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Shamelessly inspired by this question (sorry), but I got this question after Michael gave a answer - my mathematician would essentially have to relearn maths, as mathematics in Ancient Greece is completely different to what we have now.

So the question that came to me was - how far could a mathematician go back in time, and have to spend as less time as possible in relearning stuff? If he goes back in time for this question, he has to relearn a lot of things.

Background: The main character has realised that he can travel back in time voluntarily, and he wishes to travel back in time to a time-period where he can participate in the beginning of maths, but without relearning as much as possible.

Magic: To make things clear, I'll add this in. The magic allows him to communicate in the time-periods language easily. He can understand it effortlessly, and it stops the other people 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. (It stops them from digging to deeply, even if he does not know what they think is common sense.) They also have given him food and a place to stay.

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    $\begingroup$ 1 2 3 many. caveman maths? $\endgroup$ – Donald Hobson Aug 10 '16 at 10:53
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    $\begingroup$ @DonaldHobson : They could barely speak, let alone do maths lol. $\endgroup$ – King of Snakes Aug 10 '16 at 10:59
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    $\begingroup$ You postulate that "The magic allows him to communicate in the time-periods language easily." Well, if so, then the magic allows a very difficult thing - translation of human language with all its inevitable dependence on context and shared culture - but does not allow a basically rather easy thing: picking up a new terminology to describe what are, by definition, universal truths. Maths is maths. Space aliens could understand it. The only difference between mathematics as known to the ancient Greeks and mathematics now is that back then there was less to learn. $\endgroup$ – Lostinfrance Aug 10 '16 at 12:04
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    $\begingroup$ @Lostinfrance He could say that the mathematician is accompanied to the destination by an anthropologist who is very familiar with the language of the time, but happens to be unfamiliar with their mathematics. $\endgroup$ – Superbest Aug 10 '16 at 22:38
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    $\begingroup$ One thing to consider is that going very far back in time may not be the best use of his time. If I was your mathematician, I would probably travel to the 18th century, even if I could understand perfectly older maths. Because I think it's where I could have the most impact. $\endgroup$ – Tryss Aug 11 '16 at 1:52

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It strongly depends which area of maths you're talking about.

  • Category theory is basically new, so before the 1950s or so, it just didn't exist in anything like its modern form.
  • Combinatorics has been around for a long time, but before Erdös it looked very different.
  • Before Newton and Leibniz, the notion of calculus wasn't very clear, and its notation would make it very difficult for us modern-day people to work with.
  • Before Cauchy, they didn't really have what we would refer to as a "rigorous" foundation of analysis, and the relevant language changed substantially since Cauchy to take into account the new approach to rigour.
  • There was a time, even some point after the Renaissance IIRC, when mathematicians were still not really sold on this whole "rigour" thing, and the art of defining things crisply so as to deduce (nearly) incontrovertible stuff about them. The entire mindset of mathematics is different now.

A first-year undergraduate going back before Newton could, if their ideas were taken seriously, revolutionise multiple areas of maths simply because we now know (and take for granted) the correct ways of thinking about certain fields of study. Conversely, of course, the first-year undergraduate would have a hard time following the maths of the day, because the technical language and frameworks are so strongly unfamiliar. The only frameworks I can think of which haven't changed much post-Renaissance are Euclidean geometry and arithmetic, though of course geometry and number theory have advanced substantially since then.

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  • $\begingroup$ If my answer doesn't have atleast 5 votes more than yours (by tomorrow), then i will accept this :) $\endgroup$ – King of Snakes Aug 12 '16 at 9:31
  • $\begingroup$ Funny, after I said this, I got 3 downvotes. Anyways, your answer is accepted ^_^ $\endgroup$ – King of Snakes Aug 13 '16 at 3:39
  • $\begingroup$ I was out yesterday so couldn't have coordinated that even if I wanted to :P $\endgroup$ – Patrick Stevens Aug 13 '16 at 7:59
  • $\begingroup$ Eh its fine. I have 42 upvotes and only 3 downvotes, not a problem anyways. Your answer is good 👍 mines was because of a comment I made, and I decided to post it as a answer $\endgroup$ – King of Snakes Aug 13 '16 at 10:24
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Technically, if he went all the way back to the beginning (where it was just starting), he won't have to relearn anything (as nothing had been invented). He can simply use his own knowledge, and it will become the norm - others will learn whatever he writes, and he won't have to learn anything new.

