How does 2020 have complicated technology, if most people don’t know how it works? One big part is specialization: across the world, quite a few people do know what bicycles look like.

I find this example pretty uncompelling. While I probably couldn't draw you a bicycle if you gave me 5 minutes of time (which is what the experiments I've seen are doing), I am very confident I could understand and draw you a bicycle if you gave me a day or two of time in which I could experiment and propagate all the various disparate fact I know about bicycles into my beliefs of what they actually look like. I just don't think my ability to draw you a diagram in five minutes is particularly predictive of my ability to reconstruct a relevant technology given substantially more time and feedback.

I agree with 1 and 2 to some extent, but still hold to my original opinions about ease of world takeover.

History has a few precedents for this; in particular, the conquistadors were precedents for what can happen when a few people from a more technologically advanced civilization find themselves alone inside a gigantic but technologically less advanced civilization. Some anecdotes: Cortez gave instructions to his native allies for how to make better spears, and they were able to produce lots of them in time for the final battles. Cortez brought with him a shipbuilder named Martin Lopez. Lopez knew enough about shipbuilding that under his direction the natives of Tlaxcala were able to construct a fleet of ships on Lake Texcoco, with which to assault Tenochtitlan. To be fair, he did have a supply of the most important components, cannibalized from the ships they had used to get to Mexico. But still. Ships are much more complicated than bicycles, and manufacturing them isn't as simple as bolting bits of wood together either. They also built a trebuchet during the siege (though embarrassingly they screwed up the first shot, launching the stone straight up into the air and down on themselves!). They also built a mobile fortress thing (a wooden tank?) earlier to escape the city, IIRC. They also made their own gunpowder from local materials. There are probably other things too that I'm forgetting.

Here are some more thoughts:

--I think probably not everyone could do this; having the right skills and preparation would be important. I think the whole "connecticut yankee in king arthur's court" thing is plausible, but only for the right sort of connecticut yankee, who prepared well beforehand!

--Like I said, I think the farther you go back in history, the harder it would be. However, this is not primarily for the reason you mention (that to get up to 21st century tech, you'd need to gradually work your way up through many iterations of gradually more advanced artefacts and production methods.) It's entirely true that you'd need to gradually work your way up like that; however, you don't need to go nearly as far as the 21st century in order to take over the world. All you need to do is go slightly farther than the local tech level, and then you can take over the local region, and then you can use that region's military to take over as much of the world as can be reached using whatever technology is available at the time. (Assuming you have the political and social acumen, of course--like I said, you need to be a particular sort of person with particular skills.) The primary reason it would be harder to take over the world the farther back you go is that, given the lower tech level, it would be harder to reach the whole world in time. (If we loosen the time constraints, and give you say 200 years to take over the whole world, then maybe you could do it even in the Stone Age. But if instead you were sent back to 1700, you'd only need 20 years or so, maybe less. And by you I don't mean you, or me, but rather someone from our time with the right skills.

--I agree that a single person sent back in time would have a lot more difficulty than Cortez did. Having a few hundred men + their equipment is a big deal; it's two order of magnitude better than just yourself. Since you need to scale up to the whole world, you need to cross, like, eight orders of magnitude, and starting with 2 already means you are a quarter of the way there. Less abstractly, a single person would have significantly less knowledge, less equipment, and less credibility/impressiveness, while being much more vulnerable, than a group of a few hundred.

--I agree that lots (most?) knowledge is lost over time, rather than stored somewhere. We still don't know how they made Damascus steel or Greek fire, etc. However, for taking over the world you don't need to know everything, just some things--enough to give you an advantage. Moroever, as Habryka said, it's in general much easier to rediscover something than to discover it. If you already know something is possible, you are 90% of the way there. Then if you remember what it looked like and roughly how it worked, that's 99%. (And again, this doesn't need to be true for everything, only for some things.)

Mathematics taught at the high school level today is beyond what could be done in 1200. Several caveats:

• A random adult from a 1st world country doesn't necessarily recall their high school mathematics.
• I'm sure 1200-era mathematicians had a lot of specialized knowledge not conveyed by modern high school. For a small example, they would probably be better at Euclid-style proofs than high school graduates. For a larger example, I believe mathematicians in the middle ages were expected to know a lot about astrology, as this was part of how they made money (not sure about 1200 exactly). 

However, it seems possible that a smart person from today could make a rather good career as a mathematician in 1200, by reconstructing as much of modern mathematics as they could manage.

Of particular interest would be the theory of computation, because the construction of a mechanical computer might be accomplished much earlier -- although, the construction of suitable clockwork would be required.

 Other sciences would also be of interest, but advances would in general be more difficult to prove or convincingly establish. For example, you could accelerate science a little by writing a book about the cell theory, evolutionary theory, atomic theory, etc. But those theories may not be accepted within your lifetime.

One could write about probability theory along with the scientific method, but I don't know to what extent that would fuel an early scientific revolution -- to what extent was the scientific revolution bottlenecked by the ideas, vs the fertile ground of the time period?

Overall, I think the model you present in the post is most true of technology, less true of science, and least true of mathematics. Mathematics most fits the picture of:

1. Artifacts are not central at all; it's mostly about actual knowledge in heads, and written in books.
2. The knowledge is not very context-sensitive; EG it does not depend on what tools are around.

