If you traveled into the past, say medieval times, without any modern devices with you and without any books, only you and your mind, would you be able to advance the human race? What invention would you be able to build using only your knowledge?
I tried this mind experiment and wasn’t impressed with my results. I mean maybe I would be able to build an engine of some kind or I would try to make a gunpowder (although I wouldn’t know where to get potassium nitrate (I know now, I checked)).
Almost any school child could introduce them to decimal numbers, place value, and zero. That’s a big advance, that wasn’t made until the late Renaissance in the West.
I’d say math in general would be most people’s biggest contribution, without special tools or equipment. That and personal hygiene and the germ theory of disease.
Although some handymen with special skill could probably go a long way with the tools and materials available in the middle ages. I just can’t think of how at the moment.
A lot of our modern advances are based on significant infrastructure that a single person couldn’t possibly reproduce in the past. So important ideas are our best bet.
I think it depends a lot of the circumstances – particularly whether or not you end up in a situation where you’re be able to use future knowledge safely, and whether anyone present would believe you. The Church at the organized end, and superstition at the disorganized one, present a serious danger to anybody demonstrating unusual ideas in the Middle Ages.
I think the best case scenario would be something like popping into a Royal Society meeting a few hundred years later. Enough of the groundwork of science will have been laid that they’d listen to you, and you wouldn’t be nearly so “on the spot” for results – these are people for whom you can say “this is how something works, but I’m not sure of the details” and they could run with it.
In the former circumstance, I doubt I could get anyone to listen to me, and any change I made would end up local and potentially lost a few years later.
In the latter, I could provide things like the Bohr model of the atom, many of the basics of chemistry, cellular biology, germ theory, astronomy, and evolution, and they’d have the ability to check my stories for validity (In many cases I’d only be advancing them by a few hundred years, anyway). More intriguingly, I’m pretty sure that I could push computer science into the past significantly – by 1500, they already had looms that were “programmable” by what amount to punch cards to produce various patterns. Even without things like the transistor, I think I could show them what (initially mechanical) computers could do for them. I’d be a lot less capable of things like steam engines and electronics, but making people aware that it was possible to create such things would go a long way in the right company.
The OP might enoy Michael Crieghton’s (sp) book Timeline, which takes a swing at this.
Hygiene, definitely. Forget the lancing wounds, leeches, or bloodletting. Make people wash themselves often, cook food thoroughly before eating, boil your water before drinking, that kind of stuff. I know penicillin comes from mold but I have no idea how to make it other than leave out some stale bread and rub the mold onto a wound. But the other stuff I can totally do.
Agreed, this would be the first thing to start with. Then teach them to control their rat population and get rid of those pesky fleas that are carrying the plague around.
I think that I could build a simple steam engine, develop gun powder and come up with a movable type printing press if I had the resources. Perhaps I could first invent chocolate covered donuts and use the profits to fiance my imagination. I could then also introduce them to the concept of franchises and mass production.
Food was pretty thoroughly cooked and people didn’t drink water when they could get beer, ale, or wine (which was often served watered). They even made breakfast beer, which had a lower alchohol content. Water, especially cold (unboiled) water, could unbalance your humors, leading to illness.
They won’t do it. There was a meme in the middle ages that cleanliness amounted to paying too much attention to your current life instead of your future one in heaven. Our “cleanliness is next to godliness” came along later. The church would be a dead hand on any real innovation. If you could have just got the idea across that one way to approach any question is by experiment you would have advanced civilization immensely. But I don’t think you could have.
When Fibonacci (the same) introduced decimal notation into Europe, it was rejected by many as being “too hard”, compared to good old roman numerals. Of course it did eventually catch on, but that wasn’t medieval times by then.
I could certainly build a simple galvanic cell and demonstrate it, but it would doubtless be denounced as “work of the devil”. If I could talk to a blacksmith (how’s your Anglo-Saxon?), I could doubtless fashion a primitive generator.