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    $\begingroup$ Technically, if he went all the way back to the beginning, he could teach them as much as he wanted -- and unless he's teaching them the basics they'll probably forget everything again. Higher mathematics only make sense in conjunction with sciences like Astronomy, Physics or Chemistry, or then engineering as an applied science. So he'd basically have to bootstrap those too, and all the while take care that he doesn't accidentally insult any believes in any Gods or other religions. I really don't think that he'd be able to really make a difference there... $\endgroup$ – subrunner Aug 10 '16 at 12:13
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    $\begingroup$ @KingOfSnakes - no, he will have written a bunch of dangerous gibberish, and be burned at the stake, on a pyre of his own books. $\endgroup$ – AndreiROM Aug 10 '16 at 12:36
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    $\begingroup$ @KingofSnakes - and if he keeps quiet on those other topics, his maths will have no application at all. How many people today are willing to listen to a Math lecture simply for the brain exercise and not for anything with purpose? $\endgroup$ – subrunner Aug 10 '16 at 12:43
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    $\begingroup$ @AndreiROM : Show us one single example of someone being burned at the stake solely for their mathematical works, or retract your comment! Come on, is this Worldbuilding.se or some random clickbait site gathering nonsense "factoids"?? $\endgroup$ – vsz Aug 10 '16 at 17:37
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    $\begingroup$ I dispute this, simply because it considers maths as being a concept in vacu. Maths is embedded heavily in all parts of civilization for a reason. The ability to count days led to the calendar and consequently farming. Going back to the "beginning" would mean going back before "language" and "civilization". $\endgroup$ – Aron Aug 11 '16 at 5:21
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I'd say that math itself would be understandable to a modern mathematician (worth the name) in any period.

What might be a problem is terminology and convention. If your magic extends to not only translating general language, but also makes an immediate translation from I + I = II to 1 + 1 = 2 the protagonist won't have much of a problem. Reinventing classical math from it's origin is generally taught as an excercise in pregraduate math classes.

The thing about math (and logic) after all is that a thousand year old proved theorem is still as valid today. This is in stark contrast to other sciences where proofs are merely "until further notice".

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    $\begingroup$ Thats not really true. The symbols for addition and subtraction did not exist at one point. That makes it really hard to understand maths for a modern mathematician. $\endgroup$ – King of Snakes Aug 10 '16 at 11:09
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    $\begingroup$ Nah... The symbols might not have existed, but the concept of addition certainly did. "Og have one rock. Og get another rock. Og now have two rocks." So it's just really a matter of communication/translation. Since "magic" handles translation it's just a matter of how much magic to apply. $\endgroup$ – Guran Aug 10 '16 at 11:34
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    $\begingroup$ @KingofSnakes The symbol may have not existed, but then again - books weren't as commonplace either, so there wouldn't even be a place where you can note the absence. $\endgroup$ – Ordous Aug 10 '16 at 11:40
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    $\begingroup$ And if the concept exists, it's easy enough for you to just say "Look, let's put two lines like this + to say we're adding them together, and one line like this - to say we're subtracting them". "Oh" says the Caveman, "That's a neat idea". $\endgroup$ – Jon Story Aug 10 '16 at 15:08
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    $\begingroup$ Understandable, maybe, but with great difficulty. The Babylonians did their math in base 60, while the Egyptians didn't have fractions as we know them -- instead, they used sums of reciprocals (eg. what we'd write as "4/5" would be written as "1/4 + 1/2 + 1/20"). $\endgroup$ – Mark Aug 10 '16 at 21:22
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Serious scholars of mathematics tend to be educated in the history of mathematics and mathematical notation as well. As such, the question is complicated by the fact that the time traveling mathematician is not entirely helpless, he has advance knowledge.

As far back as the Renaissance, there would be hardly any difficulty. They pretty much did math like us. For example, Descartes was one of the originators of denoting variables with $x$ and $y$. Of course, if you are interested specifically in discipline X, and you arrive just when X is being invented, things will be in flux, and there will be multiple competing notations that confuse matters somewhat. Calculus is the classical example: Leibniz and Newton both independently discovered it; Newton favored $\dot x$ and $\ddot x$, while Leibniz used $dx/dy$ and $d^2 x/ dy^2$. I don't recall now how $f'(x)$ and $f''(x)$ fits into this.

