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A professor of mathematics is sent back in time to a post-Neolithic farming village. They accept him as their wise man, and he spends 30 years teaching them all he knows about mathematics until he dies. The ancient people preserve most of this knowledge in stone carvings and baked mud tablets, and continue to teach their descendants the secrets of basic and advanced mathematics.

Now, the question is... how greatly might this advanced mathematics change history? There is no need for a precise answer, but I would like to understand just how big a difference something like this could make.

Or, would we see very little difference, in terms of technology and culture?

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    $\begingroup$ Where was he? That will make a lot of difference to the answer, although I think in reality you're up against it in the POB and Too Broad departments both. $\endgroup$ – Ash Jul 30 '18 at 10:23
  • $\begingroup$ Neolithic societies built some amazing structures ... The Stonehenge complex for example. Neolithic peoples were not just Hunter gatherer tribes, but also settled farming communities and some decent sized cities. Who is to say your professor don't have something to do with some of these? $\endgroup$ – pojo-guy Jul 30 '18 at 11:45
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    $\begingroup$ only a comment because I don't actually know if post-Neolithic people would already know about this, but: what about geometry and trig for navigation? $\endgroup$ – MissMonicaE Jul 30 '18 at 15:53
  • $\begingroup$ When the accepted answer starts with "I think", you can be pretty certain that the question is POB. $\endgroup$ – Aify Jul 30 '18 at 16:54
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    $\begingroup$ First thing that comes to my mind is trigonometry. $\endgroup$ – Parag S. Chandakkar Jul 30 '18 at 17:44

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I think the knowledge would simply be lost in time unless they actually have use for it.

A tribe of post-neolithic settlers have very little use for differential equations and stuff of that level. And even simpler stuff like linear algebra would not be helpful to them.

Basic math is easily applicable in everyday life. Even in simple non-currency-based trading. Or in planning, etc..
So there is an obvious use for it.

Teaching math is hard
Anybody who ever took a class in advanced mathematics at university level will have seen that a decent chunk of educated people studying to become engineers, mathematicians or other scientists can struggle really hard to learn that. And they spend most of their time doing that. Tribespeople do not have that much free time.
If you assume these tribespeople have almost no basis before that you will not get far in teaching them math.

Advanced math needs applications to not be forgotten easily
If you were to teach them very basic physics with the use of math, you might inspire early engineering. Have people built simple mechanism if the natural resources allow that.
Helping them understand gravity, force and all that could be very useful for them - If you show them the usefulness of this knowledge.


Conclusion:
It is of utmost importance to make use of the knowledge. Post-neolithic tribes have very little free time. Spending a lot of time and energy on studying something that does not help them survive is a waste of time and would cause trouble for the tribe in the long run.

You might inspire early engineering, but not with just abstract math. People need some physics education, too. (An average mathematician should be well educated enough to do that.)


Addendum:
Because of a discussion in the comments I felt like it is worth to add this.
Richard Dawkins originated the word meme in the context of evolution in 1976. He proposed the idea that like genes, ideas that benefit a population (like the crafting of specific tools) will persist throughout the generations. He referred to these ideas as memes. Memes will persist throughout the generations only if they continue to be useful.

If an idea is useless or even harmful in regards of survival it will not spread as well and eventually die out. Sometimes long useful memes become replaced by better ones in the ever changing situations the enviroment forms.

So the knowledge your professor brings them requires to be useful for that reason. There is no time and resource to be wasted on a luxury that does not benefit survival if those are not expendable. And the ideas will only reliably persist as memes throughout the generations if they continue to prove themselves as useful or they will soon be discarded

