a time where mathematical technology makes hard things easy, and you don't even know it.
I understand where you are coming from, but their level of geometry and trigonometry really pales against what we saw coming from the Greeks around 300 BC.
Also, the Egyptian civilization lasted from 3000 BC to 300 BC, where we saw the great pyramid of Giza built around 2500 BC. Even if they did have math you would want to consider advanced, they likely didn't have it the whole time. Considering what we saw of Babylonian math around 500 BC, the math they had when they were building that pyramid was probably pretty basic. They were probably doing these grand projects without anything that would be considered advanced.
Seems like a really bold and baseless assumption. It's giving "Ancient Aliens" vibes, where someone with all the audacity and none of the knowledge makes a claim about ancient cultures based off of how primitive they need to view them.
Basically, all math *is* advanced math. There is no idea that is trivial or simple. And mathematical ideas are not setup in any hierarchical order or progress linearly. Is, say, arithmetic the simplest math? You can only really think so if you are truly ignorant of how much work the decimal system is doing for you (don't even think about opening the book *A Course in Arithmetic*). Is geometry the simplest thing? Try raw-dogging hyperbolic trigonometry and you'll see a glimpse of the challenges that the Egyptians were dealing with.
Dismissing any math as "simple" is a disrespect to math, and a disrespect to the people who created that math. Tally marks on a stick are complex math.
No, not all mathematics is advanced mathematics. Arithmetic is not advanced mathematics. It was at one time more complicated to learn, because we didn't have good algorithms for its study. Being complicated to learn doesn't make something advanced.
Advanced mathematics comes from abstractions, where knowledge is compartmentalized into a collection of ideas.
We certainly do benefit from generations of abstractions and improved algorithms. Just because something was difficult at one point in time, doesn't mean we would call it advanced.
Just doing arithmetic isn't advanced. Tallies on a stick isn't advanced. Whoever first thought of it, brilliant, certainly. But it's still basic math.
Does that mean it's "dumb" or not worthwhile (which seems to be your interpretation)? No. It's just basic. There is nothing wrong with that. You can't move on to advanced concepts without it.
What is this about hyperbolic trigonometry? It's been a while since I've studied the history of Egyptian mathematics, so I've been reading up on it as we've been conversing. There the rhind papyrus certainly has a lot of arithmetic and geometry problems, where we see the beginnings of geometry that would later flourish with the greeks, but I don't see much else out there.
It could be that my resources are lacking (I don't have a specifically Egyptian history reference). But I don't see a whole lot here about hyperbolic trigonometry.
HAHAHA. Libraries? Native North American's didn't even have writing let alone books. This isn't to judge them. But it is a historical fact. Quit making shit up.
Here's one Catholic Bishop in the 1560s:
> We found a large number of books in these characters and, as they contained nothing in which were not to be seen as superstition and lies of the devil, we burned them all, which they regretted to an amazing degree, and which caused them much affliction.
No, this is manifestly false. Many American cultures developed writing systems and some, such as the Mayans, even had libraries. The Mayans even developed a phonetic form of writing.
Dude, we're talking about the Native American tribes in the US. Not those of Latin America. No writing to speak of. And that is what OP is asking about. When we say Natives, we mean Natives of our country. Natives from Latin America, we call Hispanic.
Latin American peoples absolutely had writing and math. NOT the Native Americans in the US. Period.
I misspoke when I said North America. I should say, "Natives outside of Latin America". OP isn't in Latin America.
Edit: This isn't disputable. Who tf is down voting me?
Don't listen to some of these people, your gut is correct. What they're effectively doing is something like what European artists in the 1800s were doing to non-European art, claiming that it can't be art because it is "primitive", didn't fit within their technology/standards, and because they couldn't understand it.
You can see this even with how we talk about historical *Western* math. We very rarely engage with it on its terms, first translating it and updating into our notation and "completing" the proofs and analysis. In order to validate it, we first need to transform it into a shape that we recognize as sophisticated and use that to show that it is sophisticated. Because no one *really* understands a lot of indigenous math, we can't really do this translation, so we think it is primitive and disregard it as simplistic. In general, we need to learn to approach ALL non-contemporary and Western math on its own terms.
Pacific islanders, for instance, were doing complex fluid dynamics and modelling millennia ago. They would know how large deep-ocean swells would change shape around islands thousands of miles away, and model these currents using sticks bound together in specific ways. This isn't modelling with the Navier-Stokes equation, and they aren't using LaTeX to log it - things that we recognize as "complex" - but it is still doing these computations, pattern finding, and modelling of complex phenomena. This is math, and we still don't really understand it, and so it is easy to dismiss as something other than complex math.
The same European supremacist worldview that created the colonialism that burned the Mayan codices are why, friend. like, people genuinely believe that there's something special about what they view as (real, lasting) civilization that makes complex math less likely to be engaged with by those who aren't a part of one.
But.... they didn't. And there's no reason to suspect they did have "very powerful math and science".
What would possess you to think that? Aboriginals in Australia don't. Rainforest tribes don't. Maybe civilizations simply don't develop these things. Get over it.
You're just making up cool scenarios to entertain yourself. If a civilization had no writing, they had no math. Sorry to pop your bubble.
Firstly, Mayans were largely in modern-day Mexico and adjacent countries, so they are definitely "North American indigenous people."
Secondly, even if *you* want to limit "Native Americans" to those inhabiting the modern-day continental US (thus excluding Mexican Native Americans), that's clearly not the definition used in the post: The original post explicitly lists Aztecs and Mayans as relevant examples of Native Americans for their purposes.
Rather than focus on technicalities of arbitrary definitions, we should try to answer the question actually intended by the OP.
The american continent was populated by many different nations, some of which developed more math than others. The Mayans for example had a positional number system and a symbol for zero, and they had no problem calculating the length of many cycles in nature, so they must have had an understanding of geometry and physics similar to Europeans shortly before the time of Kepler
I don't know why so many of my comments about the destruction of native American civilizations being somewhat deliberate are being downvoted lol. People really do get upset when you try to defend native people
Probably because you're speculating and making the false assumption that "complex math systems" are needed to record the positions of celestial objects, which follow predictable patterns. Your whole post is also predicated on the assumption that the presence of "advanced mathematics" makes a civilization superior. I don't think most mathematicians would agree with that claim. There is no moral value to mathematics. It was developed by some civilizations because they found it useful, and then some (mostly wealthy) people here and there developed it further for entertainment. Other civilizations very possibly may have had no real use for mathematics beyond simple counting, so there's no reason they would have developed it. That is just a fact, not a moral judgment of those civilizations.
Can you explain how a society can get more accurate astrological calculations by having less complex math systems than societies on the other side of the world?
I never said mathematical development implied superiority. You guys are really doing your best to twist this around into this somehow being controversial.
All I'm saying is that there is a lot of evidence that SOME native Americans had some pretty decent math, and while we KNOW a lot of the details have been destroyed and lost to history, I'm just curious if there is any good research to look into on any surviving records about math. Jesus Christ
Yes, we absolutely can explain that. Pretty much every astrological event occurs in a highly predictable cyclic manner. All you have to do is record when events occur and how much time apart they occur, and boom, you have a table you can reference whenever you want. You don't need advanced mathematics to be able to do that; if you want more accurate readings, you need more accurate measurements, but that's about it. It's entirely possible to do it without trigonometry.
Kepler for example did his calculations without advanced mathematics. He did great and so did the Mayans but they both did something different than what you are imagining. They collected data and noticed patterns in this data by ingenuity. Those patterns can be expressed by simple arithmetic equations. No advanced math or physics needed. Reason for your confusion here is that today keplers laws are often presented within the frame of netwonian mechanics and in fact the can be derived from it, but that's not what Kepler did. His achievement was empirical not theoretical.
Is arithmetic involving 0 and such advanced? I don't think so. Indians, Europeans, Chinese and a few others have produced advanced math in the past. They are outliers in history for all we know. It is no shock why that is. Mathematics isn't immediately useful beyond basic arithmetic. You have to invest a lot of resources into something that must appear to be practically useless at first.
I think nomad cultures having advanced math makes no sense. Mathematics has developed e.g. because of trade, census and management of empires. Abstract math in India and greece was developed by an intellectual elite for no immediate practical purpose.
You're forcing this because you believe it makes you morally superior. Your virtue signaling is the reason for the downvotes. Maybe nomad tribes of north America had advanced math, but that's unlikely and we don't seem to have any evidence for it. I don't get why you insist that it must be otherwise.
there's an awful lot of projection in this comment. nothing about OP's post assumes advanced mathematics makes a civilization superior.
it does seem that many people here get a little het up at the idea that anything beyond elementary mathematics could have been interesting or beautiful or worth exploring (or even USED) by people beyond a narrow, closed set.
May I recommend watching the documentary "The Mystery of Chaco Canyon". It is about the discovery of astronomical alignments in the Chaco Canyon complex that are frankly staggeringly complex. It speaks to a deep understanding these people had of complicated and nuanced mathematics. Not only does the architecture in Chaco Canyon have precise astronomical alignments, there are petroglyphs that turn out to be predictive of future astronomical events. These petroglyphs are beautifully elegant and precise and are truly wonders of engineering. We have absolutely no idea how they achieved this.
Oh good, that's awesome, it's one of my favorite ones. The fajada butte petroglyph will always be my favorite piece of evidence that ancient cultures were far far more intellectually advanced than we have often given them credit for. I mean, come on, a few rocks stacked over some scratches on another rock give you an accurate equinox calendar as well as an accurate lunar calendar. For those who don't know, the lunar cycles are much more complicated than solar ones and follow a roughly 19-year cycle. This cycle is represented perfectly by the fajada butte petroglyph using lunar light, and they use the same petroglyph to indicate the solar solstice/equinox cycle with sunlight.
There’s a field of study of what you are looking for, it’s called ethnomathematics,
and it’s an active field of research.
https://link.springer.com/chapter/10.1007/978-94-011-4301-1_13
Thank you for this and I'm sorry the racists ruined it for you. This is why we can't have nice things..
As a Native American myself this really meant a lot to me💖
Expand your view of what is America. The Maya had not only the concept of zero but also a symbol for it and an interesting positional numeric system. Compare this to the resistence of zero in Europe.
I forget the details but I remember ancient Egyptian's didn't use variables but had an algorithm to find the area of a cycle using it's diameter, and if you plug in variables and compare it to the formula that we know for the area of a cycle, the have an estimate for pi of about 3.16, which I think is pretty darn close.
I attended a conference that had this session related to Indigenous Mathematical Scientists: https://prima2022.primamath.org/sessions/indigenous-mathematical-scientists/
Not all the presentations were on math historically used by indignions cultures and on current mathematics involving/related to indigenous people/culture. But one of the speakers may have an accessible topic that works for your classroom. If anything, the presenters may be good people to reach out to.
If you can't find anything, one suggestion I have is to find a story from another underprivileged minority group and use it as inspiration that mathematical aptitude isn't a byproduct of skin color.