Calculus is an interesting exmaple for another reason, though: Typically common calculus textbooks will include this little story about the discovery of calculus, and many professors enjoy retelling the tale when teaching Calculus I. This nicely illustrates my point about mathematicians having a head start due to their knowledge of math history.

In Middle Ages things get slightly funky: This is when Arabic numerals were brought to Europe (in India and Arabia they had been in use for centuries), so before that time, things might start out a bit confusing to a modern mathematician. However, the alternatives like Roman numerals or Mayan numbers are really not complicated (I personally find Mayan numbers more elegant than ours), and anyone who is interested in math can easily learn them in a few hours, if they don't know them already. Mathematical thinking also seems comprehensible, going by examples like Fibonacci.

In Dark Ages Europe it may be difficult to discuss math due to the odd attitudes people may have towards learning in general, but assuming you do find a cooperative scholar, the math itself should be easy. Of course, concepts such as variables or equations (as we know them) had not been established yet, so discussion may be a bit cumbersome ("thus we find that number which is thrice that number of which the square is a tenth of the number which...") until the interlocutors have been "enlightened" with modern algebra.

I'm not sure what state Roman mathematics was in, but surely they would be intimately familiar with the heritage they received from the Greeks. Discussing math with the Greeks might be positively pleasant: The Ancient Greeks pioneered so much of our philosophy, logic, math and geometry, and their writings show an exceptional clarity of reason. Not having knowledge of modern notation, perhaps the way in which they talk about math would be a bit strange, but the essence of their thoughts should be familiar, given that to this day we have students of mathematics retrace their footsteps when learning. Incidentally, Greek mathematics is very readable today. I believe Euclid's elements was commonly used as a textbook of geometry up in many places up until a couple decades ago.

Further back, there are some mathematical manuals from Egypt that give a fascinating glimpse of truly ancient mathematics. The Rhind papyrus appears to be some kind of math textbook, complete with example problems and reference tables, very similar to textbooks we use today. Going through the problems, many of them are very obviously similar to elementary math we learn in school: Calculations of volume, work, fractions, and so forth. There is also a great example of where the difficulty might be: Egyptians, apparently, liked to represent non-whole numbers as sums of fractions with quotient 1, eg. $3+1/6+1/7+1/13$. This bizarre notation can lead to some very confusing numbers (they got confused too, and they made tables of fractions to cope with it) for a modern reader. Their units are also quite strange. But once you start systematically writing down the fractions and equation in a neat manner, as should be instinctive after a modern education in math, everything is very clear.

There seems to have been a similar situation in ancient China. Taking as example their approach to calculating areas and circumferences of circles we can see where the biggest issues might be: These people had not yet standardised $\pi$, nor did they fully understand its nature, so familiar formulas like $\pi r^2$ become $730/232 \cdot r \cdot r$, or $2\cdot \sqrt 10 \cdot r$. At first, these weird constants might throw you off (they are approximations of $\pi$ that haven't been needed for centuries) but it's not something that anyone educated in math would not know, or couldn't figure out. I would guess that in other cultures, such as ancient Amerindians, the math would likewise be straightforward besides a few trivial oddities.

In general, math is different from language. Unlike languages, all human mathematics appears to converge to some universal (or possibly metaphysical) truth -- it is no wonder the Greeks venerated it religiously. Whereas language is for the most part arbitrary and varies greatly between cultures, math is convergent: "Great minds think alike". Mathematicians are in addition taught how to analyse and dissect problems into math - while typically people are not taught research linguistics in school. So I would say that besides communication itself, and strange archaic notations, math from any time period past would not pose any problems to a modern, educated person besides obviously the issue of them not having discovered certain useful things yet (but those can easily be taught). It is interesting that in Contact, the aliens begin their message with elementary math, as this ensures that it will be readily comprehensible. It is a very relevant comment on the relation of humanity with mathematics.

