In other news, I have been working on writing up more category theory notes for fun and stuff. Here’s how you prove the four lemma in an arbitrary abelian category (where you can’t push around elements).

Make a puzzle out of this >:]

is it not already one?

FRIENDLY REMINDER: Months before RFRA, over a year before HEA 1337, Gov. Mike Pence unilaterally repealed clauses of the state constitution of Indiana, overrode the landslide majority will of the voters of this state, seized powers not enumerated to him and superseded the authority of duly elected members of the executive and legislative branches.

If you’re not in Indiana, you might not have heard of that because it didn’t get national coverage because it wasn’t as sexy outside the state as (completely unenforceable and meaningless) laws against LGBT rights and abortion, but it happened and actually affected the daily lives of every single one of 6.6 million Hoosiers ages one day to one hundred years.

This is not hyperbole. It is fact and was reported on by the “Indianapolis Star,” the “Bloomington Herald-Times,” the “South Bend Tribune,” the “Fort Wayne Journal Gazette,” the “Evansville Courier & Press,” and other daily newspapers and TV and radio stations in the state.

Pass it on.

- http://satyronline.tumblr.com/

—–

my impression of pence when i covered a talk of his shortly before he ran for governor was that he was bland and forgetable, without even the malicious buffoonery of a guy like scott walker, but i guess it’s the under the radar types who do the most damage. not that there’s much you can do to make indiana worse.

Can I get a link to a source on this?

Here’s a selection of the art that’ll be in my new Benign Kingdom art book! The Kickstarter for my book, in a series with Becky, Meredith, and Dustin, is going on RIGHT NOW and I hope you’ll have a look!

I very rarely go back and look at all the stuff I’ve made like this, but there’s a lot of it and a lot of different kinds of it. Here’s an angel, a devil, a marble, a bad-luck dog, a robot, some Homestucks, an orrery/armillary, a goblin, a gnome, a sun god.

Thank you for helping to spread the word, friends!

BENIGN KINGDOM 2016!

Just over ONE WEEK LEFT to back the new Benign Kingdom Kickstarter! thank you so much everyone.

Is Unsong a Superstimulus?

The following should not be taken seriously, but it should probably be taken as something.

So the new chapter of Unsong (Scott Alexander’s ongoing web serial novel) is out, and it is extremely hilarious. It is also ludicrously high-context. I would not have expected a spiders Georg reference to show up in the same paragraph as a reference to The Northern Caves, and yet both occur before the chapter even technically starts. This is really common for Unsong, though. The density of puns, ingroup jokes, referential name-dropping, and similar techniques is astounding. I convinced a friend who is not involved with rationalism or rationalist adjacent to read it, and he assures me it is still quite good if you don’t get all the references, but for me, the references are the main thing there.

I was thinking about this while reading the newest chapter, and I remembered how Scott Alexander also recently wrote an article about writing techniques. While that article was about nonfiction and really had nothing to do with this concept itself, it did cause me to form a thought about how this style of writing is a heavily complementary to a specific kind of audience.

I also just recently read Crystal Society, and there was a comment in it about how references are a strange form of humor. It seems to me, though, that references meet a very similar psychological desire to the joy that people get from solving an interesting problem. Understanding a high context pun or getting an ingroup reference or joke requires having a rare and specific set of knowledge and being able to make inferential leaps using that knowledge.

Unsong likely has a fairly intelligent reader base. It also likely has a reader base that is nearly entirely a part of a specific subculture. This means that they will share a large quantity of cultural memes and will generally be able to identify those memes under fairly heavy alteration.

If you had a story which was being sold to a community like that, and the only thing you cared about was getting as large of a percentage as possible of that community in your reader base, then it seems like this would be one of the best possible ways to do it. You make as many complex references and puns as you can. You make them layered and inter-dependant. You make some of them easy and some ridiculously difficult. You reference other people’s content. You reference references from other people’s content. You do all of this on a fractal level so that any division of your work is itself also a reference.

And the result is that your readers feel tremendously clever. Every time they read more of your story they feel, perhaps on a subconscious and intuitive level, but they feel it all the same, that this story is proof of how smart they are.

I don’t mean to insinuate that anyone is doing anything shady here. Particularly not Scott, who I have great respect for. However, it does seem like an easy opportunity which points to something that may be significant. Feeling clever is an important part of human motivation, at least in my experience, and if this is hijacking the same response, then I think it points to a phenomena similar to that of superstimuli.

Of course, I don’t really know enough of the science on this topic to properly say if my assessment is fully correct. Or rather, I know it is not fully correct, but I don’t know if it is mostly correct. If anyone has the appropriate base in evolutionary psychology or what have you, I would be very appreciative if you could give your own analysis.

But what I really want to know is whether this effect is positive or negative, if it exists at all. Most superstimuli are considered to be a problem because they hijack normal reward structures and turn them to less valuable behaviors. For instance, the reward structure to encourage us to not starve to death instead causes us to gorge ourselves on candy. This occurs in large part because the strongest possible forms of those stimuli are the artificial ones, and gaining tolerance to the lower levels causes us to seek out stronger forms. But in the case of feeling clever, the strongest form is not reading a book with a lot of references. It’s an easily accessible form, but significantly weaker than, say, a major scientific discovery.