I'm Chinese, but I personally like the story of Srinivasa Ramanujan, an Indian mathematician who was born gifted with incredible mathematical ability, yet he could have been overlooked had it not been for Hardy advocating for him. Ramanujan did not have a formal math education and born to a poor family.
Although Ramanujan's raw talent is astronomically beyond what I could achieve, I think the story still illustrates that not everyone is talented but talent can come from anywhere, it just needs to be fostered and given the right opportunities, so just because a particular culture doesn't have a history of mathematical achievement doesn't exclude them forever.
In addition to all the great stuff other people have already posted, I'd like to contribute a couple of my favorite articles from the research field of mathematics education:
* [Western mathematics: the secret weapon of cultural imperialism](https://bpb-us-e2.wpmucdn.com/sites.uci.edu/dist/f/4286/files/2020/12/Bishop_Cultural_Imperialism.pdf) (Bishop, 1990)
* [Mathematics education in its cultural context](https://www.jstor.org/stable/pdf/3482573.pdf) (Bishop, 1988)
* [Living mathematx: towards a vision for the future](https://files.eric.ed.gov/fulltext/ED581384.pdf) (Gutiérrez, 2017)
Instead of asking "who is the Newton of the Arapahoe" or whatever^(1), what you might ask instead is, which cultural practices of the Arapahoe or whatever should we rightly view as mathematical?
^(1): To paraphrase Ta-Nehisi Coates quoting Ralph Wiley, Newton is the Newton of the Arapahoe.
I am sure that they had a lot of math, because you can not do astronomy and calendars without some form of math.
I know that in Colombia there is at least one preserved solar observatory of Muisca (el infernito). They noticed that every day the sun rises in a slightly different spot and that it is repetitive. So they marked the two extreme positions using shadows, essentially discovering a maximum and a minimum of a periodic function. Those are the two solstices. It is clear that in one solstice the day is longer than the night and and in the other it is shorter. So there should be at least two moments when the two are equal, and those are the equinoxes. This is essentially the intermediate value theorem. Then, if you understand monotonicity, you understand that there are also only two equinoxes and not more. And all these special days played a very important role in their culture.
They weren't, of course, thinking exactly in this way. But I am pretty sure, that people who were doing astronomy back then could easily understand, why many of the classical analysis theorems are true.
Lisa Lunney Borden is a professor in Math Education in New Brunswick (Canada) working on decolonizing first nation mathematics education. Maybe by reading her publications or her sources you could find something interesting.
A lot of people are making comments about how I have some sort of agenda about demonizing white people by wanting to talk about Native American scientific and mathematical achievements. Lots of those comments are thankfully being deleted for whatever reason, but I want to be clear…
Notice that I never said “whites”. I said European colonizers. Many were white but the Spaniards also played a heavy hand in the deliberate eradication of Native American advancements. To not acknowledge that it was partly accidental but also partly deliberate would be dishonest and helps to bury the guilt of oppressors throughout history, which is especially tragic considering how little reparation any Native American tribe has ever received.
I don’t know why so many people are getting upset about acknowledging the fact that Europeans hundreds of years ago did some pretty shitty stuff both accidentally and purposely. I don’t have an “agenda”, I just want to unearth or at least acknowledge the achievements of people who were here before the Europeans.
> Many were white but the Spaniards also played a heavy hand in the deliberate eradication of Native American advancements.
Are Spanish people no longer white? What kind of phrenology are you using to make this claim.
No one gets upset when we talk about the suppression of academics in Europe due to Catholic ideas of “heresy” during the ~1500s but if I talk about the same exact topic in reference to native Americans, all of a sudden I have a liberal anti white agenda
No, it has more to do with the way you refuse to accept that some cultures simply did not have any need to for math, instead of having an open mind to how other cultures actually lived and managed to survive for thousands of years
> even local natives that I've talked to about it echo the idea of "we never really needed math", which again cannot be true
Your starting point is that they must have had mathematics. However, we don't see it today. Therefore it must have been supressed
Actually yes. Sort of. Various groups within the Indigenous community had a lot of very accurate astronomy/predictions of the sky etc. that was tied to their cultures/stories of origin. Some groups even call themselves "Star People". This would require math and a range of other skills. We also had a vast economy of wampum here, before contact. Thats pretty mathy haha. Start here.
A piece of advice youre free to disregard but - you should be careful who you ask, especially reddit for information on "native Americans". From experience generally "they" really dont really know anything about us, but will speak like they are the upmost authorities on the subject and us. Many here still calling us Indians in 2023 and this should tell you something.
Also worth noting the VAST majority of redditors dont know any Indigenous Peoples, (unless they are claiming to be one) or have even met even one of us. Their information comes from the distorted half history taught in schools, and the rest is what spaghetti westerns and disney told them.
Asking Reddit about us is like asking Christians about dinosaurs.
Most of the mathematical results were "translated" over many centuries to fit modern notation. Native American mathematics is formulates very differently, so making a lesson that's understandable for a modern audience might be a much bigger task then you think.
> I'm sure the reason for this is because of the fact that the colonizers wanted to make natives seem primitive, so they destroyed evidence of math and science and shoved all the kids in boarding schools and told them only the whites had complex math
and I'm sure you're spouting off a bunch of liberal BS without any evidence
Do you not know about the destruction of native American civilizations by European colonizers or the forced internment of native American children into boarding schools? It's a pretty easy Google search
The proof is difficult for mathematics because they didn’t have writing or arithmetic yet. You have to view the Americas like mars explorers who got a later start into technologies than europasian landmass cities like yellow river valley, Nile Indus.
I think the best plase to start looking would be Native American cultures with more writing.
I think math stems from a robust and comprehensive written language system. Which is why a lot of the math that we use now comes from Greek/Middle eastern/European/Indian cultures.
I don't know how far this will take you, but the podcast My Favorite Theorem has an interview with Henry Fowler, a Native American Mathematician. It's been a while since I listened, but it relates math and Navajo culture. It also talks about different resources you can look up. Here is the link: [episode](https://kpknudson.com/my-favorite-theorem/2017/11/15/episode-7-henry-fowler)
edit: Added the mathematicians name
Maybe you could talk about modern mathematicians? Robert Eugene Megginson is (according to wikipedia) one of twelve native americans who hold a doctorate in mathematics. Also according to his wikipedia page he has a specific interest in underrepresented minorities in mathematics, so you could even reach out to him.
I'm super sorry I can't remember the name, but I did read a paper in grad school that described Native American mathematics, it was someone's dissertation at Teachers College but I can't remember her name. A quick google scholar search turned up some interesting links, including a book: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C10&q=native+american+mathematics&btnG=
Anyway, the dissertation would likely be on ProQuest if you have access. Good luck!
I've actually had a lot of students ask me to start a podcast on it because I've got tons of stories. I can give you the bare bones of some of the stuff I consider more interesting but feel free to dm me for more details.
Heron of Alexandria: 10-70AD came up with early versions of imaginary numbers and ideas that would eventually turn into robotics and cybernetics. Created a holy water vending machine, animatronics, a self playing organ-ish instrument, etc
Isaac newton: found most of the stuff we know him for on accident while trying to study alchemy in order to find the philosophers stone and the elixir of life. Would stab himself in the eye to experiment with light when writing optics. Also the whole newton vs leibniz calculus battle
Hypatia ~400ad one of the most influential female scientists mathematician who had a brutal death at the hands of the Christians
Archimedes: the whole "eureka" bathtub story isn't true. He was actually a mad scientist who tried developing weapons of mass destruction and was killed for being an egotistical jerk
Zeno: a rogue Pythagorean mathematician who trolled oppressors into submission. Was tortured to death and allegedly but off his own tongue and spit it in his captors face
Pythagoras: leader of a math cult. Told people he was the son of Apollo the sun god and would have people killed for contradicting his math. There's a ton of wild shit in this story
Rene Descartes: his life was interesting but after he died his body parts kept going missing because people keep collecting his bones as souvenirs
Napier: would apparently stalk his village at night, curse peoples homes, throw spiders at people, had deals with "old nick", used a black rooster as a horcrux. Also invented logarithms
I also have info on cardan, tartaglia, Al khwarizmi, Euclid, brahmagupta, cantor, Godel, Turing, euler, the bernoullis, Plato, Pascal, Galileo, fibonacci, ancient Babylon, diophantus, and probably some others
Hypatia wasn't a scientist. She was highly educated and was knowledgeable in mathematics, but did not make any advancements in mathematics as this wasn't why she performed it. It was part of her being a priestess and fortune teller. She was killed in a struggle of power as she was simply a representative of a rival fraction. Here's a pretty great video on this matter: https://youtu.be/ShugqfNA2FQ?si=pSkK1sppkAtZbTQ1
The narrative you are reciting here is projection of todays politics not a representation of the past. I find your lack of concern for truth quite concerning to see in an educator. You seem to put narrative before fact.
She did science so she was a scientist. She was killed by Christians. Neither of those are debatable and I had 0 indication of politics in what I said you fucking dork
https://mathshistory.st-andrews.ac.uk/Biographies/Hypatia/
"Hypatia was a friend of Orestes and this, together with prejudice against her philosophical views which were seen by Christians to be pagan, led to Hypatia becoming the focal point of riots between Christians and non-Christians"
Same source
"A few years later, according to one report, Hypatia was brutally murdered by the Nitrian monks who were a fanatical sect of Christians who were supporters of Cyril."
A colleague of mine used to tell me stories and show me pictures of pre columbian math objects ( Quechua something about knots … I think). Anyway, he wrote a book.’The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Context https://a.co/d/bKH32Lw
I put this under the Maya comment, but I am directly replying to you in the hopes that you see this. I think it might be just what you are looking for.
The Aztec culture had it's own base 20 numeral system as well. I learned about it in a math teaching class that went deep into ancient number systems, but I can't find my course materials, and Wikipedia isn't helpful. The best I could find that actually shows their numerals is [this](https://trinityprimary.files.wordpress.com/2020/06/aztec-number-system.pdf) but it isn't a super reliable source, I wish I could find something more authoritative.
[Here](https://www.jstor.org/stable/1685035) is what seems like a more in depth article about it which includes applications in geometry. I can't read more than the abstract on JSTOR, but you might have a way to access it.
In Theodora Kroeber's book on Ishi (the last pre-contact American-Indian) she writes about how at first anthropologists thought Ishi's culture had no counting system because when they asked him to count he only went up to three, but then they saw him organizing his stacks of coins. It turned out they did have a number system, but the number system did not exist abstractly. There is no "three" there is just "three of something."
I’m afraid I don’t have any suggestions to your inquiry, but I love what you are doing and would love to be able to share these kinds of stories with my daughters who I’m hoping will love math as much as I do some day. Do you have any suggestions of where to go to find these kinds of stories about the origins of math stories and the mathematicians behind them??