Even the math of the near future would likewise be manageable; but once you go thousands of years into the future, it may turn out that a massive revolution in math has completely changed the way people think about it (not to mention possible impact of ubiquitous computation technology and AI). By the way, historically mental/manual arithmetic was a vital skill for many mathematicians, but modern scholars are blissfully freed of this burden by calculators. A time traveler may have trouble being taken seriously at first, due to his poor calculation and rote memorization skills, but this barrier should be easy to overcome once the merit of his knowledge is demonstrated.

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    $\begingroup$ With regards as to the cumbersomeness of discussing mathematics, it is precisely the reason why Al-Khawarizmi wrote the famous algebra book (he didn't technically invent it but was the first to comprehensively document it). It turns out that Islamic inheritance rules are complicated enough as to require the solution of multiple simultaneous quadratic equations. Thus having a shorthand (algebra) to manipulate the meanings of the calculations make it easier to figure out who inherited what percentage of the estate. $\endgroup$ – slebetman Aug 11 '16 at 6:10
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Speaking as a mathematician, I would say (since you've magically removed the language barrier) any time will do.

Most every math department has an undergraduate course about the history of math. In my experience, these courses involve solving problems in the way they would be solved from whatever period is being covered. This involves grappling with lack of notation and modern tools. Even unskilled, average students can quickly adapt their modern knowledge to solve problems in old ways - matching (and sometimes surpassing) the experts of that time.

In the end, concepts are important. Notations are mere tools. An expert craftsman can still make amazing works of art with really lousy tools. A skilled violinist can use a toy violin to make beautiful music - don't get me wrong - better tools yield better results, but the player is more important than what is being played.

If I could go back in time to any period 100 or more years ago, I could become the world's greatest mathematician. Easily.

Conversely, if Newton or Gauss or Archimedes were raised from the dead (again removing the language barrier), they could be dominant mathematicians within a few years of catching up. However, I'm not sure they could do the same in the realm of Physics.

The raw skillset making one a great mathematician has not changed since the beginning of time. Mathematics is timeless.

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First of all a lot of the answers seem to confuse semantically mathematics - the scientific model build by mathematicians - with the reality that mathematics tries to describe. Mathematics, the model, is all about the notations being used, the notations are what allows us to work collaboratively and create a model which we use to generalize the world, whilst the underlying reality it attempts to model in an extremely abstract and simplified way has not. The exact same is true for any other scientific discipline.

Either way, that means that if you go back far enough you won't understand a thing about mathematics as the model simply will look alien to you. Take for example a look at this beautifully clear representation of polynomials in 150 AD (ignore the brackets at the beginning and end):

enter image description here

Source: http://www.stephenwolfram.com/publications/mathematical-notation-past-future/

Or lets move forward to 1600 and see how François Viète did it by that point

enter image description here

Source: http://www.stephenwolfram.com/publications/mathematical-notation-past-future/

At least that looks somewhat comprehensible, but it will still be quite the puzzle to figure out. So

In the early to mid-1600s there was kind of revolution in math notation, and things very quickly started looking quite modern. Square root signs got invented: previously Rx—the symbol we use now for medical prescriptions—was what was usually used. And generally algebraic notation as we know it today got established.

Source: http://www.stephenwolfram.com/publications/mathematical-notation-past-future/

So that's I think the most reasonable answer: before 1600 your mathematician will have to learn the mathematical model used by the people he wishes to interact with. Around 1600 he can specifically chose to interact with the people who establish modern mathematical notation, e.g. he could go out of his way to visit William Oughtred. Still though, this will still take a whole lot of trouble as a lot of notations were put forward in that time. So after around 1700 or 1800 he could join the mathematical world without learning an entirely new mathematical model.

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Probably back to beginning of modern mathematics but maybe as far back as the renaissance, reading up on the history of mathematics the renaissance seems to be the beginnings of mathematics as we understand it today but the science of it really came to the fore in the 19th century.

It depends on his field of study within mathematics as well, looking at the list of mathematical achievements if he's a calculus guy he could go back to Newton's time.

