(...and also permission from Eliezer, who has the right to veto this whole idea should he so desire.) 

In the hope of launching a collaborative project, I've set up Wordpress sites for translations of the Sequences into French, Italian, and Spanish; and to get things started, I've put up my own attempts at translating the first post, The Martial Art of Rationality. 

I'm looking for collaborators. At the very least, I'm hoping to find folks who would be willing to proofread my translations and help improve them -- in particular to help me purge them of the inevitable mistakes, non-native shibboleths, and typos with which they're bound to be infused. Beyond that, it would of course be nice to have people contribute translations of their own of entire posts. (I'll keep plugging away myself no doubt, but if I'm the only one, it will take a very long time to complete.)  

If you'd like to translate articles yourself into one of these languages, send me a PM so I can add you as an author on the appropriate blog(s). Proofreading my work can be done either privately or in comments here on LW; I'd probably prefer to reserve the comments sections of the actual sites for discussion of the posts' content (in the target languages), although this isn't an inflexible demand. 

Also, one would like to add more languages, of course... 

One must always invert.

- Carl Gustav Jacobi

I'm grateful to orthonormal for mentioning the following math problem, because it allowed me to have a significant confusion-dissolving insight (actually going on two, but I'll only discuss one in this post), as well as providing an example of how bad I am at math:

"[I]f you want to average 40 mph on a trip, and you averaged 20 mph for the first half of the route, how fast do you have to go on the second half of the route?" 

When I read this, my first thought was "Huh? If you spend an hour going 20 mph and then spend another hour going 60 mph, you've just gone 80 miles in 2 hours -- for an average speed of 40 mph, just as desired. So what do you people mean it's impossible?"

As you can see, my confusion resulted from interpreting "half of the route" to refer to the total time of the journey, rather than the total distance.

This misinterpretation reveals something fundamental about how I (I know better by now than to say "we") think about speed.

In my mind, speed is a mapping from times to distances. The way to compare different speeds is by holding time constant and looking at the different distances traversed in that fixed time. (I know I'm not a total mutant in this regard, because even other people tend to visually represent speeds as little arrows of varying length, with greater lengths corresponding to higher speeds.)

In particular, I don't think of it as a mapping from distances to times. I don't find it natural to compare speeds by imagining a fixed distance corresponding to different travel times. Which explains why I find this problem so difficult, and other people's explanations so unilluminating: they tend to begin with something along the lines of "let d be the total distance traveled", upon which my brain experiences an error message that is perhaps best verbalized as something like "wait, what? Who said anything about a fixed distance? If speeds are varying, distances have to be varying, too!" 

If speed is a mapping from times to distances, then the way that you add speeds together and multiply them by numbers (the operations involved in averaging) is by performing the same operations on corresponding distances. (This is an instance of the general definition in mathematics of addition of functions: (f+g)(x) = f(x)+g(x), and similarly for multiplication by numbers: (af)(x) = a*f(x).) In concrete terms, what this means is that in order to add 30 mph and 20 mph together, all you have to do is add 30 and 20 and then stick "mph" on the result. Likewise with averages: provided the times involved are the same, if your speeds are 20 mph and 60 mph, your average speed is 40 mph. 

You cannot do these operations nearly so easily, however, if distance is being held fixed and time varying. Why not? Because if our mapping is from times to distances, then finding the time that corresponds to a given distance requires us to invert that mapping, and there's no easy way to invert the sum of two mappings (we can't for example just add the inverses of the mappings themselves). As a result, I find it difficult to understand the notion of "speed" while thinking of time as a dependent variable. 

And that, at least for me, is why this problem is confusing: the statement doesn't contain a prominent warning saying "Attention! Whereas you normally think of speed as the being the (longness-of-)distance traveled in a given time, here you need to think of it as the (shortness-of-)time required to travel a given distance. In other words, the question is actually about inverse speed, even though it talks about 'speed'."

Only when I have "inverse speed" in my vocabulary, can I then solve the problem -- which, properly formulated, would read: "If you want your inverse speed for the whole trip to be 1/40 hpm, and your inverse speed for the first half is 1/20 hpm, how 'slow' (i.e. inversely-fast) do you have to go on the second half?" 

Solution: Now it makes sense to begin with "let d be the total distance"! For inverse speed, unlike speed, accepts distances as inputs (and produces times as outputs). So, instead of distance = speed*time -- or, as I would rather have it, distance = speed(time) -- we have the formula time = speed^-1(distance). Just as the original formula converts questions about speed to questions about distance, this new formula conveniently converts our question about inverse speeds to a question about times: we'll find the time required for the whole journey, the time required for the first half, subtract to find the time required for the second half, then finally convert this back to an inverse speed.