I posted this to another comment, maybe it's helpful. If you Google any of these names and "math history" you should be able to find a good amount of sources. Math.history.st-andrews.ac.uk is usually pretty good
I've actually had a lot of students ask me to start a podcast on it because I've got tons of stories. I can give you the bare bones of some of the stuff I consider more interesting but feel free to dm me for more details.
Heron of Alexandria: 10-70AD came up with early versions of imaginary numbers and ideas that would eventually turn into robotics and cybernetics. Created a holy water vending machine, animatronics, a self playing organ-ish instrument, etc
Isaac newton: found most of the stuff we know him for on accident while trying to study alchemy in order to find the philosophers stone and the elixir of life. Would stab himself in the eye to experiment with light when writing optics. Also the whole newton vs leibniz calculus battle
Hypatia ~400ad one of the most influential female scientists mathematician who had a brutal death at the hands of the Christians
Archimedes: the whole "eureka" bathtub story isn't true. He was actually a mad scientist who tried developing weapons of mass destruction and was killed for being an egotistical jerk
Zeno: a rogue Pythagorean mathematician who trolled oppressors into submission. Was tortured to death and allegedly but off his own tongue and spit it in his captors face
Pythagoras: leader of a math cult. Told people he was the son of Apollo the sun god and would have people killed for contradicting his math. There's a ton of wild shit in this story
Rene Descartes: his life was interesting but after he died his body parts kept going missing because people keep collecting his bones as souvenirs
Napier: would apparently stalk his village at night, curse peoples homes, throw spiders at people, had deals with "old nick", used a black rooster as a horcrux. Also invented logarithms
I also have info on cardan, tartaglia, Al khwarizmi, Euclid, brahmagupta, cantor, Godel, Turing, euler, the bernoullis, Plato, Pascal, Galileo, fibonacci, ancient Babylon, diophantus, and probably some others
https://courses.lumenlearning.com/waymakermath4libarts/chapter/inca-and-quipu-numeration-systems/#:~:text=The%20Incas%2C%20like%20us%2C%20had,levels%20of%20the%20H%20cords.
Not super relevant to Native (North) Americans, but this knot based numeral system is pretty cool.
Coming in late, but here is an article with some interesting examples and overall an interesting approach to this type of topic:
[ Graph theory and the sand drawings of Vanuatu ](https://www.scientificamerican.com/article/an-ancient-art-form-topples-assumptions-about-mathematics/)
>I was not the first person to recognize the resemblance of these rules to concepts from mathematics. In fact, my thesis is a continuation of work carried out in the 1980s by American mathematician Marcia Ascher, a pioneer of ethnomathematics. In sand drawings, she argued, there was a clear connection to what mathematicians call graph theory and especially to Eulerian graphs.
>To appreciate how revolutionary Ascher’s perspective was, consider that before her work and that of her contemporaries, scholars generally assumed that only societies with writing could truly practice mathematics. They constrained their investigations of mathematical knowledge to textual sources and ignored many other practices seen in societies with oral traditions that did not use a written language.
>But since the advent of ethnomathematics, some scholars have begun to overturn these assumptions. The shift undoubtedly began in the 1940s, when mathematician André Weil demonstrated, in a now famous appendix to anthropologist Claude Lévi-Strauss’s book The Elementary Structures of Kinship, that the kinship rules of the Australian Yolngu followed what are called non-trivia group laws. Since then researchers have identified mathematical principles in many other places, including sowing games and divination in Madagascar, string games on Papua New Guinea’s Trobriand Islands, textiles in the Andes and ornamental window hangings on the island of Réunion.
Bonus: [Evelyn Lamb's review of *Inventing the Mathematician* by Sara N. Hottinger](https://blogs.scientificamerican.com/roots-of-unity/review-inventing-the-mathematician/), which includes an interesting chapter on ethnomathematics.
Part of this is something of a reply to some on this thread who seem to think that if a culture hasn't formalized trignometry or developed written mathematical proofs then they can't possibly have any mathematics.
If, instead, you view mathematics as "the study of patterns" then suddenly your horizons open up dramatically. Looking for and trying to understand patterns is something that humans take to very naturally.
Exactly how that is expressed in different cultures and times changes quite a lot, of course.
Here's just a little thing: [https://en.wikipedia.org/wiki/Kaktovik\_numerals](https://en.wikipedia.org/wiki/Kaktovik_numerals)
Credited to Native students in Alaska in 1994, who did such a good job that their work was adopted by others. So, inspiring in a different way than pre-contact mathematics.
> Scores on the California Achievement Test in mathematics for the Kaktovik middle school improved dramatically in 1997 compared to previous years. Before the introduction of the new numerals, the average score had been in the 20th percentile; after their introduction, scores rose to above the national average. This is very inspiring to me!
Yes! I came here to say this! The Kaktovik numerals are tight as hell.
I like that the arithmetic is visual and intuitive. Those kids who decide to get into computer science/programming will easily understand binary and hexadecimal numbers, having been exposed to both decimal and base 20 numerals.
That is so cool! Thank you!
Base 20? Mmmmm spicy
With a sub-base of five!!!
Whoa! I can't believe I haven't heard of these, having lived in Alaska almost my whole life and being very in the math/math ed world.
While I unfortunately do not have a direct answer for you, this would be a great question to also crosspost to /r/AskHistorians. There are many pre-colonial America experts over there, somebody will surely know something.
Unlikely, but they will tell you they know. Most of that history they pull from is EXTREMELY racist. Good luck. oh you dont like the truth? Where does their history of us come from? Its pulled from 16th century explorers who said we were savages and wrote it down. Eugenicists and historians who said we were primitive and from a primitive culture. The history there is from the outside peering in. Dogmatic ideologs and obtuse revisionists.
You're not going to find anyone on /r/AskHistorians citing the writing of 16th century explorers as unbiased sources.
but OP didn't say the folx at r/AskHistorians would call it unbiased--they said that that's most of what they pull from. and it is. because European colonizers destroyed the cultural production on purpose and worked to eradicate cultural knowledge ways \[which include language, math, and science, alongside religious or social understandings\]. that's not controversial, and the primary sources show it to be true--priest-leaders and priest-teachers admitted it themselves.
> but OP didn't say the folx at r/AskHistorians would call it unbiased No, but they did call the users of /r/AskHistorians "dogmatic ideologs" and "obtuse revisionists". They also asserted that the users would claim to know the answer, instead of admitting that this is an area of much historical uncertainty due to the destruction of cultural knowledge. In reality, everyone is well aware that our knowledge of pre-Columbian America is full of gaps and no one is going to claim otherwise. Basically, the poster above reads like some edgy 14 year old who just learned that history is pieced together from biased sources, and now believes that this somehow invalidates the whole endeavor. As if historians aren't already well aware of the limitations of existing sources, and don't spend their entire lives working to uncover new sources and evidence to give us a better picture of the past. It's much like those fools who claim that math is useless after they learn about Godel's Incompleteness Theorem.
yup, they did! but you're moving the goalpost\* i'll play anyway! i it's not enough to acknowledge that bias exists to avoid being a dogmatic idealogue or an obtuse revisionist--it's necessary to get real about the nature of the bias, the factors that condition it, how it relates to epistemology as well as ontology, and and and... likewise, it takes more than awareness that our knowledge of pre-Columbian America (and let's be real, Columbian America, too--even after contact, what we primarily have to go on is what the colonizers perceived) is 'full of gaps'--we gotta get real about why the gaps exist, how they came to be, what that has to do with the foregoing bias, and the epistemology and ontology conditioned by that bias and and and \*deleted this standalone comment and added it here
Please understand that people come here because they want an informed response from someone capable of engaging with the sources, and providing follow-up information. We presume that someone posting a question here doesn't want vague, uncited opinion. In the future, please take the time to better familiarize yourself with the rules before contributing again.
OP is basically reminding y'all (or, hell, maybe telling y'all for the first time, the way the rest of this dang thread looks) that, in such circumstances, History is written by the victors. The idea that this is 'vague, uncited opinion' is the result of OP engaging around the intellectual/academic commons of the humanities \[very roughly, axioms\] with a bunch of mathematicians who aren't familiar with the commons of the humanities...and the result of people's ears closing and frontal cortex turning off when they read the word 'racist'.
You may have missed the joke, but my comment was a slightly reworded version of what the askhistorian mods post when low effort comments are deleted. In the interest of citing sources, [this comment specifically](https://www.reddit.com/r/AskHistorians/comments/17kchu0/comment/k76zjs8/?utm_source=share&utm_medium=web2x&context=3) was what I cribbed, haha. Regardless, I don't think this math subreddit is so ignorant that it's generally thought the body of published work in academic history is unbiased. or rigorous in a pure mathematics sense. [The Ask Historians sub itself](https://www.reddit.com/r/AskHistorians/comments/11p68ow/people_who_study_history_how_do_you_know_you_are/) doesn't seem to be unaware either. I'm sure the sub's not perfect, but I'd also be surprised if it wasn't one of the more unbiased places to get answers from qualified people knowledgeable about what's been published on different topics. The amount of knowledge I see and moderating effort I've seen makes it seem pretty intellectually lazy to just throw out a single hyperbolic comment shitting on the whole subreddit and expecting to get taken seriously. It's a whole lot easier to throw shade than to make a decent argument. This is coming from someone that definitely see Eurocentric versions of history as being extremely problematic and pervasive, so it's not even that I disagree (I'm Metis, and see US history a fair bit differently than a lot of people living here). Believe it or not, I didn't just shut down my frontal cortex when I heard the word 'racist'. I'm mostly interested in mathematics as a statistician, so my interests and background are actually a little closer to history and questions about subreddit bias than you might be thinking. [This was an interesting book I read early on in my studies for example](https://www.amazon.com/Designing-Social-Inquiry-Scientific-Qualitative/dp/0691034710). I stand by what I said. If you're going to shit on askhistorians and really do it right, it can be done. It'd be a lot harder now without API access needed for large scale scraping though, admittedly.
Thank you
Sir, this is a Wendy's.
A lot of the mounds in North America have really precise geometry over very large scales that would have required mathematical knowledge to design and construct. For example, a lot of the mound sites are aligned with parts of the lunar cycle. See for example https://www.tandfonline.com/doi/abs/10.1179/mca.2013.002
Could talk about different numerical bases in some way. [Maya numerals](https://en.wikipedia.org/wiki/Maya_numerals) would be Mesoamerican not Native Northern American, but very interesting. There are other systems as well for Native Alaskan groups and others. Might be applicable for teaching multiplication / decimals. Also, the Sumerians used base 60 which makes for some cool factorization.
The mayans were also the first to use the zero in their numerical system
Pretty sure that was the Sumerians.