Would that mean that under ideal circumstances, a growing cultural reliance upon referential humor will make the world more productive? Or does it just mean that Unsong is morally equivalent to wireheading? Anyone else have thoughts on the subject?

I think Homestuck may be a precedent for what you're talking about, although a great deal of the referential stuff is entirely internal

via trolley problem memes

i… guess not?, it’s not like it’ll run out in either case,and according to the figure the person density is the same on both tracks (tags via @itsbenedict​)

That can’t actually be right though, because then both tracks would be countable. The density of the lower track can’t be constant, my gut feeling is that it can’t even have a well defined density everywhere in the sense of “finite number of people per finite amount of track” but I’m not sure of that one.

I’m really tired so this may be nonsense, but you may be able to do it by putting a measure on the long line?

2* aleph-null is still aleph null though right? aleph one is so much bigger than aleph null that it should definitely be you send it down the aleph-null track.

At any finite time, though, the offshoot would have hit twice as many people as the original.  You’ll never see the difference between the two infinities reflected in reality.  (Say, if you didn’t know whether the length of a given track was aleph_0 or aleph_1, you’d never be able to tell by just looking at what had happened so far.)

I guess one might care about the different infinities if one though that what mattered was not who had been hit at any given time, but how many people, in total, were in the state of “will be hit eventually.”  Even then, though, you’d have to come up with some sort of utilitarian aggregation rule that behaves well over infinite cardinals (and doesn’t treat them all as the same).

This is obviously not a problem with any kind of practical importance, but it’s pretty fun to think about

I think it may be hard to find a definition of “will be hit eventually” which makes that dilemma work.

Suppose we have some monotonically increasing function f : ℝ → (Sets of people), such that f(t) is the set of people that have been hit at time t. Then it seems the set of people that will eventually be hit is ⋃{f(t) | t∈ℝ}. By monotonicity, this is the same as ⋃{f(t) | t∈ℕ}.

But now, if at any finite time the function f for the lower track is smaller than the the function for the upper track, then f(t) is always countable, so the “eventual” set is a countable union of countable sets.

Conversly, if you really want to fit ℵ₁ people on the lower track, then it seems you need to pack them really densely (like, one on every point of the continuum), so at some point the number of victims of the lower train will overtake the upper one…

Since people apparently had an actual discussion about this…I guess I’ll expand on the earlier shitpost.

Youzicha’s post is basically it. The picture makes it look like people are tied to the aleph-1 track one by one, in order. Which means that for each person you can ask “where is he in the line of people on the track?” and get an actual answer out. And that’s exactly what being countable, i.e. aleph-0 in cardinality, i.e. not aleph-1.

In contrast, suppose you have aleph-1 people tied to the line. If the density of people is constant, then any non-empty interval of the line contains aleph-1 people (because the line is the union of countably many such intervals) and thus the train kills aleph-1 people as soon as it starts moving.

If the density isn’t constant I’m not even sure how you can write down these definitions (I’m not convinced “density” in the previous paragraph is well-defined). But all that “the track contains aleph-1 people” would mean is that some interval contains aleph-1 people. If the train is going to hit such an interval, it obviously kills “more” in the aleph-1 track. If it doesn’t hit such an interval, the fact that the track “contains aleph-1 people” is not relevant to anything.

Yeah – tbh I feel kinda stupid now for babbling about this without noticing how weird it is that people are lined up on the aleph_1 track.

I had been thinking about whether you could put a “person density” on the track that’s finite-valued (i.e. no Dirac deltas) and gives you the number of people on an interval when integrated.  Of course this isn’t the same thing as the problem as pictured, but I think you may be able to set this up in a way that preserves the spirit of the problem as written?  You put a measure on the topological long line, so that locally everything is the same as on the other track, but the order types are different and there’s some sense in which the track “contains aleph_1 people.”  (This seems like a counterexample to your third paragraph?)

I have a very amateur hazy bullshit understanding of all of this, so this may not work, but in either case I am curious because I’d like to understand this stuff better.  For instance I don’t really understand the relationship between countability, concepts like total order and well-order, and “people are lined up on the track one by one.”  Clearly there are well-ordered sets that aren’t countable (and thus can’t be “counted” in the one-by-one sense), but I have a hard time visualizing this or imagining what an “admissible” picture of it might look like in the trolley picture (if any exists).

an uncountable well-ordered set can be "counted" in the one-by-one sense (depending on what you mean, I guess), it's just that if you start reading them off in order and keep going forever, there are elements that you will never read at any point in the future.

I must have made a post about this long ago, but Esther’s started reading Homestuck (:D) and that reminded me of it

In this early page of Homestuck, we have one of our usual misattributed quotes:

Of course, Mark Twain didn’t say that.  It’s a line from Timon of Athens, a relatively obscure Shakespeare play, and thus most famous not for being a Shakespeare quote, but for providing the title of Nabokov’s novel Pale Fire.  The novel is named after a poem that plays a central role in it, which has a couplet describing its own naming:

It seems much more likely that Andrew Hussie was aware of the “arrant thief” quote as “from Pale Fire” rather than “from Timon of Athens.”  Has he read Pale Fire, or just heard the quote?  No idea.