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  • $\begingroup$ Maybe it's a short story $\endgroup$ – DonQuiKong Jul 30 '18 at 18:11
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    $\begingroup$ I would disagree quite adamantly on the amount of free time available. One of the primary outcomes of the agrarian revolution was that people were able to undertake other trades because of the surplus of food created from farming. It was essentially from the Neolithic phase that the Bronze Age came about as people were now able to concentrate spare time on alternative pursuits such as extracting metals. $\endgroup$ – EnE_ Jul 31 '18 at 4:58
  • $\begingroup$ @Ene yes, they had some free time. But as you pointed out they undertook other trades - most of which useful in the long run. Do you really think advanced math that was taught 10 generation ago and never applied once will stay part of the tools people use and pass on to the next generation? Are you familiar with Dawkin's term meme? A meme, like a gene, is an idea that proved useful to the population and is passed on to next generations. Same were useless or harmful memes. E.g. Sacrificing babies at random to appease the gods. Some did it, but it didn't last. $\endgroup$ – ArtificialSoul Jul 31 '18 at 5:52
  • $\begingroup$ Memes don't have the same selective pressures as genes, because memes can be passed to those that aren't your kin. For example, a meme for celibacy can thrive in the meme pool, but a gene for celibacy certainly cannot. Memes can exist, for example, purely because the people have a psychological inclination towards it, even if it hinders their survival and reproduction. It certainly helps the meme's propagation if it reinforces survival or reproduction but it's not required. $\endgroup$ – Bridgeburners Jul 31 '18 at 14:40
  • $\begingroup$ @Bridgeburners That is entirely correct, but if survival is hard to achieve then useless or harmful memes are either dropped or lead to extinction. When a drought comes and all of the village people need to work every single second they have to somehow get food they will take little care of math. If the situation persists then math will be lost in the long run. $\endgroup$ – ArtificialSoul Jul 31 '18 at 14:48
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A math professor is not a math exercise book, then why would he limits his help to math? He most likely also knows a good amount of physic and informatic.

He could obviously store some of his advanced math theorems and physic theories (carved rocks), for the times when some people will actually be able to understand them and use them but i think that's not the most relevant change that he will provoke.

He can introduce the concept of "zero", the Pythagorean theorem, and all the milestones we usually teach in our child/teen-age math classes but also these efforts won't be the most relevant ones.

Way more important for neolithic humans: he likely also have a lot of general scientific knowledge that will allow huge, tremendously fast bootstrap from neolitic tech level to a something at least similar to Middle Age.

He can introduce the neolithic people to some tech like:

  • Metal casting and coal production
  • Gun/black powder
  • The existance of America and general world geography
  • Basic hygene precautions and antibiotics principles
  • Crop rotation and animal trained plow
  • Modern sailing in which the sail is actually an airfoil
  • Darwinian evolution (useful for both farm and animal breeding)
  • Architectural devices (cranes, leverages), and architectural solutions (round arch)
  • Steam

The tech development is not linear, is exponential, therefore a "small" tech jump happened thousands and thousands of years ago will result in a huge difference in the today tech level. Since we are not talkin about a simple or small jump but rather a huge leap instead, the today tech won't even be close to the current one. It's not relevant that the person sent back in time is a math professor, every scholar person will surely provoke such a huge leap that is impossibile to foresee where will it lead to. We can only be sure that the difference would be gigantic.

ps: is not required to have a detailed knwoledge about a concept or a theory. Also knowing that "something works this way" will provide something priceless, for future scientists, and even a simple hint that points the right path/theory could help tremendously (see antibyotics, or vaccines).

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    $\begingroup$ Being a mathematician who has actually thought through the scenarios for what I'd do if I was moved back in time to various eras, this is pretty much my exact plan. Teaching early people combinatorics won't do crap for them. Teaching them to wash their hands, basic genetic inheritance, germ theory of disease, gunpowder and basic machines will give them a jump to within the last 500 years. (Oh and a little political science: All people are people, treat them properly would help too.) $\endgroup$ – JKreft Jul 30 '18 at 19:22
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    $\begingroup$ Except for hygiene, I am not sure if these are immediately useful (what would you do with the knowledge that Americas exist) or achievable in a lifetime (it takes a lot of resources to start smelting metals to cast them, making steam is easy, building machines to use steam is hard.. and so on.) $\endgroup$ – Dhara Jul 31 '18 at 12:05
  • $\begingroup$ @Dhara: not only America, but also China, Japan, Australia and so on.. And not only places, he can teach also the other civs specific traits. Then you can choose to trade with them (you also know how to drastically improve vessel range) or to conquest them (with gunpowder, military tactics or reverse hygiene aka bacteriological warfare). I'm not saying that the professor will conquer most part of the world itself, i'm saying that the lucky civ that was enlighted will do. Probably in few centuries, but eventually they will. Speaking of a neolithic civ, it's something huge. $\endgroup$ – theGarz Jul 31 '18 at 12:15
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    $\begingroup$ @theGarz, giving map of the world to neolithic people would be as useful as giving map of habitable planets in the galaxy to the modern people: it's pretty cool and all, but also pretty much useless. Knowledge of things near you would be always more useful than knowledge about people on the other side of known world. $\endgroup$ – user28434 Jul 31 '18 at 15:01
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    $\begingroup$ @theGarz, we are talking about neolithic civilisations here, when they will get to americas from their england all your maps and interactions manuals will be most likely outdated or pre-dated(or about societies of your alternative-history world). And tons and tons of gold may mean nothing, like our world mesoamericans were sitting on them for ages, and so what did it to them? $\endgroup$ – user28434 Aug 1 '18 at 9:10
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The most impact would probably be his knowledge about basic physics.