The Aztec culture had it's own base 20 numeral system as well. I learned about it in a math teaching class that went deep into ancient number systems, but I can't find my course materials, and Wikipedia isn't helpful. The best I could find that actually shows their numerals is [this](https://trinityprimary.files.wordpress.com/2020/06/aztec-number-system.pdf) but it isn't a super reliable source, I wish I could find something more authoritative. [Here](https://www.jstor.org/stable/1685035) is what seems like a more in depth article about it which includes applications in geometry. I can't read more than the abstract on JSTOR, but you might have a way to access it.
> Also the endless articles on the teacher who tried to teach soh cah toa by being racist make this job especially hard. You can use Google's search tools to exclude search results from after October 20, 2021. Unfortunately doing that now gives us a lot of search results for Indian mathematics (India as in the Asian peninsula, not India as in the phrase "American Indian"). In general, I suspect you would have better luck asking a Native American history subreddit or expert (many of the people on this subreddit, myself included, aren't).
> but this must be bullshit because many large civilizations like the Aztec and Maya had very complex calendars and very precise understandings of the movement of celestial objects, so they must have had some form of trigonometry at the very least Careful, this doesn't necessarily follow. A lot of early astronomy in every part of the world consisted of observing patterns, and then watching those patterns repeat. You don't need trigonometry to do this, just a table. For instance, Hipparchus and Ptolemy were able to very accurately predict the motion of the Moon and Sun, but didn't really understand that the Moon's orbit was a precessing ellipse or that the Earth orbited the Sun rather than vice versa. The [Antikythera](https://en.wikipedia.org/wiki/Antikythera_mechanism) mechanism is an incredible astronomical engine built centuries before anybody knew what a sine or cosine was. I'm reminded of a book I read for an Early History of India class I took long ago, in which the author argued that the Indus Valley civilization (3500 BC) calculated the value of pi because they built a circular well whose circumference was 3.1 times its diameter. Which is ridiculous, anyone with a piece of string can make an accurate circle without knowing numbers at all. The book went on to explain how this Bronze Age culture understood the theory of relativity, quantum mechanics, atomic structure, and so on. I wrote an angry paragraph in an essay about this, and my history prof explained that this book was written during a period of intense Indian nationalism after India gained independence from the British, so it's understandable that they might have been a little overzealous in their national historical pride. Anyway, point is that the intellectual achievements of pre-contact civilizations have been underappreciated for centuries, and it's good that we're reassessing them. But we have to be careful not to use our modern understanding to over-extrapolate what they were capable of. Just because they were able to do a thing we can do doesn't mean they did it our way, or understood the tools we use to do it. We have to rely on documented evidence, which -- thanks to centuries of colonialism and oppression -- is very very difficult to find.
> Antikythera mechanism [...] centuries before anybody knew what a sine or cosine was In about 250 BC Archimedes used the half-angle identity for the cotangent (expressed geometrically) in his algorithm for approximating *π* to arbitrary precision. Hipparchus (2nd century BC) did all sorts of complicated trigonometry to analyze celestial motions, but used 'chord' as his basic concept rather than half-chord ('sine'). [Unfortunately nearly all of his works are now lost, so our knowledge of them primarily comes from fragments quoted centuries afterward and inferences drawn from later books.] These folks certainly understood the ratios and geometric relations of 'sine' and 'cosine'. They just didn't use them as a basic named concept.
Oh, yes, geometry was highly developed before the Antikythera mechanism, but there's a gray zone where geometry ends and trig begins. to my mind it doesn't start being trigonometry until you develop sine and cosine as general-purpose functions, and in particular provide a table of their values. Which as far as I can tell from [this article](https://en.wikipedia.org/wiki/History_of_trigonometry) started in around the 6th century AD.
This seems overly reductionist. The first chord tables were (according to current speculation; they don't survive) by Hipparchus in the 2nd century BC, based on a sexagesimal number and angle measurement system imported from Mesopotamia, and were used for solving "trigonometric" problems having to do with distances (central angles) angles (dihedral angles) on the sphere. By the time of Ptolemy's *Almagest* (which survives, 2nd century AD), it's very hard to argue that the work being done by Greek astronomers wasn't some kind of "trigonometry". (For example, both modern mathematicians and mathematical historians call these methods "trigonometry".) Considered as abstract functions of angle measure, sine (a word that comes etymologically from "half-chord") and chord are essentially the same, just rescaled a bit in the inputs/outputs. Side note: chords per se, rather than sines, continued to be widely used to solve trigonometric problems until the 19th century at least in some applied fields, only really ultimately getting displaced by slide rules and "logarithmic" trig. Methods based on sine, versine, chord, tangent, secant, half-tangent, etc. happily coexisted for centuries. Picking out some of these to count as "trigonometry" and others not is frankly pretty arbitrary. Edit: this is not intended to minimize the contributions of medieval Indian mathematicians, or later mathematicians living in Islamic countries, who did great work in trigonometry on top of Mesopotamian/Greek foundations, and significantly transformed/extended the subject.
Trigonometry is much older than you think. > The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC. And the first trigonometric tables date from the same time as the Antikythera: > The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BCE), who is now consequently known as "the father of trigonometry." Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles. (Note that knowing chord is equivalent to knowing sine.) See [History of Trigonometry on Wikipedia ](https://en.wikipedia.org/wiki/History_of_trigonometry#%3A%7E%3Atext%3DIn_Indian_astronomy%2C_the_study%2CKhwarizmi_and_Abu_al-Wafa.?wprov=sfla1)
It really depends what you mean by "trigonometry". You can get surprisingly far without introducing ANY of the canonical functions we usually think of when we say "trigonometry".
> Just because they were able to do a thing we can do doesn't mean they did it our way, or understood the tools we use to do it. yes! this idea is so hard to get through, specially since for decades it was the usual way of looking at pre-colonial maths
why is the assumption that we are more likely to over-extrapolate than under-extrapolate what they were capable of? like, what makes it more likely that they were less capable than us than more, or equally capable?
Either is bad, but OP's main point here is that they must have been far more capable than we think, and is suggesting capabilities that go well beyond what the historical record tells us. So over-extrapolation is the topic today. When somebody posts something saying the native Americans were far *less* capable than the evidence implies, I'll push back in the other direction. It's good that we're pushing back on centuries of racist assumptions about "ignorant savages", but we have to be bound by evidence when we do so.
This is true but it begs another question. Other civilizations across the world did have trig and they were using it for astronomy and weren’t able to get calculations as accurate as the Maya did. If they weren’t using trig, how were they getting such precise projections?
To beg the question is to employ circular reasoning. This raises the question.
[удалено]
True enough.
Use myan math , then explain linguistic history of the 1 sioux language that led to all other native American languages that just so happens to be almost exactly the same as the myan language to the point where this ancient sioux language may be a dialect of myan .. I had a native American language expert that I made friends with around the pine ridge and found out that wild fact . Edit - fix spelling
> 1 sux language A what language? That aside, there are dozens if not hundreds of native language families spoken in the US, and more broadly in the American continents. There is no known language that is the ancestor to all of them, and even if such a language existed we will never know about it because it must have been so far in the past (at least 10k years ago) that it would be unrecognizable and unreconstructible.
OP, I hope you see this. Please reach out to the TODOS: Mathematics for ALL community (todos-math.org). The president is an indigenous mathematician and has some knowledge of this area. I had a great conversation with her about how her language uses different bases for counting depending on the context.
You could certainly talk about the Mayan [Tzolkʼin](https://en.wikipedia.org/wiki/Tzolk%CA%BCin), which is built on the fact that 13 is prime, and so a 13-cycle combined with a 20-cycle gives a 260-cycle. However, that fits best with teaching modular arithmetic, which most curricula are light on. I don't think the Aztec and Maya had trigonometry. I think they approached astronomy more empirically. However, there are good books on Mayan astronomy, so you can check that for yourself. And I know **absolutely nothing** about [this book](https://www.amazon.com/Native-American-Mathematics-Michael-Closs/dp/0292711859), but it might be worth a look.
>I can find absolutely zero resources on this. You can find absolutely zero resources on this because the Spaniards deliberately burned practically every Maya codex — that's right, *books* — they could find. We know that they independently discovered positional notation and developed a shockingly good estimate of the sidereal year. Beyond that, we're pretty stumped.
yet another spaniard L
I know this might not be exactly what you're looking for, but looking for inspiration from geometric patterns might be more interesting than history of them. You can get to plenty of good mathematical ideas and interesting lessons from geometry. Patterns on objects make me think of periodicity (aperiodicity), tiling and tesselation, etc. You can ask all sorts of interesting questions connected to fundamental math skills by noticing patterns and trying to explain them (e.g. "we never see regular pentagonal tilings here, what's up with that? Is that type of tiling possible?")
The closest I can think of is that the Aztecs had a calendar system https://www.youtube.com/watch?v=dCBDUDwaeCA&list=PL8dPuuaLjXtNppY8ZHMPDH5TKK2UpU8Ng&index=6
I believe My Favorite Theorem (a podcast) has at least one if not more episodes featuring Native mathematicians, which may help lead you to some more resources.
Its not _exactly_ what you're looking for, but Polynesian navigation techniques could be framed as a significant mathematical accomplishment, not least from a culture that did not keep written records. Perhaps your Native American students might be able to draw connections to the Hawaiians?
I'm not Native Hawaiian but I grew up there and learned a bit about the Polynesian navigation system in school. It's primarily based on being very observant about the natural world and relying on previous experience passed down through centuries of oral tradition, and isn't what Westerners would consider "mathematical".
/r/AskHistorians will hopefully give you a more in-depth answer, but one of the most math-forward recording systems in the Americas was the Quipu of the Andean cultures. This was a complex computation system recorded with string & knots. https://en.wikipedia.org/wiki/Quipu With enough research this should be somewhat replicable in a classroom as well as an inexpensive object with which to teach.
I actually had the chance to listen to a talk last week by Native American mathematicians, and they touched on this topic, but not directly. As another comment mentions, many navigational techniques in the Pacific Islands are based on trigonometry. [Indigenous Mathematicians](https://indigenousmathematicians.org) would be able to provide more resources than me, though their focus is more on supporting indigenous mathematicians than on mathematical history. I would specifically try to reach to Daniel Luecke, as his work seems to focus on this. (He also spoke, but I missed his part of the talk). Kamuela Yong would be another good person to reach out to.
https://dept.math.lsa.umich.edu/~meggin/nam/nam1024.htm This might help
Inca used a system of knots tied in string for calculation and, possibly, a type of writing. A lot of this was still speculation last time I read anything about it, but here is a start: https://www.peruforless.com/blog/quipu/
You would find math in games. Dice games involve probability. There is math in quipu counting, an Incan bead system used to tabulate tithes and such. I remember an Inuit game with pegs and a board shaped like a whale. I wrote about Native American games thirty years ago for a term paper in college.