But where things get eerie is when we look at the previous page, which has the command “John: Admire harlequins.”  This sounds very much like another imperative:

I honestly believe this was probably not a coincidence.

The resulting story has a number of Nabokovian motifs: butterflies (the wings of the God Tier trolls), chess, puzzles whose resolution lies in seemingly innocuous details (”diluvial”).  (All of this is almost certainly a coincidence.)

this is not a coincidence because nothing is ever a coincidence

honestly that reminds me of the mentality you complained about from "notes from the tilt-a-whirl"

If you reach the tenth floor, you may either choose to get off the elevator or to stay on it. If you choose to get off, and if the woman entered the elevator on the fifth floor, she will ask you, “Where are you going?” Do NOT answer her. Do NOT look at her.

this is from the Holders series, isn’t it? (I haven’t read the specific one, I’m just guessing based on tone.)

I guessed the same thing! But google says it's not.

I’m too tired to make a proper post about this, but maybe that is appropriate because my thoughts about it are not very clear

Basically … I think that in the sort of circles I have been in, there is not enough recognition of intellectual arrogance except when it falls into certain familiar, stereotypical patterns.  There are certain notions of what an intellectually arrogant person looks like – something that roughly amounts to the “fedora Redditor” archetype.  Someone who talks about how “rational” or “logical” they are, makes a point of caring about things like proper grammar or having a high IQ or being an atheist, that sort of thing.

To be honest, I’m not really sure I’ve ever encountered this guy?  Maybe it’s because I spent my time on tumblr and not on, say, Reddit.  But even if I accept that this guy exists and is as annoying as they say, I feel like I see lots of intellectual arrogance going unnoticed because it doesn’t fit a pattern like this.

In particular, I’ve met a lot of people who … really give me this vibe of thinking that their opinions are special, that they have some magic characteristic that makes them more insightful than most people and anoints their opinions as uncommonly interesting and legitimate and worth hearing.  They’re often from “elite” backgrounds, often have intellectual parents, and seem to have spent all of their lives being treated as “one of the interesting people.”  If these people were less socially competent – or less interested in social approval – they might become Euphoric Reddit Fedora Guy.  Instead, they become, well … 

They don’t lose the feeling that their opinions are special or that they’re smart.  But they aggressively define themselves in opposition to the cliches about the sort of person who thinks themselves that way.  They got high scores on standardized tests (and internalized the implied message), but are quick to tell you that they really think standardized tests are bullshit, and that anyone who suggests otherwise is [something bad].  They have highly refined tastes and judge others for not living up to them, but are canny enough to frame those judgments in superficially anti-elitist terms.  (The popular stuff they dislike is problematic or regressive or based on dangerous “myths” about people and society, never mind that it’s liked by huge numbers of people from less privileged or elite backgrounds than theirs.  The stuff they do like is politically and personally right-on, even if only people who went to elite liberal arts college have ever heard of it.)

The really big problem here is that the thing that results from resisting the cliche may in fact be a worse version of the cliche.  The silver lining of the classic intellectual elitist – the “I have a high IQ let’s have a rational argument and discuss classical music, shall we“ person – is that their self-image requires them to take you seriously, as long as you talk the way they want.  They think of themselves as dispassionate and logical, and hence if you disagree in a way that scans as dispassionate and logical, they can’t dismiss you out of hand, because they want to look like someone who would take you down with an argument instead of a dismissal.

There is a case to be made (and it has been made, a million times) about how this sort of demand on one’s interlocutors is stifling and overly restrictive.  But at least it gives you, in principle, a way in – you may not always be able to get these people to treat you as a human being, but there is a clear, well-defined secret password that lets you do it, so long as you have the energy (etc.) to use it.  (I said earlier that I’d never met the Reddit fedora guy, but I have met a number of people fitting the broader archetype that that narrower archetype belongs to, and we actually got on well once I figured out how their input syntax worked.)

But when a person who’s been told they’re smart and special all their lives internalizes the critique of the classic intellectual elitist, you tend to get the worst of both worlds.  They hear the standard critique of “rational, dispassionate debate” as an ideal, and think “great!  Now I can act like my opinions are special endlessly, without any built-in escape hatch.“  They can now guiltlessly dismiss people who question them or their sense of superiority, because we all know that dismissing people is OK sometimes, that “don’t dismiss anyone who’s making a dispassionate, rational argument” is an flawed ideal.

So you try to reach some sort of rock bottom where you and the person can admit you’re both human and continue from there, and your attempts to start from simple raw materials only get responses that amount to “lol i don’t need to explain myself to a pleb like you.”  They’re special, you’re not.  Retreating to bland dispassionate reason has been, among many other things, the traditional way of saying “look, we both put our pants on one leg at a time just like everyone else” – but now you don’t even have recourse to that.  There is no escape.