There are no archeological proofs that post-Neolithic farmers had cranes or pulleys. But these rather simple tools are an enormous help in constructing buildings. He could revolutionize their irrigation system (or introduce them to irrigation if they still relied on rain exclusively). He could show them tools like the plow, scythe, wheel barrow, wind mill or water mill and improve their design of the wheel.

The simple knowledge of the three field crop rotation would make this one tribe be the most powerfull community all around.

The (for us) simple act of converting thoughts into symbols that can later be converted back into thoughts and knowledge could result in another parallel development of script (there were several of those throughout history). As a math professor he has a vast range of symbols to choose from, but realisticly he would choose the hieroglyphic approach instead of phonetic writing.

The thing with all these great, yet simple, accomplishments of human history is that you don't need mathematics to use them. The villagers would probably think that the gods gave this wise man visions of godly tools. After his death, they would either replicate his designs as exactly as possible or begin to experiment with slightly different designs and maybe find improvements by accident.

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  • $\begingroup$ Hmm, I think I have a comparative level of education to that of our hypothetical math professor -- and I wouldn't have the first idea how to go about making a plowshare or a scythe. I've heard of three-field crop rotation, but I wouldn't have any idea which crops to rotate, or in which direction, or what the benefits of it would be (so how would I convince the villagers to even try it out?) And some of the Bronze age artifacts I've seen look like they had better wheels than I'd know how to design. $\endgroup$ – Henning Makholm Jul 31 '18 at 11:04
  • $\begingroup$ @HenningMakholm Maybe it's a matter of the educational system rather than the level of education. During my school time, topics like cranes, pulleys and crop rotation were taught sometime in the first 10 years of school (= basic education level). Basic scythes and plows are rather simple to recreate if you know how they look and work. With wheels and troughs available, I could probably cobble together something resembling a wheel barrow. A functioning windmill is a task for someone stuck in time though ;). You might be right concerning the wheel, I was too lazy to investigate. $\endgroup$ – Elmy Jul 31 '18 at 11:18
  • $\begingroup$ Crop rotation is useful when you export your crops. If you plant crops that suck the nutrients out of the soil and then you send the crops away, you need a way to replace those nutrients. If you just eat the food and return the nutrients to your fields, you mostly don't need crop rotation after all. $\endgroup$ – J Thomas Jul 31 '18 at 15:14
  • $\begingroup$ @JThomas Crop rotation is still used today and was nessecary until early modern times to avoid starvation! Medieval crops yielded much less than modern crops and food couldn't be transported in great amounts. In the traditional crop rotation system legumes and manure (dropped by lifestock in the fallow) restore nutriens in the soil naturally which leads to better harvests. $\endgroup$ – Elmy Jul 31 '18 at 16:48
  • $\begingroup$ @YElm You can dig a small garden with a hoe, with a wooden, bone, or stone blade. Kill everything and mix up some deeper dirt that hasn't had the minerals sucked out of it yet. And you can individually plant big seeds with a seed drill; poke a hole in the dirt as deep as you want it, drop one or more seeds in, and cover it up. There's a limit how much of that you can do without metal, and if you add your own wastes back onto your own garden, the minerals will work out without rotation. It's civilization that requires crop rotation, and civilization invents it. $\endgroup$ – J Thomas Jul 31 '18 at 18:05
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He's a mathematics professor who really likes tea. This leads him to start making infusions of various plants, boiling the water and seeing how they taste. The key here is that he drinks boiled water, which is many times safer than (non-alcoholic) drinking water and would have a dramatic effect on the lives of any tribe once they started enjoying the tea too.

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  • $\begingroup$ This is a great answer! Like parents pass values more than actual knowledge, the maths teacher passes on valuing mind doping drugs (coffee, tea, ...) over mind numbing ones (alcohol), this would have a long lasting effect on the tribe he joined ! And since he put them on the science/ tech bandwagon this could have a lasting effect $\endgroup$ – Auberon Vacher Jul 31 '18 at 4:51
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    $\begingroup$ The usefulness of this depends on where the civilization is based. From my understanding, the worst water contamination was due to large population of humans and animals living in close quarters, dumping their waste in the water. If the civ is near a running source of water (river) and is small, polluted drinking water isn't the first concern $\endgroup$ – Dhara Jul 31 '18 at 13:46
  • $\begingroup$ @Dhara that's true, but as for changing history, and not just the immediate tribe, having a culture which boils its water would allow for the civilization to grow faster and healthier over thousands of years because their cities wouldn't have the same issues. Having a habit of handwashing would probably also help, too. $\endgroup$ – David Rice Jul 31 '18 at 18:15
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Here are some fields that could be jump started by your scenario:

Astronomy: Accurately predicting seasons, defining a calendar to mark dates to sow crops.

