cousin_it:
From my experience teaching math to kids, I think it's actually more difficult to go from zero to (1) and (2) than to go from those to (4), because the hard part is learning how to think rigorously at all.
That sure depends on what we consider to be "zero"!
I do know some very smart people for whom (1)-(3) would be a breeze, but who couldn't prove a theorem if their lives depended on it. (In my experience, lots of such people can be found among programmers and engineers.) I have the impression that quite a few people on LW are in a not too dissimilar position, in the sense that they could easily harness their general intelligence to develop the right intuitions for solving problems of the sorts (1)-(3) reliably, but training themselves for formal math would be a much harder step.
Maybe I'm also biased due to my own position. I can easily pass the tests (1)-(3) (out of curiosity, I just tried writing a quine in C -- I thought of the basic idea in about 5 minutes, and it took me 10-15 min. more to sort out the mess with the escape characters). But although I had a decent knowledge of the basics of math foundations some years ago (to the point where I was proving theorems in exams in graduate courses), scraping the rust off of it to the point where I could constructively contribute to the discussions here would require a significant time investment (which I still hope to do as time permits).
There is no serious descussion of quantum mechanics (or physics in general) on LW. I'd be glad if there was.
Lots of discussions here touch on MWI and make MWI-related assumptions. While one can grasp the basic idea of MWI without knowing the actual math of the quantum theory, such knowledge is pretty pointless, since it basically involves taking a controversial view on pure faith. (I am familiar with the basics of QM, but I don't think my knowledge is still anywhere near the level where it would make sense to stick my head out with judgments about such things.)
By the way, there is an interesting ongoing physics discussion, just in case you missed it:
http://lesswrong.com/lw/2sl/the_irrationality_game/2qiu
While one can grasp the basic idea of MWI without knowing the actual math of the quantum theory, such knowledge is pretty pointless, since it basically involves taking a controversial view on pure faith.
Are you going to claim that you believe into AI going FOOM based on the actual math? Why would you care about how founded MWI is if you accept the basic premise of risk from AI to an extent that you donate to some institute with Singularity in its name when not even gravitational singularities are proven beyond the point that people would ground a movement around them...
I just got a PM with this question: "What would be the minimum intellectual investment necessary to be able to fruitfully take part in the discussion of decision theory on LW?" This is not the first time I've been asked that. Our new discussion section looks like the perfect place to post my answer:
1) Learn enough game theory to correctly find Nash equilibria in 2x2 games all by yourself.
2) Learn enough probability theory to correctly solve Monty Hall, Monty Fall, Monty Crawl all by yourself.
3) Learn enough programming to write a working quine (in any language of your choice) all by yourself.
4) Learn enough logic to correctly solve the closing puzzle from Eliezer's cartoon guide.
Then you're all set. Should take you a few days if you've studied math before, a few weeks if you haven't. No special texts needed beyond Wikipedia and Google.