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Somewhat based on this question, let's say a normal person from our world for some reason goes back into time - to the Hellenic Greece around 490 BC and somehow manages to become a philosopher. My question is, how will he compare to a scholar? Will he be able to achieve more, because of his knowledge?

The Character: The unlucky character is a person of above-average intellect (130), doing a PhD in Maths. He has not memorised vast amounts of information, but he enjoys research in maths, even if it is not related to his topic (which is why I haven't given the topic). Assume that the magic that brought him back into the past allows him to survive there, and communicate. The Ancient Greeks think he is a travelling scholar, but due to the magic, they don't question too closely.

Will he be able to revolutionise maths? He can't just say, "This is called integration" as he has to prove it.

Magic: To make things clear, I'll add this in. The magic allows him to communicate in Greek easily. He can understand it effortlessly, and it stops the Greeks from asking him very incriminating questions (like where are you from, etc). They simply think he is a travelling scholar and leave it at that. They also have given him food and a place to stay.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Serban Tanasa Aug 10 '16 at 13:58
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    $\begingroup$ What mathematician worth his salt couldn't give at least a rudimentary proof of integration at a moment's notice? And he's supposed to be getting a PhD? For shame! Maybe it's just how I was taught, but I always thought you weren't supposed to memorize mathematical tricks without understanding why they work. $\endgroup$ – Superbest Aug 10 '16 at 22:44
  • $\begingroup$ Despite me linking this question to my more popular one, this somehow has only 490 views, while that one has 6833 views XD $\endgroup$ – King of Snakes Aug 12 '16 at 5:18
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Something that most people would think nothing of would revolutionize math in that age.

The number 0. Many mathematicians view it as key to most modern mathematics.

You could also introduce the idea of counting in various bases and information theory pretty easily.

You could standardize units of measure.

Let's not talk about irrational numbers, infinites, imaginary numbers, set theory, etc, or non-euclidean geometry... Anything Principia Mathematica...

Any one of these would totally revolutionize Ancient Greek thinking, let alone mathematics. It might be that only someone with a decent understanding could introduce it though, because most of us don't know the names of these things or how they work and are more second nature to us today.

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    $\begingroup$ If the legend is true, it is better for our hero to not mention irrational numbers to Pythagoras. Also, infinite sets was a big piece to swallow for Cantor's fellow mathematicians. It is likely that the Greek scholars would look at our time traveler as a fool. $\endgroup$ – Taladris Sep 26 '16 at 15:27
  • $\begingroup$ @Taladris some would see the traveler as a fool, but infinite sets do interesting things like resolve Zeno's paradoxes. The key is whether the traveler could get enough attention for thoughtful people to listen in the first place. $\endgroup$ – Deolater Sep 26 '16 at 18:01
  • $\begingroup$ It's okay to mention irrational numbers around Pythagoras, as long as you have access to a bean field. $\endgroup$ – papidave Mar 25 '17 at 12:35
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My answer differs from the other ones: a revolution does not occur ex nihilo and most of the time, the work of mathematicians that history remembers was preceded by the work of people that have been forgotten now (bare the specialists of history of sciences and mathematics). Life is hard for precursors. It was not easy for the ones who truly made a mathematical breakthrough, being one step ahead of their contemporaries:

  1. According to the legend, Pythagoreans that revealed the existence of irrational numbers were forced to commit suicide.
  2. Cantor's theory of infinite sets was not understood at his time, which lead him to depression (though some believe his depression was mostly due to the loss of his daughter).
  3. Riemann's non Euclidean geometries were looked down, until Einstein used it for its theory of relativity.

Remember also that mathematical notations, even the ones that we seems elementary like +,-,etc... are rather new, so everything that modern mathematicians would write would look esoteric. Therefore, introducing modern concepts to Greek scholars would not be as easy as one could imagine.

Our time travelling mathematician would feel quite lonely if he expects to transmit his abstract knowledge.

However, he can make use at his great advantage of one large area of mathematics: applied mathematics! Demonstrate by example! He can employ modern applied knowledge to daily life, at least the part that doesn't need heavy computations, to pull himself above the crowd of commoners:

  1. Probability and statistics can make him rich, with all advantages that come from it: wealth, slaves, women(*), army, political power,...
  2. He can use statistics and its predictive power to become an augur or a religious leader.
  3. His knowledge of conic sections could help him improve optic, astronomy, probably one of the few domains where his fellow Greek scholars would understand him.
  4. He can use his knowledge of geometry for military purposes. After all, the tiny island of Syracuse was kept safe from invasion of its powerful neighbours thanks to Archimedes' science.
  5. Modern cryptography will help him to maintain an efficient network of spies.