Geography: Accurately measuring land and making precise maps.

Navigation (resulting from both above): Maps and calculating speed and time makes long distance travel far more reliable.

Architecture (needs some physics): far bigger and taller structures even without new materials

Trade, Taxation and other administrative tasks would cerainly profit and all sorts of discoveries that result from applying the above. Other than this, I doubt early Farmers could immediately gain from advanced mathematics. But you could also try to look at the historical rise of math to pick other discoveries that suit low-tech society.

Long term changes would depend on how society handles these changes.

As a sidenote: Today's science is highly dependent on our modern society, especially the more theoretical sciences like math or theoretical physics. A history professor specialized in your timeframe might be your best bet if you want a high impact on the society, because he could start a more "natural" advance.

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  • $\begingroup$ I think your trade and taxation section could be expanded because such a society might very well invent money much earlier than otherwise. $\endgroup$ – Todd Wilcox Jul 30 '18 at 18:00
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Worldview matters

If this math professor is like many that I studied under, he probably doesn't know anything about metallurgy or agriculture or a host of other applied fields in any practical detail. He probably wouldn't know about crop rotation or how to domesticate corn or how to carve wood with stone tools or how to start a fire or how to knap flint.

But, what a math professor would have, is a scientific and modern worldview.

He would think in terms of a spherical earth in a heliocentric solar system containing other planets in a big galaxy full of stars in a bigger universe of finite age. He would see mountains and valleys and oceans in the context of plate tectonics. He would know the basic concept of what a volcano was and would be aware of the possibilities of meteors and comets. He would see rocks as assemblages of certain kinds of minerals.

He might not be good at predicting weather, but would understand that it flows from chaotic deterministic systems, that air is a mixture of gases, and that water evaporates, is carried in clouds and produces rain when the temperature and pressure are right.

He would have a germ theory of disease, a crude Mendelian concept of genetic heredity, and understand the concepts of atoms and molecules. He would have a sense of how predator-prey models work and the basic mechanisms behind evolution.

He would naturally think in terms of currency, the time value of money, and supply and demand. He would recognize core concepts like the non-zero sum nature of economic interaction. He would be alert to the possibility of domestication of plants and animals.

His natural philosophy would be more Aristotle than Plato. He would understand the scientific method and be able to advocate for it.

He would know instinctively what kinds of social organization and etiquette and diplomacy lead to productive interaction and which do not. He would know how to teach people to operate productively in much larger groups than they had ever imagined.

He would have a good knack for distinguishing what kind of investigations and new ideas were likely to be fruitful and which were not.

For example, wheels were used in toys by the Sumerians and Aztecs for centuries before anyone thought to make an industrial scale version with economic advantage, but our professor would see those opportunities to generalize existing knowledge that might seem like dead ends to others.

Also, the math professor would be literate, would insist on training children to be literate from a young age when it is easiest to learn (in the real world people didn't start to learn to read and write for the first time until middle school or high school age and these skills were restricted to a narrow class of scholars until the late Bronze Age or early Iron Age, the Minoans and earliest Chinese dynasties, for example, had no literate elementary school age children). This technology could provide huge advantages.

These worldview differences wouldn't have direct application, but would set his band of people on a straight and narrow path towards unprecedented progress relative to those of people with a muddled and inaccurate understanding of the big picture of how the world worked.

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Mathematics is probably the most complicated, but also the most basic type of science for your scenario. Knowledge in mathematics appears worthless to most people (ask high school students!), unless they see an actual application. A professor of theoretical logic or number theory will likely have no influence teaching his field. A high school teacher used to teaching mathematics to a far less educated audience than the university professor would likely have more influence, if he sticks to very practical examples for every day usage of the tribe(all this is ignoring the language barrier which increases the difficulty of teaching illiterates). An expert in teaching will actually be more valuable than an expert in advanced mathematics.