So if d is the total distance, the total time required for the journey is (1/40)*d = d/40. The time required for the first half of the journey is (1/20)*(d/2) = d/40. So the time required for the second half is d/40 - d/40 = 0. Hence the inverse speed must be 0.

So we're being asked to travel a nonzero distance in zero time -- which happens to be an impossibility.

Problem solved. 

Now, here's the interesting thing: I'll bet there are people reading this who (despite my best efforts) found the above explanation difficult to follow -- and yet had no trouble solving the problem themselves. And I'll bet there are probably also people who consider my explanation to be an example of belaboring the obvious.

I have a term for people in these categories: I call them "good at math". What unites them is the ability to produce correct solutions to problems like this without having to expend significant effort figuring out the sort of stuff I explained above.

If for any reason anyone is ever tempted to describe me as "good at math", I will invite them to reflect on the fact that an explicit understanding of the concept of "inverse speed" as described above (i.e. as a function that sends distances to times) was a necessary prerequisite for my being able to solve this problem, and then to consider that problems of this sort are customarily taught in middle- or high school, by middle- and high school teachers.

No indeed, I was not sorted into the tribe of "good at math".

I should find some sort of prize to award to anyone who can explain how to solve "mixing" problems in a manner I find comprehensible. (You know the type: how much of x% concentration do you add to your y% concentration to get z% concentration? et similia.)

For the past few days I've been pondering the question of how best to respond to paulfchristiano's recent posts and comments questioning the value of mathematical research. I don't think I can do it concisely, in a single post; bridging the inferential distance may require something more like a sequence of posts. I may end up writing such a sequence eventually, since it would involve ideas I've actually been wanting to write up for some time, and which are actually relevant to more than just the specific questions at issue here (whether society should sponsor mathematics, and given that it does, whether paulfchristiano or anyone else in the LW readership should pursue it). 

However, as the preceding parenthetical hints at, I'm actually somewhat conflicted about whether I should even bother. Although I believe that mathematical research should be conducted by somebody, it's not at all clear to me that the discipline needs more people beyond those who already "get" its importance, and are out there doing it rather than writing skeptical posts like paulfchristiano's. It seems perfectly plausible to me that those who feel as paulfchristiano does should just leave the profession and do something else that feels more "important" to them. This is surely the best practical solution on an individual level for those who think they have a better idea than existing institutions of where the most promising research directions lie, at least until Hansonian prediction markets are (ever) implemented.

Nevertheless, for those interested in the society-level question of whether mathematics (as such) may be justifiably pursued by anyone, or any community of people, as a professional occupation (which is quite distinct from the question of whether e.g. paulfchristiano should personally pursue it), I recommend, at least as a start, grappling with the arguments put forward by the best mathematicians in their own words. I think this essay by Timothy Gowers (a Fields Medalist), titled "The Importance of Mathematics", is a good place to begin. I would particularly draw the attention of those like paulfchristiano, who think they have a good idea of which branches of mathematics are useful and which aren't, to the following passage, from pp.8-9 (unfortunately the illustrations are missing, but the point being made is pretty clear nonetheless):

So - mathematicians can tell their governments - if you cut funding to pure mathematical
research, you run the risk of losing out on unexpected benefits, which historically
have been by far the most important.

However, the miserly finance minister need not be convinced quite yet. It may be very
hard to identify positively the areas of mathematics likely to lead to practical benefits, but
that does not rule out the possibility of identifying negatively the areas that will quite
clearly be useless, or at least useless for the next two hundred years. In fact, the finance
minister does not even need to be certain that they will be useless. If a large area of
mathematics has only a one in ten thousand chance of producing economic benefit in the
next fifty years, then perhaps that at least could be cut.

You will not be surprised to hear me say that this policy would still be completely
misguided. A major reason, one that has been commented on many times and is implied
by the subtitle of this conference, "A Celebration of the Universality of Mathematical
Thought", is that mathematics is very interconnected, far more so than it appears on the
surface. The picture in the back of the finance minister's mind might be something like
Figure 4. According to this picture, mathematics is divided into several subdisciplines, of
varying degrees of practicality, and it is a simple matter to cut funding to the less practical
ones.

A more realistic picture, though still outrageously simplified, is given in Figure 5.
(Just for the purposes of comparison, Figure 6 shows Figures 4 and 5 superimposed.) The
nodes of Figure 5 represent small areas of mathematical activity and the lines joining them
represent interrelationships between those areas. The small areas of activity form clusters
where there are more of these interrelationships, and these clusters can perhaps be thought
of as subdisciplines. However, the boundaries of these clusters are not precise, and many
of the interrelationships are between clusters rather than within them.