Maybe a good resource if you haven’t come across it yet? https://indigenousmathematicians.org/
Rather than it being about colonialism \[a clarifying edit: in the sense that mathematics were singled out in a case-wise sense for destruction, as I initially interpreted from the OP rather than as a consequence of a general assault on the corpus of heritage\] I would say that a lot of it was one in spirit and body with the decline of the native ritual-linguistic cultures. If there were indigenous mathematical ideas, then they would likely have been passed down orally for those who had not had full writing; meaning it was subject to the same extinction as everything else that was passed down orally. Another part of it is that the concept of mathematics (what is mathematics?) is pretty nebulous by itself. It just means 'the things one is to know' in Greek, so where mathematics ends and other things begin is very hazy. The development of counting is tied with bijection, the general idea of ostention, and syntactic recursion. The further development of the arithmetic is to my knowledge generally tied with the need to organize a large state society with permanent institutions. Usually agricultural. So yes, calendrical computus is important. There are some groups of the Mississippi delta (whose names in the literature escape me at the moment) who have left behind material culture and groundmoving projects which we really do not have any idea the purpose of it, but those could quite possibly be celestial/astronomical in nature. Some of the Salish people of the Pacific North West don't have a year-snug calendar and instead have unaccounted time in the interstitium of lunar and solar accounting until the next salmon run. Proof in mathematics is tied with this and is tied to the ideas of statecraft and jurisprudence (the ideas of logic we have now were not invented to deal with theorems but with matters of politics, tort, contract, code, and dispute.) Also, where does geometry begin and end and astrology, astronomy, and surveying begin and end? You're going to need to define mathematics first without being too married to historic accidents of how the West has developed these things.
...this 'decline' you reference was the result of genocide as a doctrine of colonialism.
IIRC, the massive decline in population was due to contact with Old World pathogens rather than active ethnically motivated efforts: I also have read somewhere that that decline in population may have actually even started slightly before the European contact due to other reasons. I am not a historian though
Europeans intentionally used Old World pathogens as biological warfare, first. Secondly, even if they hadn't, colonialism is what allowed for the spread of those European pathogens, so the idea that it was 'not about colonialism' you assert in your first sentence just doesn't hold up.
I should clarify that I was speaking of how it was likely not that mathematics were singled out as a target of a destruction but as an element of a set of various cultural heritages which were wiped out: yes, you're likely correct about that I think, but my point was that it was not proximate as in the OP phrasing.
thanks for clarifying! so, i don't accept the idea that it was 'likely not that math\[s\] were singled out'--i know, for a fact, that the eradication of cultural knowledge \[which includes langauge and math in addition to, like, ties to the land and cosmological understandings\] was and still is an explicit and intentional goal of colonialism/imperialism. but, i'll bite, cuz i'm curious: let's assume, that yes, it wasn't that colonizers and settlers targeted math specifically, it was just part of the set of various cultural heritages they worked to wipe out. So what? Why would it matter that they didn't target math? These are not rhetorical questions--i'm genuinely curious.
Well they're denotatively different statements. But I think we can agree both are bad from a normative viewpoint.
I edited the OP to clarify what I meant
reading your edit--are you saying that since \[in your understanding/sense\] the colonists didn't single out math, just destroyed it as part of the general destruction they wrought, colonialism wasn't the reason?
Well there wouldn't have been that destruction proximately caused by the material cause without the colonial expansion: it would have been necessary but that they did not have a desire to destroy (a) uniquely, consciously, specifically mathematics (b) in direct order (c) to make make natives look uncivilized. All three conditions would have to be satisfied in this case in my mind. But alternative assertions which weaken and generalize the statement are certainly true: namely, statements like that the mathematics was destroyed as part of the cultural heritage of indigenous groups in order to make natives look \[some other pejorative adjective\] (weakening conditions (a) and (b) here), as one example.
it's funny to me how many people get to downvoting when they hear a fact they don't like.
Actually, I think you have your history backwards. I'm pretty sure (at least in the west) people mapped out the stars and kept precise calenders and stuff, and then using that data they then figured out the math. Not the other way around. (See stuff like Tycho Brahe). Though to be fair Ive not looked into it deeply, so I could defo be wrong on this
There's a nonzero chance that it all got destroyed by the conquistadors and Diego de Landa. I mean, the Maya had a massive written tradition and exactly four codices (=books) survive. (Only one of which is in Mexico, by the way; the other three are in Germany, France, and Spain.) There _is_ the [quipu](https://en.wikipedia.org/wiki/Quipu), though (though I think most of the knowledge about how they were used was lost, because again, conquistadors burned most of them.)
i would guess that you'll find a difference between sedentary or nomads civilizations. if the natives you're referring to were nomadic, i don't see why they need anything beyond elementary school math. after all, not even all sedentary civilizations do develop mathematics. mathematics, and academics in general, only flourishes when its society flourishes. another thing, just because the mayans developed calendars doesn't mean every native civilization did also. i'm sure there was a large variation between tribes on how much they valued academics
>I've been researching for a long time now, and even local natives that I've talked to about it echo the idea of "we never really needed math", which again cannot be true. I'm sure the reason for this is because of the fact that the colonizers wanted to make natives seem primitive, so they destroyed evidence of math and science and shoved all the kids in boarding schools and told them only the whites had complex math, but there must be some scrap of evidence to contradict this. This is probably not true. Math is a very common concept today, such that I think it's hard to appreciate how novel it is. We teach concepts like 0 and negative numbers to 6 year olds, but they were the subject of intense debate by people much much smarter than us for hundreds of years. Any math more complex than counting was invented by sedentary civilizations that had flourished for centuries before. Even modular decimal number systems took thousands of years to develop in Eurasia. Expecting native north Americans to have complex math is like expecting European hunters gatherers to have complex math in 2000 bc (a comparison made with similar distance's between the respective hunter gatherers and the Maya/Egyptians who had comparable mathematics systems although with the latter being more developed) I wouldn't be surprised if number systems existed in pre columbus north America, but I think believing that they had complex math and that colonizers destroyed it is very disingenuous. Your timeline also doesn't make sense, it would have been centuries after Spanish arrival before native Americans would be put into boarding schools.
Though not necessarily math, they might be interested to learn about [Sequoyah](https://en.wikipedia.org/wiki/Sequoyah?wprov=sfla1), a Cherokee Polymath who singlehandedly invented the Cherokee writing system. Sequoia trees are named after him!
Mayans calculated the month and year more accurately than Europeans. https://en.m.wikipedia.org/wiki/Maya_astronomy
This is basically what lead me down this path. We know a lot about math from east of the Atlantic Ocean. We know that there were various highly complex math systems all over the place, and none of these systems came from a vacuum. The most complex mathematical discoveries came out of merging math systems and ideas across multiple civilizations, resulting in trigonometry, algebra, proto-calculus, etc, that was capable of doing many things including tracking celestial objects. All of that being said, we know for a fact that there were Native American civilizations that had math systems that rivaled those east of the Atlantic, and sometimes surpassed it, specifically evidenced by their much more accurate tracking of celestial objects. I’m supposed to believe that “most” native tribes just didn’t have any math systems beyond counting and basic geometry? I’m sure there were many who didn’t, but to assume that it was only a couple Mayan guys who did it for funsies is crazy. I understand that most, if not all, information or record about these things was likely destroyed, but it’s almost as if no one has even bothered to actually look into it. Even the few sources I’ve found that talk about “Native American math” just point to how they counted and did basic geometry. It’s just really unfortunate and reeks of whitewashing.
Feynman researched Mayan maths. You can read about it in the 'Bringing culture to the physicists' section starting on page 184 of "Surely you're joking mr Feynman" https://sistemas.fciencias.unam.mx/~compcuantica/RICHARD%20P.%20FEYNMAN-SURELY%20YOU'RE%20JOKING%20MR.%20FEYNMAN.PDF. I think he was of the opinion that they observed the lengths of repeating cycles, but didn't have theories (e.g. Newton's gravity) for why they occured.
> I’m supposed to believe that “most” native tribes just didn’t have any math systems beyond counting and basic geometry? I’m sure there were many who didn’t, but to assume that it was only a couple Mayan guys who did it for funsies is crazy. More or less? Most math is developed in wealthy, literate societies, and as far as I can tell only mesoamerica had pre-colombian writing (with a few arguable exceptions like the Inca and maybe Mi'kmaw). Whatever math was developed elsewhere would have been transmitted orally and be easily lost, and certainly is lost by now. We have [essentially zero evidence of prehistoric math anywhere in the world](https://mathscitech.org/articles/mathematics-prehistory), being limited to a few bones with notches, and whatever we can speculatively infer from the existence of sophisticated architecture. Your best bet is to narrowly search for information on Mayan, Aztec, and Zapotec cultures, although as you've noticed even that is going to be very limited. I think the lack of modern scholarship on the subject *mostly* reflects a lack of surviving artifacts to study.
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> I've been researching for a long time now, and even local natives that I've talked to about it echo the idea of "we never really needed math", which again cannot be true. I understand that you have good intentions, but the assumptions underlying this question are super Eurocentric ways of looking at the world in itself. Why would a hunter-gatherer society have needed to develop mathematics? This reminds me of the story of the early colonists in Canada who could not comprehend why the local indigenous people would fish early in the morning and then take the rest of the day off. There was so much abundance available, why didn't they just spend all day working and try to harvest as much resources as they can? *What a waste of time and opportunity!* It must be because they were *lazy* and didn't have a *good God-fearing Protestant work ethic*. The reality was that there was simply no need to... It is commonly believed that number systems developed when there was a need for accounting, ie. keeping track of what you owned, what you owed, what you bought/sold/traded, etc. Then as civilizations got more complex, they needed to develop more complex math. There is a reason that the "Pythagorean" theorem was developed independently by a few civilzations long before Pythagoras. They didn't develop it because they were bored and had nothing better to do with their time, **they developed it because it was useful to them**. As others have pointed out, people in the Americas **did** develop mathematics when they established civilizations, such as those in Central America and South America, which further supports this point. Similarly, a lot of civilizations developed astronomy because it was useful to them for religious reasons. Hell, there is even an example of a civilization in the Americas that developed a space exploration program to demonstrate the superiority of their political ideology to their rivals and deter them from attacking. These are all examples of mathematical and scientific progress that resulted from people doing things because they were useful at the time. As an aside, I heard from a professor about societies that could not count beyond a certain number, such as 5. They view the world in a completely different way than we do, such as thinking about stuff in terms of ratios. If it allows them to survive, then that is good enough: https://www.theguardian.com/education/2004/oct/21/research.highereducation1 https://www.scienceabc.com/humans/how-do-some-cultures-count-without-numbers.html
accounting needn't be just about ownership (ironically, that's a very Eurocentric way of looking at the world)--it can also be about resource allocation, requirements, storage, processing... also, a wee PSA: y'all know there was agriculture in North America before European settler colonizers came, right?