This leads to the question, what knowledge is founded on basic mathematics, which is relevant in ancient ages:

Geometry is one of the most ancient disciplines of maths, because independent of number systems (including 0 or not) it helps in measuring and also partitioning farming plots and regulating the distribution of land between the inhabitants, e.g. if a father dies you can now calculate how to split the farmland between his children according to his will. The ability to draw plans helps with creating larger buildings, planning ahead construction time and material cost, making actual larger constructions works more feasible to actually realise. Inventing the wheel, planning carts and contructing vessels, such as sailing ship, galleys and so on.

Introducing currency by calculating a common value for the loaf of bread, the chicken and small tool is one of the most valuable things, as it will allow for easier exchange of goods first within the community and possibly also with neighbouring communities, once traders start appreciating that they do not have to carry all their goods along with them all the time. Teaching professional merchants and exporting the idea of currency could actually make the community one of the first trade hubs and greatly increase their wealth and living style.

Functional relations and the idea how to quickly solve certain equations have been a large challenges for mathematicians for centuries. Bringing back the knowledge of approximating some behaviours with a linear process to be able to predict what will happen in the near future and providing the ready made soltuion for quadratic equations and showing the physical processes like gravity, acceleration, speed, ... where these are applied could spark interest in at least a few to learn the 'more advanced' ways of the wise man.

Vector geometry is a simple visual way of illustrating certain natural behaviours such as currents in the sea or river and how they affect sailing ships. Also redirecting rivers for better irredation or flood prevention might fall in here.

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The Scientific Method

A lot of very interesting answers here.

The most useful skill, on the long-run, would be to teach the scientific method: base everything on observation and the confirmation of those observations through falsifiable experiments. Question everything, do not trust blindly in any superstition/witch doctor/shaman/spirits etc.

This, if ingrained deeply in their culture, will ultimately lead them to be greatly successful over a long period of time.

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Short Version : You need an expert in Manure, not mathematics.

In such a post-neolithic world (e.g early bronze age), your mathematician starts of :

  • Probably older than everyone else - average life expectancy would be about 25 at that time and even a new graduate in mathematics would be about 22.
  • A babbling idiot who could not communicate in any locally recognizable language
  • Potentially lethal to the locals as his body carries diseases internally their immune systems would be very vulnerable to.
  • Incapable of feeding, clothing or hunting in the way the locals do (if at all).
  • Socially inept to the point that they he cold easily cross a local social or religious taboo and find himself dead (or worse). Such a culture would be essentially unrecognizable to a modern human and they would be lucky to navigate through it socially without disaster after disaster.
  • Repelled by local drinking water. Ever actually look at a glass of water from a river ? Don't drink it unless you're feeling lucky - no clean tap water in those days.

So your mathematician would start out, not as a wise man, but as a useless, incompetent misanthrope and fool.

And as survival was a day-to-day struggle and charity a luxury at that time, I'd say the odds of your mathematician surviving long enough to learn basic survival skills and communicate with the locals is very small.

But lets say they do ...

What can they teach the locals to gain their respect ?

Not bloody mathematics that's for sure !

Useful skills to neolithic cultures :

  • Manure Science - what can you do with manure ?
  • Maybe basic hygiene practices - can you mathematician make basic soap ?
  • Calender skills - can your mathematician remember enough basic astronomy to help the locals predict the seasons accurately (useful for agriculture) ?
  • Maybe mapping and surveying skills.
  • The use of a plumb bob.

Ancient cultures has little or no practical use for mathematics (note that the concept of a number zero did not appear until about 300 BC and negative numbers were not developed until about 200 BC).

Again remember that survival is the only focus of day to day activity and long term planning means crop gathering, planting and rotation and the storage of grain and other food.

Lets say your mathematician strikes it lucky and comes to the attention of the equivalent of scholars of the time and gets to show some skills (area of a circle, geometric proof processes). He might get to teach some basic skills in symbolic algebra and maybe elementary calculus. But again remember that applying these skills is not the same as knowing them. It's quite possible he could teach e.g. twenty people these (for the time) very abstract and esoteric studies, but would they have any impact long term ?

It's not like he could write a book.

Books !!!

Now we're talking.

Your chap comes to the attention of high and mighty and, more to the point, the early equivalent of bureaucrats, and knowledge that you could print in a structured way (typeface) would be really useful them.

If your chap forgot the maths and simply got around to explaining how to print and keep records on paper (how do you make basic paper) or even on cloth, your chaps would bring the printed word (and the ability to educate and organize and record) to the world very early on.

