### How far back does mathematical understanding go?

*This is my answer to the same question posed on the WorldBuilding Stack Exchange. It is therefore licenced under CC-BY-SA 3.0.*

# Question

How far could a mathematician go back in time, and have to spend as less time as possible in relearning stuff?

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.

# Answer

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.