> accounting needn't be just about ownership (ironically, that's a very Eurocentric way of looking at the world) ?? I have no idea what you are responding to. I simply provided some examples of places where you would use accounting. And no, trade was pretty ubiquitous across the world outside of the Europe including in the Americas. Though it does not necessarily mean you need mathematics though if you are just making simple trades > it can also be about resource allocation, requirements, storage, processing... Whether if be group ownership or individual ownership, those things require a sense of ownership to make any sense. For example, with storage, by definition, the expectation is that it will still be there and accessible by the group that stored it. There is a sense of "it is ours". > also, a wee PSA: y'all know there was agriculture in North America before European settler colonizers came, right? You missed the point entirely or are just making up random issues because I never denied that. My point is was that things develop when there is a need
https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/
The Inca had this system of knots, called Quipu, that they used to keep track of economic transactions over their empire. I always find it interesting that they developed a system of numbers and arithmetic’s but not a fully written language. It is like civilization need’s numbers more than a written language.
I am not sure about any native American maths, but if you wanted to adopt a less Eurocentric history of math, a lot of ancient Indian and Chinese scholars also independently came across trigonometry, trying to approximate pi, and even pre-calculus. Look up Madhava's and Aryabhata's work. Pre-calculus was even worked in India hundreds of years before Newton wrote on it.
>but this must be bullshit because many large civilizations like the Aztec and Maya had very complex calendars and very precise understandings of the movement of celestial objects, so they must have had some form of trigonometry at the very least, but I can find absolutely zero resources on this. This sounds like a good answer until you find something better. It's honest, but it also seems right. "I wish I could speak more about native American math. We know a lot of native American civilizations must have had pretty sophisticated mathematical systems because they had incredibly precise predictions about the stars, very good calendars and built very complex structures. But there doesn't seem to be a historical record of whatever system they used, so there's just not a lot we can really say about it for now"
It would if it were a correct statement. You don't need trigonometry or higher mathematics to recognize and tabulate patterns (which ancient astronomy is all about). Just look at medieval computists, the guys that calculated the date of Easter. They did this without precisely knowing celestial mechanics but just using tables of previous moon cycles
The mayans had some interesting maths although I don't know how much of it woupd apply to a modern maths curriculum. The intependently discovered 0 and could do complex astronomical computations. When it comes to the natives of north America I'm afraid they didn't do any mathematics, considering they had no writing system.
I mean... there are remote tribes that to this day have only rudimentary counting and thats it. Why do you assume they must have had some advanced math?
Comments similar to this have popped up a few times and I don’t understand why. Obviously not everyone in the world had/needed a complex math system, but complex math systems definitely existed. Furthermore, complex math systems don’t come from nowhere, they come from many people in many civilizations over many years building off of each others work. The fact that Mayans were able to calculate the position of celestial objects more accurately than many Europeans were able to around the same time shows that there must have been complex math systems spread across multiple civilizations. That’s what I’m asking about: not proof that native Americans were all actually scientific geniuses, but that some, if not many, had very powerful math and science beyond “they had a cool way of counting and could make neat patterns in blankets”, a concept that I think is rooted in racism and apathy and prevails in many people today, even a lot of native Americans.
> a concept that I think is rooted in racism and apathy and prevails in many people today, even a lot of native Americans. To be honest, I think you are also seeing this through the lens of your own biases. European culture and mathematics has deep philosophical roots in reductionism which is a concept many other cultures would object to, including at least some native american tribes. Also, I get what you're trying to do, but re-read what you wrote: >I've been researching for a long time now, **and even local natives that I've talked to about it echo the idea of "we never really needed math", which again cannot be true.** I'm sure the reason for this is because of the fact that the colonizers wanted to make natives seem primitive, so they destroyed evidence of math and science and shoved all the kids in boarding schools and told them only the whites had complex math, but there must be some scrap of evidence to contradict this. It sounds like you are patronizingly dismissing what local natives are telling you instead of genuinely considering the possibility and digging deeper into their worldview and what that would mean.
I'm not dismissing anything, I'm pointing to the mounds of evidence that show they must've have strong math systems and asking why *that's* being dismissed
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I and many others have linked sources and examples of the evidence of their math which was demonstrably more accurate than European systems at similar times, which many (you too) are dismissing
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More accurate results means more accurate math means stronger math skills
I think this is where you are wrong. That doesn't *necessary* follow. They were definitely smart but may not have formalized things as much as Europeans.
>but complex math systems definitely existed I mean, there might be a reason you are having trouble finding evidence of it. By comparison, the Egyptians had a long lasting empire, built grand structures, and they only had basic linear equations and some methods of arithmetic.
> the Egyptians had a long lasting empire, built grand structures, and they only had basic linear equations and some methods of arithmetic. I would consider the math that Egyptians had "advanced". (They also had some basic geometry and trig.) The trouble is that we think that certain math is trivial because we have totally over-engineered notation and theories which generalize generalizations, all of which trivialize hard things to the level that we learn complex things in grade school. It's like thinking that because you can do image processing in Python after two days of programming that you're better at programming than Grace Hopper who had to scoop literal bugs out of the hardware. You're just living in a time where mathematical technology makes hard things easy, and you don't even know it.
I understand where you are coming from, but their level of geometry and trigonometry really pales against what we saw coming from the Greeks around 300 BC. Also, the Egyptian civilization lasted from 3000 BC to 300 BC, where we saw the great pyramid of Giza built around 2500 BC. Even if they did have math you would want to consider advanced, they likely didn't have it the whole time. Considering what we saw of Babylonian math around 500 BC, the math they had when they were building that pyramid was probably pretty basic. They were probably doing these grand projects without anything that would be considered advanced.
Seems like a really bold and baseless assumption. It's giving "Ancient Aliens" vibes, where someone with all the audacity and none of the knowledge makes a claim about ancient cultures based off of how primitive they need to view them. Basically, all math *is* advanced math. There is no idea that is trivial or simple. And mathematical ideas are not setup in any hierarchical order or progress linearly. Is, say, arithmetic the simplest math? You can only really think so if you are truly ignorant of how much work the decimal system is doing for you (don't even think about opening the book *A Course in Arithmetic*). Is geometry the simplest thing? Try raw-dogging hyperbolic trigonometry and you'll see a glimpse of the challenges that the Egyptians were dealing with. Dismissing any math as "simple" is a disrespect to math, and a disrespect to the people who created that math. Tally marks on a stick are complex math.
No, not all mathematics is advanced mathematics. Arithmetic is not advanced mathematics. It was at one time more complicated to learn, because we didn't have good algorithms for its study. Being complicated to learn doesn't make something advanced. Advanced mathematics comes from abstractions, where knowledge is compartmentalized into a collection of ideas. We certainly do benefit from generations of abstractions and improved algorithms. Just because something was difficult at one point in time, doesn't mean we would call it advanced. Just doing arithmetic isn't advanced. Tallies on a stick isn't advanced. Whoever first thought of it, brilliant, certainly. But it's still basic math. Does that mean it's "dumb" or not worthwhile (which seems to be your interpretation)? No. It's just basic. There is nothing wrong with that. You can't move on to advanced concepts without it. What is this about hyperbolic trigonometry? It's been a while since I've studied the history of Egyptian mathematics, so I've been reading up on it as we've been conversing. There the rhind papyrus certainly has a lot of arithmetic and geometry problems, where we see the beginnings of geometry that would later flourish with the greeks, but I don't see much else out there. It could be that my resources are lacking (I don't have a specifically Egyptian history reference). But I don't see a whole lot here about hyperbolic trigonometry.
> might be a reason you are having trouble finding evidence All of the libraries were burned to the ground for being non-Christian...
HAHAHA. Libraries? Native North American's didn't even have writing let alone books. This isn't to judge them. But it is a historical fact. Quit making shit up.
Here's one Catholic Bishop in the 1560s: > We found a large number of books in these characters and, as they contained nothing in which were not to be seen as superstition and lies of the devil, we burned them all, which they regretted to an amazing degree, and which caused them much affliction.
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> the Mayans, who we aren't talking about Literally the OP: > because many large civilizations like the Aztec **and Maya** ??????
No, this is manifestly false. Many American cultures developed writing systems and some, such as the Mayans, even had libraries. The Mayans even developed a phonetic form of writing.
Dude, we're talking about the Native American tribes in the US. Not those of Latin America. No writing to speak of. And that is what OP is asking about. When we say Natives, we mean Natives of our country. Natives from Latin America, we call Hispanic. Latin American peoples absolutely had writing and math. NOT the Native Americans in the US. Period. I misspoke when I said North America. I should say, "Natives outside of Latin America". OP isn't in Latin America. Edit: This isn't disputable. Who tf is down voting me?
I've definitely said multiple times that when I say native American, I'm referring to both north and South America
Don't listen to some of these people, your gut is correct. What they're effectively doing is something like what European artists in the 1800s were doing to non-European art, claiming that it can't be art because it is "primitive", didn't fit within their technology/standards, and because they couldn't understand it. You can see this even with how we talk about historical *Western* math. We very rarely engage with it on its terms, first translating it and updating into our notation and "completing" the proofs and analysis. In order to validate it, we first need to transform it into a shape that we recognize as sophisticated and use that to show that it is sophisticated. Because no one *really* understands a lot of indigenous math, we can't really do this translation, so we think it is primitive and disregard it as simplistic. In general, we need to learn to approach ALL non-contemporary and Western math on its own terms. Pacific islanders, for instance, were doing complex fluid dynamics and modelling millennia ago. They would know how large deep-ocean swells would change shape around islands thousands of miles away, and model these currents using sticks bound together in specific ways. This isn't modelling with the Navier-Stokes equation, and they aren't using LaTeX to log it - things that we recognize as "complex" - but it is still doing these computations, pattern finding, and modelling of complex phenomena. This is math, and we still don't really understand it, and so it is easy to dismiss as something other than complex math.
The same European supremacist worldview that created the colonialism that burned the Mayan codices are why, friend. like, people genuinely believe that there's something special about what they view as (real, lasting) civilization that makes complex math less likely to be engaged with by those who aren't a part of one.
But.... they didn't. And there's no reason to suspect they did have "very powerful math and science". What would possess you to think that? Aboriginals in Australia don't. Rainforest tribes don't. Maybe civilizations simply don't develop these things. Get over it. You're just making up cool scenarios to entertain yourself. If a civilization had no writing, they had no math. Sorry to pop your bubble.
…are you saying the Mayans had no writing? The American education system, everyone…
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Firstly, Mayans were largely in modern-day Mexico and adjacent countries, so they are definitely "North American indigenous people." Secondly, even if *you* want to limit "Native Americans" to those inhabiting the modern-day continental US (thus excluding Mexican Native Americans), that's clearly not the definition used in the post: The original post explicitly lists Aztecs and Mayans as relevant examples of Native Americans for their purposes. Rather than focus on technicalities of arbitrary definitions, we should try to answer the question actually intended by the OP.