Maybe it’s just that I went to a liberal arts college myself.  But I feel like I see this person everywhere, and they’re never really challenged.  There are no thinkpieces making fun of them, no snappy memetic terms for them.  They are there, being every elitist jerk anyone ever hated except with a slight twist (and arguably worse), and no one seems to mind.

tyvm for posting, it got me thinking & I think I'm guilty of this, or at least of treating my thoughts as Special :\

How to read math. You’d be surprised how far this will get you.

EDIT: Some corrections

I maybe already commented on this on NTNSB but since it’s come up recently…

One of the interesting things about notation in higher mathematics is that a lot of the easiest errors you can spot immediately, without any conceptual understanding of what is actually happening, because the symbols you strung together just literally don’t make sense. I’ve taken to calling it a “grammatical error”. For instance, consider the common calculus mistake of “distributing the derivative”:

$$(fg)’(x)=f’(x)g’(x)$$

This is a reasonable-looking formula, but it’s false. In Calc I, you have to sort of reason your way to realizing that this isn’t how it works. You do $\partial x^2$, or you draw a square with sides $f+\Delta f$, or whatever. But if you know that the differential of a smooth map $f:G\to\Bbb H$ on a Lie group at is a map between the tangent manifolds $f’:TG\to TH$, then you know immediately that this formula can’t be correct, since even if $f’(x)$ and $g’(x)$ are in the same tangent space, it’s still just a vector space: you don’t have a multiplication structure there.

The sentence isn’t just wrong, it’s grammatically wrong, because it asks me to multiply two things that live in a space that isn’t equipped with a multiplication.

When learning a new subject that has some algebraic teeth to it, I’ve often found it useful to pay special attention to this grammatical structure. In fact, I was doing it a lot recently for my Lie theory class, and I still remember one example fairly vividly, because it caused me a lot of headaches before I knew that the symbol on the left is defined to be the expression on the right:

$$\left(\frac{\partial}{\partial x^j}\right)_{\!p}(f) = \partial_j(f\circ x^{-1})(x(p)).$$

Now, when writing this thing down, I know that the $x$ and the $x^{-1}$ are supposed to be next to each other so that they “cancel each other out”, but I forget which one goes where about half the time. The grammar helps me out here, because the $\partial_j$ on the outside is an ordinary partial derivative, which means it needs to take a function $\Bbb R^n\to\Bbb R^m$, so it had better be that the domain is Euclidean. Since $x$ is the coordinate map, which means $x$ is a partial function $G\to R^n$, it had better be that the domain of the $f\circ [x/x^{-1}]$ is $\Bbb R^n$. Oh, but that domain is just the domain of either $x$ or $x^{-1}$, so no way it can be $x$, because $G$ has all sorts of weird curviness.

In a more down-to-earth example, I saved myself a lot of embarrassment on a recent combinatorics assignment by realizing that I was trying to prove $\sum_{S\in B_n} S(y+1)^{|S|} = \sum_{T\subseteq S} Ty^{|S|}$. It turns out that adding sets to each other is generally not what you should be doing. (The context made it more obscure, but at the end of the day this was the basic issue).

I know everyone already yelled ‘TYPES’ but I’d just like to reiterate that the notion of “purely syntactic classification of expressions with a particular use case in rejecting nonsense” is pretty much the DEFINITION of ‘types’

Honestly the reason that I’m not frantically writing a followup post is that I pretty much know nothing about types; I learned what a dependent function type was once, but then I couldn’t handle the notation overload anymore and I never followed up on it.

I’d almost surely reblog a high-quality post or sequence that tried to give a basic intro to type theory.

I like explaining things! But ‘type theory’ can look pretty different depending on what sense you’re using it in, and depending on what kind of intro you mean I might or might not be qualified to talk about it. Are you interested in type theory as it applies to languages & programming broadly, type theory as a foundational system for mathematics, or both? And do you mean ‘overview of the high level ideas’, or ‘actual educational material on par with skimming established resources’?

Primarily as a foundations of maths.

I think I’m always looking for high-level ideas, but done with as much explanation of notation as the constraints allow. (Technically there’s no length restriction on posts, but 2 pages is the practical limit I always try to keep)  I have no current interest in producing type theory but I would like to be able to read some of the basics at a word-to-word level. If you can squeeze in some of the more promising directions the field is going in, all the better.

I've been reviewing my understanding of type theory since you posted this and I've come to the conclusion that it isn't yet developed enough for me to be able to write an introduction that's both correct and comprehensible. I can probably answer or clarify specific points, but I don't think I see the big picture quite enough to structure an explanation from scratch.

Sorry :|

snake me up (snake me up inside)

sorry but it's "snx (snail x)" not snake; the snake form is "snx, or snake x,"

Take a look at this picture. Do you know who it is?

Most people haven’t heard of him.

But you should have. When you see his face or hear his name you should get as sick in your stomach as when you read about Mussolini or Hitler or see one of their pictures. You see, he killed over 10 million people in the Congo.

His name is King Leopold II of Belgium.