Woodblock printing did not appear until something like 200 AD, so if your mathematician got that going in the 4000 BC range, it would certainly have helped speed up development of many things (like the invention of taxation and revenue collection - yay !)

Now assuming your chap helps develop printing he's also now in a position to plagiarize the work of every mathematician he can remember and pass it on to what would become a very different (but probably equally bloody) history !

Another Mathematician becomes a Bureaucrat

Your chap will now follow (eh, precede) Issac Newton as a government employee par excellence by helping them become better (the first !) paper pushers and probably also help the local head honcho realize the benefits of a sound economy based on tax collection (and the need for regrettably violent punishments for avoidance and tax exemption for senior government figures and researchers in science and engineering).

And this is a good thing from your chaps point of view as he'd starve to death otherwise !

A Ripe Old Age

He could live as old as 35, maybe even 40 if he plays his cards right.

But only if his health care reforms (like soap and cleaner-than-mud drinking water) come into vogue.

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    $\begingroup$ Your first point about age is plain wrong: The average life expectancy is brought down by a huge infant mortality, even in a neolithic society there lots of people actually between 20 and 40 years old and reaching an age of 60 or 70 is not really unusual. $\endgroup$ – jknappen Jul 30 '18 at 16:40
  • $\begingroup$ @jknappen I'd be interested in a source for that and particularly some way to quantify "lots" compared to e.g. a modern life expectancy profile. $\endgroup$ – StephenG Jul 30 '18 at 17:56
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    $\begingroup$ Run your favourite search engine for life expectancy neolithic and find sources galore, e.g., my first hit ancient-origins.net/news-evolution-human-origins/… $\endgroup$ – jknappen Jul 30 '18 at 18:08
  • $\begingroup$ @jknappen no need to go that far back. Infant mortality rate still differs a lot per country (or more causally by how developed it's medical infrastructure is). $\endgroup$ – JJJ Jul 31 '18 at 4:12
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He probably would not teach them advanced maths. As we know from the XKCD Purity strip you can have all the things thanks to math.

So he would probably advance them in the field of building: the angles, the structural strength which lead to: materials (weight, density) which lead to: chemistry (after all, Avogadro and atomic number are maths). Also he can help them come up with the sextant, the watch, boats and carts. He may help build them build tools (calculating the best surface to apply the least amount of force to get the best result).

The rest is just speculation about what knowledge the professor should have to think about furnaces, metallurgy, cultivation. Depending on whether he remembers and can come up with a way to calculate best buoyancy with best sail surface. Or how to make a propeller. Or Archimedes' screw.

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"he spends 30 years teaching them all he knows about mathematics until he dies." Assuming, for the purposes of this answer, that he teaches them mathematics, as opposed to doing the practical thing and becoming the engineer that they could probably make most use of in the short term:

I think that this largely depends on where he was, and which post-Neolithic people he fell in with. I say this, because in order to make use of the mathematical tablets and such, there would need to be a leisure class, since, if one spends all of one's time hunting and gathering, there isn't much time for mathematical pursuits.

However, Let's suppose that he were to travel back to, say, only-slightly prehistoric Egypt (too long ago and the tablets would likely be lost). As the dynastic periods begin and certain classes begin to have time for scholarly pursuits, it is likely that some smart fellow or another (sorry ladies, this is ancient Egypt we're talking about) would find these tablets, and begin to piece together their meaning and the oral traditions surrounding them.

Now, it is well known that mathematics, and especially basic mathematics tends to be easily extensible to real world applications. I imagine, therefore, that this same smart fellow would soon realize that the whole geometry thing was actually useful, would decide that the remainder of the tablets on analysis and what have you might therefore also be useful, and a whole engineering discipline would arise on the shoulders of the mathematics.

(side note) OTOH, a lot of our knowledge of the real world and practical implementations in it comes not from mathematics, but from science. The scientific method, as we know it, was not exactly common knowledge in the ancient world, AFA we K. As much as I think ancient civilizations would be able to create from some mathematical teachings handed down, essentially, from the gods, I think that simply teaching them the scientific method would produce WAY more advancement, more quickly. Bonus, it wouldn't require 20 years to do it, it could probably be accomplished in an afternoon, maybe a week to really drill it in there, and maybe a month to cover most common pitfalls. Indeed, hunter gatherers could start using the information right away to discover, for example, that human sacrifice actually DOESN'T make it rain, despite the fact that it rained when ol' Tommy Stabbington killed poor Bobby Holeysides just before that hurricane came through....

