Wiki Contributions


Announcing the DWATV Discord

It seems to be "responsive"; I wasn't seeing it on desktop either, until I made the window wider.

gwillen's Shortform

A 1-vector is just a regular vector. A 2-vector (or "bivector") is a quantity associated with a two-dimensional "direction", which is an oriented plane. And so on.

Ok, but how do you actually "and so on" the orientability here? I have not actually tried to picture how you orient a 3-vector in a higher space. And I'm suspicious about my analogy between 1-vector and 2-vector orientation until I can picture that. (You can orient a plane by picking one of the two halves it divides a 3-d volume into, but you normally orient a line by thinking about the ends, not the sides where it divides the plane. Does that matter?)

gwillen's Shortform

What's up with all the foo-vectors?

This is an attempt to succinctly (hah) answer a question I keep having to refresh my memory about: What's up with (vectors, bivectors, axial vectors / pseudovectors, multivectors, the cross product, etc.?) How do they relate to each other?


Multivectors or k-vectors are a generalization of vectors. Vectors have a length and a direction, and can be thought of as one-dimensional; k-vectors generalize vectors to arbitrary dimension k. In this framework, a scalar -- a quantity without direction -- can be thought of as a 0-vector. A 1-vector is just a regular vector. A 2-vector (or "bivector") is a quantity associated with a two-dimensional "direction", which is an oriented plane. And so on.

What does it mean for a plane to be "oriented"? It means we pick one side to be the "right side" and the other to be the "wrong side". (In the same way, a vector is an "oriented line", which has a "right end" where we draw the arrowhead.)

The exterior ("wedge") product

We get multivectors from vectors using the exterior product, or "wedge product". In the 3-dimensional setting, the wedge product smells almost exactly like the cross product -- it takes two vectors and gives back a bivector, whose magnitude is the area of a parallelogram formed by those two vectors, and whose orientation depends on the relative directions of the two vectors. (I'm being deliberately vague here to avoid saying anything false; I could say "according to the right-hand rule" to get the general point across, but a later point will be that the left-right choice here is arbitrary, and could have been chosen the other way.)


In an n-dimensional space, a pseudovector (or axial vector) is an (n-1)-vector -- that is, an n-minus-one-dimensional multivector. (A pseudoscalar is an n-vector.) Consider a 3-dimensional space: A bivector picks out two dimensions of it (an oriented plane), but picking two out of three dimensions leaves just one dimension remaining un-picked. So every bivector (a plane with magnitude and orientation) can be matched up with some vector (with the same magnitude, and pointing normal to the plane in the direction of its orientation.)

So in 3-dimensional space, a bivector is a pseudovector, because it is very nearly equivalent to a regular vector. (And a trivector is a pseudoscalar -- there is only one possible basis-trivector, since there are only three dimensions and it has to span all of them. So a pseudoscalar only has a magnitude, and no meaningful direction, just like a regular scalar.)


Why did I say "very nearly equivalent" -- what's the "pseudo" part about? This is trickier to explain, and while it will work fine as a refresher for myself, I don't know if I will get it across fully to anybody else, but I'll try.

Consider unit vectors pointing along the X, Y, and Z axes. Consider also a bivector X ^ Y, which has unit magnitude, and is oriented with the "right side" pointing in the same direction as our Z vector.

Now, flip the whole space around as though you are looking at it in a mirror. You can do this by e.g. negating any of our three vectors. In the resulting space, the X ^ Y bivector is now oriented in the opposite direction from the Z vector. (Thinking about the right-hand rule, consider that a right hand viewed in the mirror looks like a left hand. So if we apply the right-hand rule to the mirrored space, it will point in the opposite direction from how it pointed in the non-mirrored space.) If you take the "pseudovector" view of it -- treating X ^ Y as something like a vector pointing along the Z axis, instead of a plane oriented towards the +Z axis -- you will see where the "psuedo" comes from. Reflecting the space in a mirror causes the vector and the pseudovector, which pointed in the same direction before, to now point in opposite directions.

If you haven't encountered this before, it's probably going to seem like sophistry or handwaving, sorry. All I can say to that is, I promise this actually makes a difference, although I cannot adequately explain why at this time.