He “owned” the Congo during his reign as the constitutional monarch of Belgium. After several failed colonial attempts in Asia and Africa, he settled on the Congo. He “bought” it and enslaved its people, turning the entire country into his own personal slave plantation. He disguised his business transactions as “philanthropic” and “scientific” efforts under the banner of the International African Society. He used their enslaved labor to extract Congolese resources and services. His reign was enforced through work camps, body mutilations, executions, torture, and his private army.

Most of us – I don’t yet know an approximate percentage but I fear its extremely high – aren’t taught about him in school. We don’t hear about him in the media. He’s not part of the widely repeated narrative of oppression (which includes things like the Holocaust during World War II). He’s part of a long history of colonialism, imperialism, slavery and genocide in Africa that would clash with the social construction of the white supremacist narrative in our schools. It doesn’t fit neatly into a capitalist curriculum. Making overtly racist remarks is (sometimes) frowned upon in polite society, but it’s quite fine not to talk about genocides in Africa perpetrated by European capitalist monarchs.

Mark Twain wrote a satire about Leopold called “King Leopold’s soliloquy; a defense of his Congo rule“, where he mocked the King’s defense of his reign of terror, largely through Leopold’s own words. It’s 49 pages long. Mark Twain is a popular author for American public schools. But like most political authors, we will often read some of their least political writings or read them without learning why the author wrote them (Orwell’s Animal Farm for example serves to re-inforce American anti-Socialist propaganda, but Orwell was an anti-capitalist revolutionary of a different kind – this is never pointed out). We can read about Huck Finn and Tom Sawyer, but King Leopold’s Soliloquy isn’t on the reading list. This isn’t by accident. Reading lists are created by boards of education in order to prepare students to follow orders and endure boredom well. From the point of view of the Education Department, Africans have no history.

When we learn about Africa, we learn about a caricaturized Egypt, about the HIV epidemic (but never its causes), about the surface level effects of the slave trade, and maybe about South African Apartheid (which of course now is long, long over). We also see lots of pictures of starving children on Christian Ministry commercials, we see safaris on animal shows, and we see pictures of deserts in films and movies. But we don’t learn about the Great African War or Leopold’s Reign of Terror during the Congolese Genocide. Nor do we learn about what the United States has done in Iraq and Afghanistan, potentially killing in upwards of 5-7 million people from bombs, sanctions, disease and starvation. Body counts are important. And we don’t count Afghans, Iraqis, or Congolese.

There’s a Wikipedia page called “Genocides in History”. The Congolese Genocide isn’t included. The Congo is mentioned though. What’s now called the Democratic Republic of the Congo is listed in reference to the Second Congo War (also called Africa’s World War and the Great War of Africa), where both sides of the multinational conflict hunted down Bambenga and ate them. Cannibalism and slavery are horrendous evils which must be entered into history and talked about for sure, but I couldn’t help thinking whose interests were served when the only mention of the Congo on the page was in reference to multi-national incidents where a tiny minority of people were  eating each other (completely devoid of the conditions which created the conflict no less). Stories which support the white supremacist narrative about the subhumanness of people in Africa are allowed to be entered into the records of history. The white guy who turned the Congo into his own personal part-plantation, part-concentration camp, part-Christian ministry and killed 10 to 15 million Conglese people in the process doesn’t make the cut.

You see, when you kill ten million Africans, you aren’t called ‘Hitler’. That is, your name doesn’t come to symbolize the living incarnation of evil. Your name and your picture don’t produce fear, hatred, and sorrow. Your victims aren’t talked about and your name isn’t remembered.

Leopold was just one part of thousands of things that helped construct white supremacy as both an ideological narrative and material reality. Of course I don’t want to pretend that in the Congo he was the source of all evil. He had generals, and foot soldiers, and managers who did his bidding and enforced his laws. It was a system. But that doesn’t negate the need to talk about the individuals who are symbolic of the system. But we don’t even get that. And since it isn’t talked about, what capitalism did to Africa, all the privileges that rich white people gained from the Congolese genocide are hidden. The victims of imperialism are made, like they usually are, invisible.

* * If you liked this post, please consider visiting the author’s Facebook page and ‘liking’ it. Thank you! * *

It’s sad I learned about him in college.

This was required summer history reason in my high school but it DEFINITELY needs to be more wide spread

I’m 28 years old and I had no fucking clue who this dude was until this very second

The funny thing about about Repeat Stuff (and Bo Burnham in general really) is that although it’s an attack on pop songs – and one I think is pretty dumb – it’s also an amazingly effective pop song.  I have probably listened to it over 100 times in the last few days, it’s just so catchy

Which is pretty weird given that (1) it’s a song whose message I find annoying and (2) it’s a song about how (other) simple, catchy pop songs are bad

I sometimes feel like Bo Burnham’s “jokes” boil down to “my audience are idiots for liking this.  You are an idiot for liking this.”  And on the one hand, it’s amazing how popular he seems to be given that he seems to loathe his audience, but on the other, it’s not my cup of tea. 

what do you think of stewart lee?

How to read math. You’d be surprised how far this will get you.