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  • $\begingroup$ Taking your 'sacrificing people for rain' example, how would they evaluate if it works or not? How many would they have to sacrifice to have enough data? Having noted all the data, how do you come to a conclusion? That already requires maths and a load of people to sacrifice. And even if you manage to do that, it might not work. Indeed to this day, many people believe in god(s) and appeal to them (or it) in some way. $\endgroup$ – JJJ Jul 31 '18 at 4:18
  • $\begingroup$ @JJJ "how many would they have to sacrifice to have enough data?"... >1 less than they sacrificed without retaining any data at all? Also, they'd need a control, so that's half of the sacrifices saved right there. "That already requires math" not a lot, really... maybe counting to a hundred and inequalities. Sure, math if you want a $p$ value, but that's not absolutely necessary. Heck, there's a 50% chance in their controlled study that there is actually LESS rain when they sacrifice people. Surely that would convince anyone. $\endgroup$ – Scott Jul 31 '18 at 13:23
  • $\begingroup$ Try convincing people who really believe in something. And actually, less rain doesn't prove anything. In the same way that throwing heads 5 times in a row doesn't prove a coin is unfair. Oh and if you set up the study badly, e.g. no sacricices first and all sacrifices after, you might find that your data points aren't independent (raining season vs a dry spell) that you might actually 'prove' that sacrifice does work. nevertheless, you can try your approach, many religious fanatics still around these days. ;) $\endgroup$ – JJJ Jul 31 '18 at 19:52
  • $\begingroup$ @JJJ "less rain doesn't prove anything" In a nothing proves anything, Popperian sort of way? Throwing heads 5 times in a row certainly lends evidence to the claim that a coin is unfair. Throwing heads 100 time in a row lends more evidence. In any case, I suppose your point that the existence of anti-science views today suggests that anti-science sentiment would exist in Neolithic cultures is fair. On the other hand, groups' anti-science sentiments has seldom stopped them from reaping the benefits that the scientific discoveries of others affords them. $\endgroup$ – Scott Jul 31 '18 at 20:19
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    $\begingroup$ Oh come on, an applied research scientist doing data analysis does (or should) know about confidence intervals ;). Here (EU) it's even in pre-uni programs. I Think you're on to something with your last comment though, start simple and win their trust, then when you have it start walking back on the crazy stuff (like sacrificing people or animals). BTW I tried convincing people not to 'feed ghosts' (because why waste good food?), even though they trusted me, it's very hard to convince them there are no ghosts. $\endgroup$ – JJJ Jul 31 '18 at 23:58
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Modern math is the result of thousands of minds working over thousands of years, each standing on the shoulders of those that went before. Even a tiny bit of knowledge that survives past a few generations compounds tremendously over time.

You premise would change absolutely everything. Every culture would be different than it is today. National boundaries would be different, languages would be different. We might even have become a spacefaring species by now.

Wherever this neolithic village was, it would become more prosperous than its neighbors, which would invite jealousy and war. Which would compel the development of military sciences, which would in turn compel the development of material and medical sciences, and your off to the races.

So your question is pretty much the same as: "Where will we be 2000 years from now or more?", because I believe that is the kind of effect it would have.

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It depends where he is.

If he lands with the bronze age beaker trades of Europe then nothing he says is going to get through, nothing, that culture got the chop from invaders coming out of what is today Mongolia and Northern China some time in the 1200s BC.

If he's anywhere in the Americas he might have some impact on those cultures and they may advance and expand, crossing the Atlantic in the other direction. But the math will be the least of his impact there, his knowledge of history would be the key, pointing the locals at industrialisation and the use of their abundant mineral resources.

Ancient China may or may not benefit from such a man, it would depend entirely on exactly where and when he landed and precisely who he met, he may have been accepted by the locals only for them to be overrun by their neighbours a generation later or he might have laid the foundations for a dynasty that lasted hundreds of years and spread without opposition China's history is chaotic enough that it's pretty impossible to know.

The Eurasian Steppes might be the one place from which his teachings would have a good chance of reaching much of the western world, there is a very long history of large migrations out of that area as populations outstripped their local resources. Math used for systematic logistics and siege warfare would almost certainly spread out with such groups.

The long term impact anywhere in the west is questionable, after Rome and the following dark age we lost a lot of what we used to know, the Renaissance was an age of rediscovery as much as anything else.

It occurs to me that while no trace of the man and his teachings need ever be found some of the great works of the ancient world may be down to his presence, Stonehenge or the Great Pyramids may in fact be his doing without anyone ever finding a trace of him.