To sum up, after an initial period of misunderstanding and ignorance, a time travelling mathematician would use his knowledge for his own personal interest.

(*) no sexism intended.

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    $\begingroup$ Interesting examples! One fairly recent and less theoretical from "my" world of electronics engineering is Black's patent on negative feedback amplifiers which was ignored for 9 years. the concept was so contrary to established beliefs that the Patent Office initially did not believe it would work. It's now one of the most fundamental concepts and I can't imagine many circuits without it. $\endgroup$ – pipe Sep 26 '16 at 17:16
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Back to school

Your mathematician would have to learn math all over again, like a school child. Do recall that you are dropping him into a place where Arabic numerals have not yet been introduced, even less so the concept of "zero".

Sure, he can do his own math like a pro, but no-one else will be able to understand what he is doing because something as simple as $1 + 1 = 2$ is — well, I will not say "Greek" ;) — incomprehensible to them.

So sure, he could, potentially, revolutionize maths (once things like food, shelter, a job and communication is solved) simply by introducing new numbers and zero. The question is if others will join him in this entirely alien way of doing maths. And there are sure to be some very pertinent questions along the way, like "How the heck did you come up with all this?!".

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    $\begingroup$ Communication problem is solved ... the magic allows him to talk to other Greek people. The rest is a problem though $\endgroup$ – King of Snakes Aug 10 '16 at 9:57
  • $\begingroup$ Added the extra information in, hope that helps. The only problem remaining is the fact that his math is completely different. $\endgroup$ – King of Snakes Aug 10 '16 at 10:00
  • $\begingroup$ @KingofSnakes Actually, if you want to get around this problem of the fact that he does not even use the same numbers, set him down not in ancient Greece but somewhere in the Arab world during the heydays of the arabic civilization, like anywhere in the Fertile Crescent. en.wikipedia.org/wiki/Fertile_Crescent The Middle East is full of rich and interesting history, you'll have a buffet of places and time periods to plonk him into. Do remember that the Middle East was as much a focal point of the world in regards to higher learning as was Egypt, Rome, Greece and China. $\endgroup$ – MichaelK Aug 10 '16 at 10:05
  • $\begingroup$ Well I've already asked this question, and If I make another question just like this but different location ... then it will probably get downvotes lol. So im gonna stick with Ancient Greece $\endgroup$ – King of Snakes Aug 10 '16 at 10:07
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    $\begingroup$ Mathematics had a different cultural role. Mathematicians were a kind of soothsayer. Pythagoras, he of the famous triangle, was a cult leader. He and his followers were into unusual stuff. Mysticism, vegetarianism, & politics. Even if magic enables communication, there will be major hurdles to get over. $\endgroup$ – a4android Aug 10 '16 at 12:54
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Each and every child who becomes a mathematician goes from no math to modern math in a period of about twenty-five years. A modern mathematician could take someone through the same process without much difficulty.

The harder problem would be to get students. You might want to start with the most advanced people of the day, but they might have the most trouble relearning the basics. Or more precisely, being convinced that learning the basics in the "new" way is valuable and more "right" than the way that they know.

One of the common jobs for scholars like Socrates, Plato, and Aristotle was tutoring the top youth of the day. For example, Socrates taught Plato who taught Aristotle who taught Alexander (the Great). If people aren't asking too many questions, your protagonist might find such a job. Consider Croton in Southern Italy (Magna Graecia). Pythagoras died around 496 BCE. A visiting scholar might well show up then or a few years later looking for him.

Athens is another possibility, although we have little knowledge of the scholars there in that period. Even Socrates, who came later, is someone about whom we know little directly. Socrates is mostly known through the writings of his student, Plato.

What you really want is to control the entire educations of a group of students. A group because you won't know who will be really important. And the entire educations because you want to control even the small aspects which might otherwise be incorrect. For example, a number system with zero is important but not what the people then would know. Or we normally prove the Pythagorean theorem with similar triangles in modern mathematics rather than with geometric squares.

The greater problem is that better mathematics won't fix the Dark Ages. Archimedes was very close to calculus with his method of exhaustion. The reason that it took two thousand years to actually get calculus wasn't because it was that difficult. It was that they turned away from scholarship. If you don't fix that, they'll just lose more advanced mathematics in the descent.

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