EDIT: Some corrections

I maybe already commented on this on NTNSB but since it’s come up recently…

One of the interesting things about notation in higher mathematics is that a lot of the easiest errors you can spot immediately, without any conceptual understanding of what is actually happening, because the symbols you strung together just literally don’t make sense. I’ve taken to calling it a “grammatical error”. For instance, consider the common calculus mistake of “distributing the derivative”:

$$(fg)’(x)=f’(x)g’(x)$$

This is a reasonable-looking formula, but it’s false. In Calc I, you have to sort of reason your way to realizing that this isn’t how it works. You do $\partial x^2$, or you draw a square with sides $f+\Delta f$, or whatever. But if you know that the differential of a smooth map $f:G\to\Bbb H$ on a Lie group at is a map between the tangent manifolds $f’:TG\to TH$, then you know immediately that this formula can’t be correct, since even if $f’(x)$ and $g’(x)$ are in the same tangent space, it’s still just a vector space: you don’t have a multiplication structure there.

The sentence isn’t just wrong, it’s grammatically wrong, because it asks me to multiply two things that live in a space that isn’t equipped with a multiplication.

When learning a new subject that has some algebraic teeth to it, I’ve often found it useful to pay special attention to this grammatical structure. In fact, I was doing it a lot recently for my Lie theory class, and I still remember one example fairly vividly, because it caused me a lot of headaches before I knew that the symbol on the left is defined to be the expression on the right:

$$\left(\frac{\partial}{\partial x^j}\right)_{\!p}(f) = \partial_j(f\circ x^{-1})(x(p)).$$

Now, when writing this thing down, I know that the $x$ and the $x^{-1}$ are supposed to be next to each other so that they “cancel each other out”, but I forget which one goes where about half the time. The grammar helps me out here, because the $\partial_j$ on the outside is an ordinary partial derivative, which means it needs to take a function $\Bbb R^n\to\Bbb R^m$, so it had better be that the domain is Euclidean. Since $x$ is the coordinate map, which means $x$ is a partial function $G\to R^n$, it had better be that the domain of the $f\circ [x/x^{-1}]$ is $\Bbb R^n$. Oh, but that domain is just the domain of either $x$ or $x^{-1}$, so no way it can be $x$, because $G$ has all sorts of weird curviness.

In a more down-to-earth example, I saved myself a lot of embarrassment on a recent combinatorics assignment by realizing that I was trying to prove $\sum_{S\in B_n} S(y+1)^{|S|} = \sum_{T\subseteq S} Ty^{|S|}$. It turns out that adding sets to each other is generally not what you should be doing. (The context made it more obscure, but at the end of the day this was the basic issue).

I know everyone already yelled ‘TYPES’ but I’d just like to reiterate that the notion of “purely syntactic classification of expressions with a particular use case in rejecting nonsense” is pretty much the DEFINITION of ‘types’

Honestly the reason that I’m not frantically writing a followup post is that I pretty much know nothing about types; I learned what a dependent function type was once, but then I couldn’t handle the notation overload anymore and I never followed up on it.

I’d almost surely reblog a high-quality post or sequence that tried to give a basic intro to type theory.

I like explaining things! But 'type theory' can look pretty different depending on what sense you're using it in, and depending on what kind of intro you mean I might or might not be qualified to talk about it. Are you interested in type theory as it applies to languages & programming broadly, type theory as a foundational system for mathematics, or both? And do you mean 'overview of the high level ideas', or 'actual educational material on par with skimming established resources'?

How to read math. You’d be surprised how far this will get you.

EDIT: Some corrections

I maybe already commented on this on NTNSB but since it’s come up recently…

One of the interesting things about notation in higher mathematics is that a lot of the easiest errors you can spot immediately, without any conceptual understanding of what is actually happening, because the symbols you strung together just literally don’t make sense. I’ve taken to calling it a “grammatical error”. For instance, consider the common calculus mistake of “distributing the derivative”:

$$(fg)’(x)=f’(x)g’(x)$$

This is a reasonable-looking formula, but it’s false. In Calc I, you have to sort of reason your way to realizing that this isn’t how it works. You do $\partial x^2$, or you draw a square with sides $f+\Delta f$, or whatever. But if you know that the differential of a smooth map $f:G\to\Bbb H$ on a Lie group at is a map between the tangent manifolds $f’:TG\to TH$, then you know immediately that this formula can’t be correct, since even if $f’(x)$ and $g’(x)$ are in the same tangent space, it’s still just a vector space: you don’t have a multiplication structure there.

The sentence isn’t just wrong, it’s grammatically wrong, because it asks me to multiply two things that live in a space that isn’t equipped with a multiplication.

When learning a new subject that has some algebraic teeth to it, I’ve often found it useful to pay special attention to this grammatical structure. In fact, I was doing it a lot recently for my Lie theory class, and I still remember one example fairly vividly, because it caused me a lot of headaches before I knew that the symbol on the left is defined to be the expression on the right:

$$\left(\frac{\partial}{\partial x^j}\right)_{\!p}(f) = \partial_j(f\circ x^{-1})(x(p)).$$

Now, when writing this thing down, I know that the $x$ and the $x^{-1}$ are supposed to be next to each other so that they “cancel each other out”, but I forget which one goes where about half the time. The grammar helps me out here, because the $\partial_j$ on the outside is an ordinary partial derivative, which means it needs to take a function $\Bbb R^n\to\Bbb R^m$, so it had better be that the domain is Euclidean. Since $x$ is the coordinate map, which means $x$ is a partial function $G\to R^n$, it had better be that the domain of the $f\circ [x/x^{-1}]$ is $\Bbb R^n$. Oh, but that domain is just the domain of either $x$ or $x^{-1}$, so no way it can be $x$, because $G$ has all sorts of weird curviness.