Science fits this picture surprisingly less than mathematics; implements are very important, EG microscopes and telescopes.

Doesn't look like anyone has mentioned Ryan North's time travel guide yet. Key points include pasteurization, pendulum clocks, and cowpox.

If I travelled back to 1200, I wonder if I would rather be in the merchant class than the scientific class. Merchants have tighter feedback loops (and I wouldn't have to sit through as much astrology). How fast did financial innovations spread? Looks like both insurance and double-entry bookkeeping took a few centuries. And now that I think of it, haven't valuable but illegible financial services been a major factor leading to persecution of lots of groups, most notably Jews? Damn, so much for feedback loops. I guess Crab Mentality and unfree markets probably put a limit on how hard you can exploit your advanced financial skills.

There's a section on building bicycles in the middle ages in "a Connecticut Yankee in King Arthur's Court" which plays a similar thought experiment and which you might enjoy. 

But to answer this question seriously: there is a form of Moore's law that has been in place at least since the late middle ages, and probably for longer, coming from the fact that any economic model that takes into account both goods and capital (the machines and infrastructure to build more and better goods) has a compounding growth component. This is true even if you ignore cultural and scientific advances and only look at physical capital.

The first steel was ridiculously costly to make, and a commoner's life savings would buy in the ballpark of a kilogram. https://acoup.blog/2020/11/06/collections-iron-how-did-they-make-it-addendum-crucible-steel-and-cast-iron/. But to produce steel more cheaply you need several tons of "manually" created steel to build a bessemer converter. The components for this giant industrial machine have to either be built by hand (again very expensive) or be precisely machined in factories, in machines. Those machines are built out of precisely put together steel (or less reliable iron) components, which need either simpler factories or ridiculously costly manual processes to make. And so on. 

A simplistic but successful way to model this is to assume that the cost of creating new capital (factories, plows, etc.) has a dependence on existing capital. This results in compounding growth. If you ignore things like population growth and accumulation of knowledge, the growth will not quite be exponential, but definitely very non-linear. 

Not only is setting this up expensive, it needs to be set up in a society where most of the population is subsistence farming, so there is very little surplus to put into building better capital, meaning that take-off would be slow. Furthermore, in addition to building up the industrial production line you would have to re-build the supply chain, since many industrial materials are geographically isolated and need to be imported.

So even if you somehow guaranteed that everyone in a middle-age society had a modern education and had a Wikipedia-level level of knowledge about the modern production line, guaranteed open trade routes, eliminated wars, etc., my guess is that it would still take multiple generations to get to modern technological levels (though certainly having modern knowledge and cultural institutions would speed things up by a lot). I seem to remember reading about an estimate of this somewhere (in the contexts of how long it would take survivors to rebuild after an apocalyptic collapse of society). I don't remember where this was and the answer they came up with, but my very un-informed range of guesses would be something between 3-10 generations assuming optimal conditions.

We are using knowledge-storage and reference devices to expand our long-term "memory." One concrete example is the list of long mathematical proofs, which I am certain nobody has memorized. Probably the details of large computer programs as well. The code for the human genome project was 3300 billion lines of code. Nobody has all that memorized, and some of the people who wrote it are probably dead.

One of humanity's great technical arts is being able to store knowledge for later retrieval, and build mechanisms that keep working when nobody can explain exactly how. We'd have lost all that accumulated knowledge as well as the physical artifacts.

Fortunately, one of the great secrets of technical creation is that a mechanism can be built by tinkering without knowing exactly how they all fit together. My guess is that a lot of early engineering comes together without a grand plan, and the accumulation and refinement of tinkering eventually results in something that functions. Knowing what can be done, and having a lot of shared cultural knowledge about how to tinker productively, would give us a huge head start.

But unfortunately we'd all die of starvation and disease before we managed to get very far!

Okay, but what is a bicycle?  

Is this?  
https://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Michauxjun.jpg/682px-Michauxjun.jpg

Is this?
https://en.wikipedia.org/wiki/File:1886_Starley_%27Rover%27_Safety_Cycle_British_Motor_Museum_09-2016_(29928044262).jpg

What about this?
https://en.wikipedia.org/wiki/File:Draisine_or_Laufmaschine,_around_1820._Archetype_of_the_Bicycle._Pic_01.jpg

Except for the pneumatic tire, about which I really have no idea, it seems pretty likely that a blacksmith and/or clockmaker in the past <i>could</i> make a bicycle.  But is it economically feasible?  How would the bicycle work on the streets/roads of the time?  Can you make a cross-country type of bicycle out of wood (i.e. less expensive but able to deal with likely terrains)?

What would people use this bicycle for?  Is it more useful than a cart?  Than a horse?  Would someone who could get around in a coach and four (i.e. rich enough to buy something that may be expensive, especially if made mostly of metal) choose to use a bicycle?

I feel like your question about physical artifacts is on the right track, but that it needs to be expanded quite a bit.  It's not just the physical artifacts for making things, but the environment in which things will be used, and that environment includes both physical artifacts and economic situation and mental memes and ways of thinking and culture and etc.

I haven't read it, but from what I've heard, the 1632 series by Eric Flint and collaborators looks into some of these questions, and at least tries to be realistic about it.  It is fiction, but I think it's the kind where some thought has gone into how things work.