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  • $\begingroup$ The funny thing is, if he goes all the way back to the very beginning - he won't have to relearn anything (because almost nothing exists) . He can invent his own stuff, and that will become the norm XD $\endgroup$ – King of Snakes Aug 10 '16 at 10:21
  • $\begingroup$ @KingofSnakes That's very true, take a small tribe and bring them into an age of modern mathematical enlightenment, causing them to leapfrog the rest of humanity. The trouble is dropping into pre-existing establishments will be having to provide proof. $\endgroup$ – Chris J Aug 10 '16 at 10:23
  • $\begingroup$ True. But considering that he is inventing maths itself, he can do the basic stuff himself (providing proofs). Also by then he will have a reputation as a legendary genius, so others won't bug him about providing proofs ☺ If he says "I have a feeling this is correct", others will believe it lol (because he has Created maths itself and knows far more) $\endgroup$ – King of Snakes Aug 10 '16 at 10:25
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    $\begingroup$ @King of Snakes: "I have discovered a truly marvellous proof of this, which this comment box is too narrow to contain." :-) $\endgroup$ – Edheldil Aug 10 '16 at 12:14
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    $\begingroup$ @Edheldil : Smaller fonts? :) $\endgroup$ – King of Snakes Aug 10 '16 at 15:16
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Saw this on the "hot" list, and thought I'd just give one more angle.

First of all, the other answers highlight very important points: Terminology and common language would be significant barriers, particularly in the recent past, where the maths were advanced but the terms and common languages were very different from now. (EDIT I see now in the original question that this can be ignored. Moving on... )

Another thing that would be difficult for a modern mathematician going back would be levels of rigor in mathematical communication. At different periods of time, less rigorous mathematical investigations and proofs were well accepted. Although any good mathematician would recognize the results of Newton's calculus, s/he may not be able to follow along at the speed of intuition or be willing to take certain results as "obvious".

The point I really wanted to highlight, though, was that at different periods of math history, very different fields were the subject of focus, and modern mathematicians do NOT know all the things that historical mathematicians knew, and especially not the common tools used to attack problems. For example, go back a hundred years and geometers knew specific details about catalogues of curves and surfaces that most modern geometers do not and are not particularly interested in. I would think that, just as a modern algebraic topologist has the mental capacity to learn another maths discipline like geometric analysis but wouldn't understand past the first five minutes of a specialist seminar on the subject, a modern mathematician going a few hundred years into the past would find that s/he would need years of study to get up to speed on the popular areas of research of the time.

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Maths will be always understandable to us (assuming we could also understand spoken language). If you see the movie Agorà, and studied some simple math and physics at high school you will understand all the statements made by Ippazia, you will say. Damn that's an ellipse! That's so easy! Why you can't understand that?

What we do not understand now is that even simple things we now assume "known", took hundreds if not thousands years to be discovered, so we will be able to understand theory math (even useless math like summing up Roman numbers), they won't be able to understand ours math.

The chance that somewhere someone invented a theorem (and proved it) that get lost and is no longer known today becomes smaller as we go back in time, so somewhere maybe 2000 years ago someone invented a rudimentary math system that today's we not know and that would be useful, but chances that was happened are really small, and anyway we will be able to understand that easily.

We will understand everything, but we will see how much value people of passed times gave to even simple equations that now (hopefully) everyone is able to solve.

Most problems for us would be about "notation".

Something simple like

3+x = 7   =>   x = 4

Was in reality written as

If we add a unknown quantity to three and obtain seven, then the unkown quantity was four.

But once we translate stuff into our notation, we are able to do maths very easy, even better, our math notation would probably spread quickly because is very usefull and probably the only real true difference with ancient times. I would claim that modern math notation and Turing machines are the 2 most important conquers after invention of the Zero.

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A first year university course in Calculus and Linear Algebra basically gets you to the early 19th century - with Calculus in the 18th and row reduction for LinAlg in the 19th. But the notation is confusing and the approach is different. As Patrick Stevens notes, before Cauchy, people did Calculus by operating with "infinitesimals" - with sometimes horrendous results. A time travelling physicist would do better than a mathematician in that regard!

I have read textbooks from the early XXth century that looked familiar and could pass for modern works (although expected a higher standard from the students), but anything prior to that would take a learning curve from your Time Traveller. Differences in notation are not trivial, since Mathematics basically lives within its symbolic representation, and differences in notation typically reflect differences in approach.

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    $\begingroup$ Not sure that just a calculus course would get you up to speed, for example the only differential equation technique you learn in calculus is separation of variables. You also don't learn to do much with complex numbers, etc $\endgroup$ – Ovi Aug 11 '16 at 0:40

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