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Cryptography

If he's a mathematician that knows cryptography, teaching a simple RSA to the world is going to change EVERYTHING.

Of all times, nation sent encrypted message to one another - and of all times, those message were 'cracked' by the enemies to take advantage of the content. With a simple encryption method that everyone can do with a little bit of effort (~10 min), but which is impossible to decrypt (at least with our current knowledge today), the whole world is going to change. The war of information would take a different face (starting with the Roman empire) and lots of dark plots would never be revealed to the light.

Sorry for the lack of examples. Since they were already load of answers, I thought I would just throw out my idea quickly without minding too much the details.

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  • $\begingroup$ This sounds like a great idea on the surface. But as an answer, it would really benefit from an example showing how simple a set of instructions it could be broken down into - that people with no mathematical basis (and potentially no writing system) could actually employ. $\endgroup$ – Bilkokuya Jul 30 '18 at 16:14
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    $\begingroup$ RSA isn't all that practical to implement without computers. You need to be able to generate huge primes, and do error-free modular exponentiation with them. Something simpler, like the Vigenère cipher might be more practical. It can be done by hand, it's more error tolerant, and with a large enough key, the lack of computers works to your advantage, since the frequency analysis becomes rather time consuming. $\endgroup$ – Ray Jul 30 '18 at 19:46
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    $\begingroup$ When most of the population is illiterate, writing is already encrypted. The simple Caesar cipher was secure in Europe until the middle ages because they hadn't discovered letter frequencies. RSA is vast overkill. $\endgroup$ – Ross Millikan Jul 31 '18 at 2:55
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    $\begingroup$ @RossMillikan, the Caesar cipher was never secure in the modern understanding of that word because it depended on secrecy of the algorithm rather than the key: if the enemy knew the algorithm there were only a couple of dozen keys to try. A substitution cipher with an arbitrary permutation, on the other hand, would be secure until people tabulated letter frequencies; and Playfair is easy enough to remember and teach but wasn't broken until the 20th century. $\endgroup$ – Peter Taylor Jul 31 '18 at 10:05
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    $\begingroup$ @Sasugasm Did you use full-size primes, on the order of 1024 bits in length, or did you use small primes like 17 and 23? It's practical to hand-calculate RSA if you're using small primes. It's also practical to break RSA if you're using small primes. $\endgroup$ – Ray Jul 31 '18 at 22:00
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I strongly second Dhara's idea. Basic mindset could make a great big difference.

Beyond that, developing a religion around mathematics that priests teach to young priests, probably requires a city. If he finds a neolithic city then he has a chance, maybe....

Maybe he could teach things that are immediately practical. If they don't already have pottery, he could teach them that. They take downed wood (because cutting down trees is hard unless you have good tools) and make pottery, and they already have everything they need to do it except the knowledge.

Do they have a source for salt? Salt is valuable, if they can get it.

If they burn wood for pottery, they can get soda ash or potash from the ashes. That's useful. They can make glass.

It's a big effort to make stone axes to cut down trees, but that effort can pay off. Quartzite etc for tree-cutting axes takes entirely different skills than flint etc for meat-cutting axes. He can teach them how (if he knows how). The time to do that is maybe after they are running out of down wood because his pottery is using up too much of their fuel.

At this point they are rich with a lot of hard work. They have pottery, glass, natron, potash, and I hope salt which they can trade. They can cut down forests on flat land to get firewood, which incidentally gets them new cropland. This might take his lifetime, but if not they have wealth and plenty of work available, they might accept immigrants and their population goes up to the point that epidemic disease matters. (When population is low and there isn't much trade, the diseases tend to come from other animals and not from other humans.) He can promote city hygiene, which was not needed before.

If they find coal, they can do a lot with that. But it isn't available everywhere. Similarly gold, copper, etc.

Do they have good leather? But then he probably doesn't know how to make good leather. Does he know how to domesticate baby cows etc? Castrate them, raise them to pull tree trunks to the kiln? Breed them for today's miniature size? Keep an auroch bull to impregnate his cows? It's a whole lot of work and a whole lot of skills. Probably too big a jump.

A lot of stuff is too big a jump. But if he can increase the prosperity enough, he might be able to create a priesthood that knows some math. They can be an elite, and volunteers get judged by how well they can pick it up. They can do proofs, and learn the most teachable proofs, and gradually expand the amount of math they can hold onto. Someday it would become valuable. The important thing would be to make sure the mathematical elite have enough children. If he manages to create a celibate math elite that creates a selection pressure against the ability to learn those skills, that would be very bad.

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