In a more down-to-earth example, I saved myself a lot of embarrassment on a recent combinatorics assignment by realizing that I was trying to prove $\sum_{S\in B_n} S(y+1)^{|S|} = \sum_{T\subseteq S} Ty^{|S|}$. It turns out that adding sets to each other is generally not what you should be doing. (The context made it more obscure, but at the end of the day this was the basic issue).

Lurking beneath all of this is, of course, Type Theory. And how I wish more higher mathematics was presented explicitly with types.

God, I can’t wait until homotopy type theory catches on enough that we can all stop pretending that everything is a set.

Why do we need HoTT for that?

We don’t, necessarily. But as long as we treat ZFC (or any other set theory) as the True Foundation of Mathematics, it will be formally true that everything is a set. HoTT is not the only foundation which avoids this, but right now it seems to be the only one with any real chance of ousting ZFC.

Ah ok. I was gearing up to have an argument about why CIC is no worse than ZFC for the purpose ZFC has in everyday nonfoundational mathematics-as-in-a-textbook.

How to read math. You’d be surprised how far this will get you.

EDIT: Some corrections

I maybe already commented on this on NTNSB but since it’s come up recently…

One of the interesting things about notation in higher mathematics is that a lot of the easiest errors you can spot immediately, without any conceptual understanding of what is actually happening, because the symbols you strung together just literally don’t make sense. I’ve taken to calling it a “grammatical error”. For instance, consider the common calculus mistake of “distributing the derivative”:

$$(fg)’(x)=f’(x)g’(x)$$

This is a reasonable-looking formula, but it’s false. In Calc I, you have to sort of reason your way to realizing that this isn’t how it works. You do $\partial x^2$, or you draw a square with sides $f+\Delta f$, or whatever. But if you know that the differential of a smooth map $f:G\to\Bbb H$ on a Lie group at is a map between the tangent manifolds $f’:TG\to TH$, then you know immediately that this formula can’t be correct, since even if $f’(x)$ and $g’(x)$ are in the same tangent space, it’s still just a vector space: you don’t have a multiplication structure there.

The sentence isn’t just wrong, it’s grammatically wrong, because it asks me to multiply two things that live in a space that isn’t equipped with a multiplication.

When learning a new subject that has some algebraic teeth to it, I’ve often found it useful to pay special attention to this grammatical structure. In fact, I was doing it a lot recently for my Lie theory class, and I still remember one example fairly vividly, because it caused me a lot of headaches before I knew that the symbol on the left is defined to be the expression on the right:

$$\left(\frac{\partial}{\partial x^j}\right)_{\!p}(f) = \partial_j(f\circ x^{-1})(x(p)).$$

Now, when writing this thing down, I know that the $x$ and the $x^{-1}$ are supposed to be next to each other so that they “cancel each other out”, but I forget which one goes where about half the time. The grammar helps me out here, because the $\partial_j$ on the outside is an ordinary partial derivative, which means it needs to take a function $\Bbb R^n\to\Bbb R^m$, so it had better be that the domain is Euclidean. Since $x$ is the coordinate map, which means $x$ is a partial function $G\to R^n$, it had better be that the domain of the $f\circ [x/x^{-1}]$ is $\Bbb R^n$. Oh, but that domain is just the domain of either $x$ or $x^{-1}$, so no way it can be $x$, because $G$ has all sorts of weird curviness.

In a more down-to-earth example, I saved myself a lot of embarrassment on a recent combinatorics assignment by realizing that I was trying to prove $\sum_{S\in B_n} S(y+1)^{|S|} = \sum_{T\subseteq S} Ty^{|S|}$. It turns out that adding sets to each other is generally not what you should be doing. (The context made it more obscure, but at the end of the day this was the basic issue).

I know everyone already yelled 'TYPES' but I'd just like to reiterate that the notion of "purely syntactic classification of expressions with a particular use case in rejecting nonsense" is pretty much the DEFINITION of 'types'

How to read math. You’d be surprised how far this will get you.

EDIT: Some corrections

I maybe already commented on this on NTNSB but since it’s come up recently…

One of the interesting things about notation in higher mathematics is that a lot of the easiest errors you can spot immediately, without any conceptual understanding of what is actually happening, because the symbols you strung together just literally don’t make sense. I’ve taken to calling it a “grammatical error”. For instance, consider the common calculus mistake of “distributing the derivative”:

$$(fg)’(x)=f’(x)g’(x)$$

This is a reasonable-looking formula, but it’s false. In Calc I, you have to sort of reason your way to realizing that this isn’t how it works. You do $\partial x^2$, or you draw a square with sides $f+\Delta f$, or whatever. But if you know that the differential of a smooth map $f:G\to\Bbb H$ on a Lie group at is a map between the tangent manifolds $f’:TG\to TH$, then you know immediately that this formula can’t be correct, since even if $f’(x)$ and $g’(x)$ are in the same tangent space, it’s still just a vector space: you don’t have a multiplication structure there.