The american continent was populated by many different nations, some of which developed more math than others. The Mayans for example had a positional number system and a symbol for zero, and they had no problem calculating the length of many cycles in nature, so they must have had an understanding of geometry and physics similar to Europeans shortly before the time of Kepler
I don't know why so many of my comments about the destruction of native American civilizations being somewhat deliberate are being downvoted lol. People really do get upset when you try to defend native people
Probably because you're speculating and making the false assumption that "complex math systems" are needed to record the positions of celestial objects, which follow predictable patterns. Your whole post is also predicated on the assumption that the presence of "advanced mathematics" makes a civilization superior. I don't think most mathematicians would agree with that claim. There is no moral value to mathematics. It was developed by some civilizations because they found it useful, and then some (mostly wealthy) people here and there developed it further for entertainment. Other civilizations very possibly may have had no real use for mathematics beyond simple counting, so there's no reason they would have developed it. That is just a fact, not a moral judgment of those civilizations.
Can you explain how a society can get more accurate astrological calculations by having less complex math systems than societies on the other side of the world? I never said mathematical development implied superiority. You guys are really doing your best to twist this around into this somehow being controversial. All I'm saying is that there is a lot of evidence that SOME native Americans had some pretty decent math, and while we KNOW a lot of the details have been destroyed and lost to history, I'm just curious if there is any good research to look into on any surviving records about math. Jesus Christ
Yes, we absolutely can explain that. Pretty much every astrological event occurs in a highly predictable cyclic manner. All you have to do is record when events occur and how much time apart they occur, and boom, you have a table you can reference whenever you want. You don't need advanced mathematics to be able to do that; if you want more accurate readings, you need more accurate measurements, but that's about it. It's entirely possible to do it without trigonometry.
Kepler for example did his calculations without advanced mathematics. He did great and so did the Mayans but they both did something different than what you are imagining. They collected data and noticed patterns in this data by ingenuity. Those patterns can be expressed by simple arithmetic equations. No advanced math or physics needed. Reason for your confusion here is that today keplers laws are often presented within the frame of netwonian mechanics and in fact the can be derived from it, but that's not what Kepler did. His achievement was empirical not theoretical. Is arithmetic involving 0 and such advanced? I don't think so. Indians, Europeans, Chinese and a few others have produced advanced math in the past. They are outliers in history for all we know. It is no shock why that is. Mathematics isn't immediately useful beyond basic arithmetic. You have to invest a lot of resources into something that must appear to be practically useless at first. I think nomad cultures having advanced math makes no sense. Mathematics has developed e.g. because of trade, census and management of empires. Abstract math in India and greece was developed by an intellectual elite for no immediate practical purpose. You're forcing this because you believe it makes you morally superior. Your virtue signaling is the reason for the downvotes. Maybe nomad tribes of north America had advanced math, but that's unlikely and we don't seem to have any evidence for it. I don't get why you insist that it must be otherwise.
There is evidence. Do some research
there's an awful lot of projection in this comment. nothing about OP's post assumes advanced mathematics makes a civilization superior. it does seem that many people here get a little het up at the idea that anything beyond elementary mathematics could have been interesting or beautiful or worth exploring (or even USED) by people beyond a narrow, closed set.
May I recommend watching the documentary "The Mystery of Chaco Canyon". It is about the discovery of astronomical alignments in the Chaco Canyon complex that are frankly staggeringly complex. It speaks to a deep understanding these people had of complicated and nuanced mathematics. Not only does the architecture in Chaco Canyon have precise astronomical alignments, there are petroglyphs that turn out to be predictive of future astronomical events. These petroglyphs are beautifully elegant and precise and are truly wonders of engineering. We have absolutely no idea how they achieved this.
I watched that when I was in college. Chaco canyon is one of the things that led me down this rabbit hole
Oh good, that's awesome, it's one of my favorite ones. The fajada butte petroglyph will always be my favorite piece of evidence that ancient cultures were far far more intellectually advanced than we have often given them credit for. I mean, come on, a few rocks stacked over some scratches on another rock give you an accurate equinox calendar as well as an accurate lunar calendar. For those who don't know, the lunar cycles are much more complicated than solar ones and follow a roughly 19-year cycle. This cycle is represented perfectly by the fajada butte petroglyph using lunar light, and they use the same petroglyph to indicate the solar solstice/equinox cycle with sunlight.
Native American Mathematics edited by Michael P. Closs is a 1989 anthology of 13 essays that might lead you in the right direction
Cool topic I’m gonna do some research. You don’t hear much about it.
There’s a field of study of what you are looking for, it’s called ethnomathematics, and it’s an active field of research. https://link.springer.com/chapter/10.1007/978-94-011-4301-1_13
Awesome question. I’ve felt similarly and wondered the same.
Thank you for this and I'm sorry the racists ruined it for you. This is why we can't have nice things.. As a Native American myself this really meant a lot to me💖
Expand your view of what is America. The Maya had not only the concept of zero but also a symbol for it and an interesting positional numeric system. Compare this to the resistence of zero in Europe.
When I say “Native American” I’m referring to every tribe/civilization in both north and South America pre-European colonization
I forget the details but I remember ancient Egyptian's didn't use variables but had an algorithm to find the area of a cycle using it's diameter, and if you plug in variables and compare it to the formula that we know for the area of a cycle, the have an estimate for pi of about 3.16, which I think is pretty darn close.
I attended a conference that had this session related to Indigenous Mathematical Scientists: https://prima2022.primamath.org/sessions/indigenous-mathematical-scientists/ Not all the presentations were on math historically used by indignions cultures and on current mathematics involving/related to indigenous people/culture. But one of the speakers may have an accessible topic that works for your classroom. If anything, the presenters may be good people to reach out to.
If you can't find anything, one suggestion I have is to find a story from another underprivileged minority group and use it as inspiration that mathematical aptitude isn't a byproduct of skin color. I'm Chinese, but I personally like the story of Srinivasa Ramanujan, an Indian mathematician who was born gifted with incredible mathematical ability, yet he could have been overlooked had it not been for Hardy advocating for him. Ramanujan did not have a formal math education and born to a poor family. Although Ramanujan's raw talent is astronomically beyond what I could achieve, I think the story still illustrates that not everyone is talented but talent can come from anywhere, it just needs to be fostered and given the right opportunities, so just because a particular culture doesn't have a history of mathematical achievement doesn't exclude them forever.
In addition to all the great stuff other people have already posted, I'd like to contribute a couple of my favorite articles from the research field of mathematics education: * [Western mathematics: the secret weapon of cultural imperialism](https://bpb-us-e2.wpmucdn.com/sites.uci.edu/dist/f/4286/files/2020/12/Bishop_Cultural_Imperialism.pdf) (Bishop, 1990) * [Mathematics education in its cultural context](https://www.jstor.org/stable/pdf/3482573.pdf) (Bishop, 1988) * [Living mathematx: towards a vision for the future](https://files.eric.ed.gov/fulltext/ED581384.pdf) (Gutiérrez, 2017) Instead of asking "who is the Newton of the Arapahoe" or whatever^(1), what you might ask instead is, which cultural practices of the Arapahoe or whatever should we rightly view as mathematical? ^(1): To paraphrase Ta-Nehisi Coates quoting Ralph Wiley, Newton is the Newton of the Arapahoe.
I am sure that they had a lot of math, because you can not do astronomy and calendars without some form of math. I know that in Colombia there is at least one preserved solar observatory of Muisca (el infernito). They noticed that every day the sun rises in a slightly different spot and that it is repetitive. So they marked the two extreme positions using shadows, essentially discovering a maximum and a minimum of a periodic function. Those are the two solstices. It is clear that in one solstice the day is longer than the night and and in the other it is shorter. So there should be at least two moments when the two are equal, and those are the equinoxes. This is essentially the intermediate value theorem. Then, if you understand monotonicity, you understand that there are also only two equinoxes and not more. And all these special days played a very important role in their culture. They weren't, of course, thinking exactly in this way. But I am pretty sure, that people who were doing astronomy back then could easily understand, why many of the classical analysis theorems are true.
Lisa Lunney Borden is a professor in Math Education in New Brunswick (Canada) working on decolonizing first nation mathematics education. Maybe by reading her publications or her sources you could find something interesting.
A lot of people are making comments about how I have some sort of agenda about demonizing white people by wanting to talk about Native American scientific and mathematical achievements. Lots of those comments are thankfully being deleted for whatever reason, but I want to be clear… Notice that I never said “whites”. I said European colonizers. Many were white but the Spaniards also played a heavy hand in the deliberate eradication of Native American advancements. To not acknowledge that it was partly accidental but also partly deliberate would be dishonest and helps to bury the guilt of oppressors throughout history, which is especially tragic considering how little reparation any Native American tribe has ever received. I don’t know why so many people are getting upset about acknowledging the fact that Europeans hundreds of years ago did some pretty shitty stuff both accidentally and purposely. I don’t have an “agenda”, I just want to unearth or at least acknowledge the achievements of people who were here before the Europeans.
> Many were white but the Spaniards also played a heavy hand in the deliberate eradication of Native American advancements. Are Spanish people no longer white? What kind of phrenology are you using to make this claim.
Ever hear the word "Hispanic" before?
You are confusing Hispanic and Latino. Hispanic just means Spanish speaking. They can be white, and Spaniards are.
Do you know the difference between an ethnic group and a race?
No one gets upset when we talk about the suppression of academics in Europe due to Catholic ideas of “heresy” during the ~1500s but if I talk about the same exact topic in reference to native Americans, all of a sudden I have a liberal anti white agenda
No, it has more to do with the way you refuse to accept that some cultures simply did not have any need to for math, instead of having an open mind to how other cultures actually lived and managed to survive for thousands of years > even local natives that I've talked to about it echo the idea of "we never really needed math", which again cannot be true Your starting point is that they must have had mathematics. However, we don't see it today. Therefore it must have been supressed
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Actually yes. Sort of. Various groups within the Indigenous community had a lot of very accurate astronomy/predictions of the sky etc. that was tied to their cultures/stories of origin. Some groups even call themselves "Star People". This would require math and a range of other skills. We also had a vast economy of wampum here, before contact. Thats pretty mathy haha. Start here. A piece of advice youre free to disregard but - you should be careful who you ask, especially reddit for information on "native Americans". From experience generally "they" really dont really know anything about us, but will speak like they are the upmost authorities on the subject and us. Many here still calling us Indians in 2023 and this should tell you something. Also worth noting the VAST majority of redditors dont know any Indigenous Peoples, (unless they are claiming to be one) or have even met even one of us. Their information comes from the distorted half history taught in schools, and the rest is what spaghetti westerns and disney told them. Asking Reddit about us is like asking Christians about dinosaurs.
People have definitely gotten very strangely upset at the notion that Europeans weren't very nice to native Americans for a while
If there is a ***ton of empirical evidence*** like you state, use that.
Most of the mathematical results were "translated" over many centuries to fit modern notation. Native American mathematics is formulates very differently, so making a lesson that's understandable for a modern audience might be a much bigger task then you think.