Mathematics? I could certainly teach them a lot of mathematics, but who would care how to, say, solve a quartic equation? First I would have to get them to accept 0, then negative numbers, then complex numbers. Probably impossible. It requires a certain mindset to think these things are interesting.
“Don’t bleed? You must be out of your mind. That is the standard of care”. Well, not in those words, they are the words of the present day opponents of evidence-based medicine. You think we are that much better than our ancestors.
I could make a hula hoop.
I might be able to fashion some kind of bicycle together, although it wouldn’t be the most comfortable of rides.
That was my first thought after reading the OP: would I even be able to communicate with people, even if my trip to the past landed me in the middle of London? Depends on whether I arrived in 1200 or 1600, I’d expect.
Let’s say you learned Latin before (assuming Latin we teach today is the same Latin that was used back then).
Gunpowder’s indeed easy. Bat guano, sulphur, charcoal. Not sure about the proper proportions but that’s where empiricism comes into play :). But then again you’d have to hop pretty far back for it to be very novel, as there were already cannon by the 13th century. The more modern applications of it, like handguns, semi-auto and so on rely on machined precision they simply couldn’t have achieved (nor can I tell them how exactly).
Other than that, I could probably become a kickass storyteller once I get the language barrier down. I’ll tell them the one about Sir Luke the Sky-walker, earl of Tatawyne, and how he rode down the trenches of his evil father’s fortress to blow it up with a magic fireball given to him by the Sage Ken Owby :D. But really, the concept of fiction for fiction’s sake is quite recent, so maybe y’all would reap a wealth of narrative benefits from my early contributions in the field.
Pal me around with a bloke who knows the local music notation, and I’ll show up Mozart with my very own Stairway to Heaven in D, too :).
Crop rotation (using a patented Crop Rotator, natch) is another easy one, as are a number of simple applications for cogs, sprockets and levers - the Greeks knew all about 'em, but the knowledge died. I guess I could design some pretty kickass siege machinery. Or just a reasonably accurate clock, which some say was a huge contribution to the Industrial Revolution. Or, hey, speaking of which : steam engines. Easy enough concept to implement, or blow oneself up with. Which of course opens the door to steam-powered siege engines ! Can y’all say mid-14th SteamPunk ?
What else… I could agitate the masses with notions of the Fundamental and Inalienable Rights of Man, that’s something right ? A short-lived something, but still.
Steam engines, AC and DC and polyphase generators and motors, germ theory, heliocentricity and galaxies and meteors and comets, telescopes, microscopes, evolution, calculus and differential equations, vacuum tubes and resistors and capacitors and inductors and filters - I think I could describe all these things well enough to get them working and get some people able to use them. The bigger challenge might be to get the right people to start paying attention.
Were would you get strong enough magnets?
And don’t forget probability. It seems obvious to us now but even the basics weren’t developed until the 17th century.
Well, if I was properly attired, I’d have a pretty good shot at rewriting history. 
Careful with that; consider what happened at the first performance of The Rite of Spring. Radical musical advances can lead to rioting.
Shirt notwithstanding, I could probably manage powered flight without too much trouble–the biggest problem would be the power. At the very least, I could probably make a horse-towed glider.
I’d advance the human race the only way I knew how: by seducing a bunch of medieval chicks and making a bunch of medieval babies.
You’d likely die of smallpox, or any number of diseases we don’t have to fear anymore, but they were constantly exposed to.
Then you wouldn’t have any chance to improve anyone.
Even knowing the language and the customs I don’t think I’d be able to make any discernible impact because I wouldn’t have the connections to do so. So let’s say I knew how to make gun powder. I wouldn’t know anyone who would provide me with the material. I also wouldn’t know how to get the material on my own and I most likely wouldn’t be able to barter anything with a blacksmith for him to make a gun for me.
I’d probably die after a month of malnutrition or from drinking bad water or something.
Well, you should focus on contacting people that have chance of being educated - starting probably with clergy and noble men.