The sentence isn’t just wrong, it’s grammatically wrong, because it asks me to multiply two things that live in a space that isn’t equipped with a multiplication.

When learning a new subject that has some algebraic teeth to it, I’ve often found it useful to pay special attention to this grammatical structure. In fact, I was doing it a lot recently for my Lie theory class, and I still remember one example fairly vividly, because it caused me a lot of headaches before I knew that the symbol on the left is defined to be the expression on the right:

$$\left(\frac{\partial}{\partial x^j}\right)_{\!p}(f) = \partial_j(f\circ x^{-1})(x(p)).$$

Now, when writing this thing down, I know that the $x$ and the $x^{-1}$ are supposed to be next to each other so that they “cancel each other out”, but I forget which one goes where about half the time. The grammar helps me out here, because the $\partial_j$ on the outside is an ordinary partial derivative, which means it needs to take a function $\Bbb R^n\to\Bbb R^m$, so it had better be that the domain is Euclidean. Since $x$ is the coordinate map, which means $x$ is a partial function $G\to R^n$, it had better be that the domain of the $f\circ [x/x^{-1}]$ is $\Bbb R^n$. Oh, but that domain is just the domain of either $x$ or $x^{-1}$, so no way it can be $x$, because $G$ has all sorts of weird curviness.

In a more down-to-earth example, I saved myself a lot of embarrassment on a recent combinatorics assignment by realizing that I was trying to prove $\sum_{S\in B_n} S(y+1)^{|S|} = \sum_{T\subseteq S} Ty^{|S|}$. It turns out that adding sets to each other is generally not what you should be doing. (The context made it more obscure, but at the end of the day this was the basic issue).

Lurking beneath all of this is, of course, Type Theory. And how I wish more higher mathematics was presented explicitly with types.

God, I can’t wait until homotopy type theory catches on enough that we can all stop pretending that everything is a set.

Why do we need HoTT for that?

some comments after finishing the northern caves

First, I’m relieved to be done with the story.  That may sound weird, since no one was forcing me to write it.  But this is the first time I’ve gotten a nontrivial amount of attention for a story while I was updating (the audience for Floornight before it was finished was around ~10 people).  This was a new kind of pressure I wasn’t used to, and while its existence wasn’t surprising, I wasn’t really prepared for it.  (This connects to some ways that I felt the story got out of my grasp, which I’ll describe under the cut.)

Also, it’s a spooky story, and it really kind of spooked me – there were a lot of times when I didn’t feel like writing it just because I wanted to write something more cheerful and less obsessed with inevitability and duty and stuff like that.  Next time I’ll write something more cheerful.

I am proud of the story, mostly on the level of prose, characterization, and setting.  I’m less proud of the plot as a whole, although I’m proud of certain parts of it.

More notes below, including some notes about how the story was conceived and written.  

Keep reading

misc thoughts and comments:

The shift in subject matter and plot direction was pretty apparent to me. I didn't specifically consciously expect something mundanely interpersonal to happen, but the payoff definitely seemed off-key from the leadup; I actually got into an argument about it with an IRC friend.

I have mixed feelings about the change in focus, overall, even apart from tonal inconsistencies. Now that I think about it, the way I feel about TNC's general direction and ultimate shape seem pretty close to how I do about Homestuck's. I think that Acts 1-4 and TNC's early interaction-focused stuff are much more cohesive, execute a core set of ideas very well, and would probably have resulted in a much better-plotted story if they weren't built on so much. They also both have a kind of addictive quality to them, associated with a sense of "oh my god this is doing the right things yesssss". Acts 5 and 6, and the later bits of TNC, feel a lot messier to me and less consistently enjoyable, but in both cases there's way too much really good shit in them for me to be comfortable saying that their stories would be better off without them. I still like them quite a bit, but the way I like them seems to have changed somehow.

Immediately after I finished the "I’ve learned a big lesson here..." paragraph I pasted it into IRC and added "alas, too late for hussie to read this post" and then I came back and I read "Those of you who have read Homestuck and are familiar with Andrew Hussie’s writing process can make the obvious connections here."

I reread the podcast, and I think I agree with most of the stuff they say before they specifically mention Spelunk 04! - was the prior bit not part of the mockery, or am I in disagreement with you?

I seriously appreciated that you avoided going supernatural, btw. That was one of my biggest concerns about where the story was going. Here's proof (this was between chapters 19 and 20 going up):

[09:49:32] <benzrf> man [09:49:35] <benzrf> if TNC goes at all [09:49:41] <benzrf> non-reality [09:49:46] <benzrf> if it starts having any fantasy elements [09:49:52] <benzrf> im gonna track down nost and bop him

(sorry for the threat of violence)

Good luck with future writing!

damn im proud of this one

A big page of comic doodles.