The cross product

This all comes around to why people say things like "the cross product doesn't really give a vector!" Because if you look at the universe in a mirror, the result of the cross product does not behave like a vector. It will not appear mirrored like regular vectors, because its direction depends on handedness, and mirrors reverse handedness.

This also explains why sometimes people say "the cross product gives a bivector" and other people say "the cross product gives a pseudovector". In 3-dimensional space, which is the only place the cross product is well-defined, the two are equivalent.

How much should you update on a COVID test result?

Thanks for the update! This is really interesting to follow along with.

We need a theory of anthropic measure binding

I would say it's extremely unclear to me that the question "what is your probability that you are agent X" in an anthropic question like this is meaningful and has a well-defined answer? You said "there are practical reasons you'd like to know", but you haven't actually concretely specified what will be done with the information.

In the process of looking for something I had previously read about this, I found the following post:

Which seems to be asking a very similar question to the one you're considering. (It mentions Ebborians, but postdates that post significantly.)

I then found the thing I was actually looking for:

Which demonstrates why "what rent the belief is paying" is critical:

If Beauty's bets about the coin get paid out once per experiment, she will do best by acting as if the probability is one half. If the bets get paid out once per awakening, acting as if the probability is one third has the best expected value.

Which says, to me, that the probability is not uniquely defined -- in the sense that a probability is really a claim about what sort of bets you would take, but in this case the way the bet is structured around different individuals/worlds is what controls the apparent "probability" you should choose to bet with.

Community Skill Building for Solstice

I'm surprised to hear you say that oratory is one of our weakest areas. Although I guess I actually think we're pretty strong in a lot of areas, so maybe 'weakest' isn't that bad. I have been quite moved by a lot of speeches at various Solstices, but I guess those were the best ones; certainly there's variance.

I think we're doing pretty well at the "first 80%" of the work, in a lot of ways, and are finding ourselves in the "last 80%" -- the detail-oriented polishing that takes a lot of effort and has diminishing returns. (Which is not to say we shouldn't do it! There are still returns to be had.)

We need a theory of anthropic measure binding

I do not have a lot of evidence or detailed thinking to support this viewpoint, but I think I agree with you. I have the general sense that anthropic probabilities like this do not necessarily have well-defined values.

What would you like from How valuable would it be to you?

One thing I'm surprised I haven't seen listed yet: Adjustment for boosters (and generally updating the vaccine adjustments, which I think are almost certainly too generous for 1- and 2-dose vaccination, once Omicron is circulating.)

Lately, I have mostly been using Microcovid as a guide for training my intuition about how important the various factors are. I don't have a lot of confidence in the actual output of the model overall right now, since it doesn't account for boosters, or for Omicron. I also have a general distrust of some of the model's simplifying assumptions about how factors interact, although I don't have anything better to substitute, other than my own intuitive judgement.

Confusion about Sequences and Review Sequences

Is there some way to see these? Or are these the same sequences Ray says he's planning to launch soon, and I should wait until then?

Omicron Post #7

In other words, Delta lasts longer in the air than the original strain (which itself could last in the air for as long as three hours)

This is tangential, but it's been fascinating to me to watch the spread of that "three hours" factoid. I believe it originally comes from this paper: (Note that "HCoV-19" was referring to what is now designated "SARS-CoV-2", i.e. COVID-19.)

I very gently criticized it at the time for having a quite misleading abstract, regarding the "three hours" figure:

This abstract, as far as I can tell, is where the "three hours" factoid originated, which has now become gospel that appears on public health pages like your link above. In reality, the abstract said they detected the virus in aerosols for "up to three hours" because that was the duration of the experiment. It's obvious from graphs, as well as their computed half-life figure -- which is coincidentally pretty close to three hours -- that if they had continued the experiment, they would have kept detecting viable virus for a multiple of that time. (Of course, this figure is also kind of meaningless anyway, since the initial quantity of virus they're experimenting on was arbitrary! Only the half-life is really meaningful, but that doesn't make good headlines.)

It feels very weird to me to have "been there at the beginning" for this. I have no specific qualifications for interpreting this paper, beyond being smart and careful. It's a little depressing (but that's the COVID story.)

(I would love for someone to correct my story here, and find some other explanation for where this factoid came from, or argue that I'm wrong about it being misleading to the point of error. I really don't like this story or what it implies.)

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