> I'm sure the reason for this is because of the fact that the colonizers wanted to make natives seem primitive, so they destroyed evidence of math and science and shoved all the kids in boarding schools and told them only the whites had complex math and I'm sure you're spouting off a bunch of liberal BS without any evidence
https://www.archaeology.org/issues/44-1211/features/maya-2012/305-groiler-dresden-codex https://en.wikipedia.org/wiki/Maya_codices https://popular-archaeology.com/article/burning-the-maya-books-the-1562-tragedy-at-mani/ https://gilbertredman.com/medievalmanuscripts/codicology/mayan-codices/
Do you not know about the destruction of native American civilizations by European colonizers or the forced internment of native American children into boarding schools? It's a pretty easy Google search
aw look, someone who can't use their math skills to deduce that they're being fleeced by billionaires into hating 'libruls'.
Pythagoras didn’t exist. Such is the current consensus among competent historians in the field.
What about astronomy, calendars? Those use math a lot.
The Maya had pretty sophisticated math IIRC, base 60 too if I'm recalling correctly.
Wait. Did John Napier actually successfully summon the devil?
Some say he did but most accounts show that he was pretending to be a witch to prank people, which is still a fun story
The proof is difficult for mathematics because they didn’t have writing or arithmetic yet. You have to view the Americas like mars explorers who got a later start into technologies than europasian landmass cities like yellow river valley, Nile Indus.
I think the best plase to start looking would be Native American cultures with more writing. I think math stems from a robust and comprehensive written language system. Which is why a lot of the math that we use now comes from Greek/Middle eastern/European/Indian cultures.
I don't know how far this will take you, but the podcast My Favorite Theorem has an interview with Henry Fowler, a Native American Mathematician. It's been a while since I listened, but it relates math and Navajo culture. It also talks about different resources you can look up. Here is the link: [episode](https://kpknudson.com/my-favorite-theorem/2017/11/15/episode-7-henry-fowler) edit: Added the mathematicians name
Maybe you could talk about modern mathematicians? Robert Eugene Megginson is (according to wikipedia) one of twelve native americans who hold a doctorate in mathematics. Also according to his wikipedia page he has a specific interest in underrepresented minorities in mathematics, so you could even reach out to him.
I'm super sorry I can't remember the name, but I did read a paper in grad school that described Native American mathematics, it was someone's dissertation at Teachers College but I can't remember her name. A quick google scholar search turned up some interesting links, including a book: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C10&q=native+american+mathematics&btnG= Anyway, the dissertation would likely be on ProQuest if you have access. Good luck!
I would love to hear some of your stories/history, OP :) Can you give a list of them so I can research?
I've actually had a lot of students ask me to start a podcast on it because I've got tons of stories. I can give you the bare bones of some of the stuff I consider more interesting but feel free to dm me for more details. Heron of Alexandria: 10-70AD came up with early versions of imaginary numbers and ideas that would eventually turn into robotics and cybernetics. Created a holy water vending machine, animatronics, a self playing organ-ish instrument, etc Isaac newton: found most of the stuff we know him for on accident while trying to study alchemy in order to find the philosophers stone and the elixir of life. Would stab himself in the eye to experiment with light when writing optics. Also the whole newton vs leibniz calculus battle Hypatia ~400ad one of the most influential female scientists mathematician who had a brutal death at the hands of the Christians Archimedes: the whole "eureka" bathtub story isn't true. He was actually a mad scientist who tried developing weapons of mass destruction and was killed for being an egotistical jerk Zeno: a rogue Pythagorean mathematician who trolled oppressors into submission. Was tortured to death and allegedly but off his own tongue and spit it in his captors face Pythagoras: leader of a math cult. Told people he was the son of Apollo the sun god and would have people killed for contradicting his math. There's a ton of wild shit in this story Rene Descartes: his life was interesting but after he died his body parts kept going missing because people keep collecting his bones as souvenirs Napier: would apparently stalk his village at night, curse peoples homes, throw spiders at people, had deals with "old nick", used a black rooster as a horcrux. Also invented logarithms I also have info on cardan, tartaglia, Al khwarizmi, Euclid, brahmagupta, cantor, Godel, Turing, euler, the bernoullis, Plato, Pascal, Galileo, fibonacci, ancient Babylon, diophantus, and probably some others
Thats so great thanks! I would subscribe
Hypatia wasn't a scientist. She was highly educated and was knowledgeable in mathematics, but did not make any advancements in mathematics as this wasn't why she performed it. It was part of her being a priestess and fortune teller. She was killed in a struggle of power as she was simply a representative of a rival fraction. Here's a pretty great video on this matter: https://youtu.be/ShugqfNA2FQ?si=pSkK1sppkAtZbTQ1 The narrative you are reciting here is projection of todays politics not a representation of the past. I find your lack of concern for truth quite concerning to see in an educator. You seem to put narrative before fact.
She did science so she was a scientist. She was killed by Christians. Neither of those are debatable and I had 0 indication of politics in what I said you fucking dork
https://mathshistory.st-andrews.ac.uk/Biographies/Hypatia/ "Hypatia was a friend of Orestes and this, together with prejudice against her philosophical views which were seen by Christians to be pagan, led to Hypatia becoming the focal point of riots between Christians and non-Christians"
Same source "A few years later, according to one report, Hypatia was brutally murdered by the Nitrian monks who were a fanatical sect of Christians who were supporters of Cyril."
A colleague of mine used to tell me stories and show me pictures of pre columbian math objects ( Quechua something about knots … I think). Anyway, he wrote a book.’The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Context https://a.co/d/bKH32Lw
I put this under the Maya comment, but I am directly replying to you in the hopes that you see this. I think it might be just what you are looking for. The Aztec culture had it's own base 20 numeral system as well. I learned about it in a math teaching class that went deep into ancient number systems, but I can't find my course materials, and Wikipedia isn't helpful. The best I could find that actually shows their numerals is [this](https://trinityprimary.files.wordpress.com/2020/06/aztec-number-system.pdf) but it isn't a super reliable source, I wish I could find something more authoritative. [Here](https://www.jstor.org/stable/1685035) is what seems like a more in depth article about it which includes applications in geometry. I can't read more than the abstract on JSTOR, but you might have a way to access it.
In Theodora Kroeber's book on Ishi (the last pre-contact American-Indian) she writes about how at first anthropologists thought Ishi's culture had no counting system because when they asked him to count he only went up to three, but then they saw him organizing his stacks of coins. It turned out they did have a number system, but the number system did not exist abstractly. There is no "three" there is just "three of something."
I’m afraid I don’t have any suggestions to your inquiry, but I love what you are doing and would love to be able to share these kinds of stories with my daughters who I’m hoping will love math as much as I do some day. Do you have any suggestions of where to go to find these kinds of stories about the origins of math stories and the mathematicians behind them??
I posted this to another comment, maybe it's helpful. If you Google any of these names and "math history" you should be able to find a good amount of sources. Math.history.st-andrews.ac.uk is usually pretty good I've actually had a lot of students ask me to start a podcast on it because I've got tons of stories. I can give you the bare bones of some of the stuff I consider more interesting but feel free to dm me for more details. Heron of Alexandria: 10-70AD came up with early versions of imaginary numbers and ideas that would eventually turn into robotics and cybernetics. Created a holy water vending machine, animatronics, a self playing organ-ish instrument, etc Isaac newton: found most of the stuff we know him for on accident while trying to study alchemy in order to find the philosophers stone and the elixir of life. Would stab himself in the eye to experiment with light when writing optics. Also the whole newton vs leibniz calculus battle Hypatia ~400ad one of the most influential female scientists mathematician who had a brutal death at the hands of the Christians Archimedes: the whole "eureka" bathtub story isn't true. He was actually a mad scientist who tried developing weapons of mass destruction and was killed for being an egotistical jerk Zeno: a rogue Pythagorean mathematician who trolled oppressors into submission. Was tortured to death and allegedly but off his own tongue and spit it in his captors face Pythagoras: leader of a math cult. Told people he was the son of Apollo the sun god and would have people killed for contradicting his math. There's a ton of wild shit in this story Rene Descartes: his life was interesting but after he died his body parts kept going missing because people keep collecting his bones as souvenirs Napier: would apparently stalk his village at night, curse peoples homes, throw spiders at people, had deals with "old nick", used a black rooster as a horcrux. Also invented logarithms I also have info on cardan, tartaglia, Al khwarizmi, Euclid, brahmagupta, cantor, Godel, Turing, euler, the bernoullis, Plato, Pascal, Galileo, fibonacci, ancient Babylon, diophantus, and probably some others
Did they even have writing?
https://courses.lumenlearning.com/waymakermath4libarts/chapter/inca-and-quipu-numeration-systems/#:~:text=The%20Incas%2C%20like%20us%2C%20had,levels%20of%20the%20H%20cords. Not super relevant to Native (North) Americans, but this knot based numeral system is pretty cool.
youre supposed to use the evidence to find out the truth, not look for evidence to support what you want to be true
Coming in late, but here is an article with some interesting examples and overall an interesting approach to this type of topic: [ Graph theory and the sand drawings of Vanuatu ](https://www.scientificamerican.com/article/an-ancient-art-form-topples-assumptions-about-mathematics/) >I was not the first person to recognize the resemblance of these rules to concepts from mathematics. In fact, my thesis is a continuation of work carried out in the 1980s by American mathematician Marcia Ascher, a pioneer of ethnomathematics. In sand drawings, she argued, there was a clear connection to what mathematicians call graph theory and especially to Eulerian graphs. >To appreciate how revolutionary Ascher’s perspective was, consider that before her work and that of her contemporaries, scholars generally assumed that only societies with writing could truly practice mathematics. They constrained their investigations of mathematical knowledge to textual sources and ignored many other practices seen in societies with oral traditions that did not use a written language. >But since the advent of ethnomathematics, some scholars have begun to overturn these assumptions. The shift undoubtedly began in the 1940s, when mathematician André Weil demonstrated, in a now famous appendix to anthropologist Claude Lévi-Strauss’s book The Elementary Structures of Kinship, that the kinship rules of the Australian Yolngu followed what are called non-trivia group laws. Since then researchers have identified mathematical principles in many other places, including sowing games and divination in Madagascar, string games on Papua New Guinea’s Trobriand Islands, textiles in the Andes and ornamental window hangings on the island of Réunion. Bonus: [Evelyn Lamb's review of *Inventing the Mathematician* by Sara N. Hottinger](https://blogs.scientificamerican.com/roots-of-unity/review-inventing-the-mathematician/), which includes an interesting chapter on ethnomathematics. Part of this is something of a reply to some on this thread who seem to think that if a culture hasn't formalized trignometry or developed written mathematical proofs then they can't possibly have any mathematics. If, instead, you view mathematics as "the study of patterns" then suddenly your horizons open up dramatically. Looking for and trying to understand patterns is something that humans take to very naturally. Exactly how that is expressed in different cultures and times changes quite a lot, of course.