In particular, if mathematicians work on difficult practical problems, they do not do so
in isolation from the rest of mathematics. Rather, they bring to the problems several tools
- mathematical tricks, rules of thumb, theorems known to be useful (in the mathematical
sense), and so on. They do not know in advance which of these tools they will use, but they
hope that after they have thought hard about a problem they will realize what is needed to
solve it. If they are lucky, they can simply apply their existing expertise straightforwardly.
More often, they will have to adapt it to some extent

(...)

Thus, a good way to think about mathematics as a whole is that it is a huge body of
knowledge, a bit like an encyclopaedia but with an enormous number of cross-references.
This knowledge is stored in books, papers, computers and the brains of thousands of
mathematicians round the world. It is not as convenient to look up a piece of mathematics
as it is to look up a word in an encyclopaedia, especially as it is not always easy to
specify exactly what it is that one wants to look up. Nevertheless, this "encyclopaedia" of
mathematics is an incredible resource. And just as, if one were to try to get rid of all the
entries in an encyclopaedia, or, to give a different comparison, all the books in a library,
that nobody ever looked up, the result would be a greatly impoverished encyclopaedia or
library, so, any attempt to purge mathematics of its less useful parts would almost certainly
be very damaging to the more useful parts as well.

For background, see here.

In a comment on the original Pascal's mugging post, Nick Tarleton writes:

[Y]ou could replace "kill 3^^^^3 people" with "create 3^^^^3 units of disutility according to your utility function". (I respectfully suggest that we all start using this form of the problem.)

Michael Vassar has suggested that we should consider any number of identical lives to have the same utility as one life. That could be a solution, as it's impossible to create 3^^^^3 distinct humans. But, this also is irrelevant to the create-3^^^^3-disutility-units form.

Coming across this again recently, it occurred to me that there might be a way to generalize Vassar's suggestion in such a way as to deal with Tarleton's more abstract formulation of the problem. I'm curious about the extent to which folks have thought about this. (Looking further through the comments on the original post, I found essentially the same idea in a comment by g, but it wasn't discussed further.)

The idea is that the Kolmogorov complexity of "3^^^^3 units of disutility" should be much higher than the Kolmogorov complexity of the number 3^^^^3. That is, the utility function should grow only according to the complexity of the scenario being evaluated, and not (say) linearly in the number of people involved. Furthermore, the domain of the utility function should consist of low-level descriptions of the state of the world, which won't refer directly to words uttered by muggers, in such a way that a mere discussion of "3^^^^3 units of disutility" by a mugger will not typically be (anywhere near) enough evidence to promote an actual "3^^^^3-disutilon" hypothesis to attention.

This seems to imply that the intuition responsible for the problem is a kind of fake simplicity, ignoring the complexity of value (negative value in this case). A confusion of levels also appears implicated (talking about utility does not itself significantly affect utility; you don't suddenly make 3^^^^3-disutilon scenarios probable by talking about "3^^^^3 disutilons").

What do folks think of this? Any obvious problems? 

One person's modus ponens is another's modus tollens.

- Common saying among philosophers and other people who know what these terms mean.

If you believe A => B, then you have to ask yourself: which do I believe more? A, or not B?

- Hal Daume III, quoted by Vladimir Nesov.

Summary: Rules of logic have counterparts in probability theory. This post discusses the probabilistic analogue of modus tollens (the rule that if A=>B is true and B is false, then A is false), which is the inequality P(A) ≤ P(B)/P(B|A). What this says, in ordinary language, is that if A strongly implies B, then proving A is approximately as difficult as proving B. 

The appeal trial for Amanda Knox and Raffaele Sollecito starts today, and so to mark the occasion I thought I'd present an observation about probabilities that occurred to me while studying the "motivation document"⁽¹⁾, or judges' report, from the first-level trial.

One of the "pillars" of the case against Knox and Sollecito is the idea that the apparent burglary in the house where the murder was committed -- a house shared by four people, namely Meredith Kercher (the victim), Amanda Knox, and two Italian women -- was staged. That is, the signs of a burglary were supposedly faked by Knox and Sollecito in order to deflect suspicion from themselves. (Unsuccessfully, of course...)

As the authors of the report, presiding judge Giancarlo Massei and his assistant Beatrice Cristiani, put it (p.44):

What has been explained up to this point leads one to conclude that the situation of disorder in Romanelli's room and the breaking of the window constitute an artificially created production, with the purpose of directing investigators toward someone without a key to the entrance, who would have had to enter the house via the window whose glass had been broken and who would then have perpetrated the violence against Meredith that caused her death.

The title says it all.