I do too. But I also have a second problem: I’ve read some philosophy. And so, when I’m frustrated with pointless arguments over definitions, my frustration becomes compounded by the fact that nobody *understands* the thing that they’re arguing about, and the only way to solve the problem is by spending even *more time* arguing over useless stuff.

Case in point, in all the time you’ve ever spent arguing over definitions, have you ever once glanced at the relevant articles in either Wikipedia or the Stanford Encyclopedia of Philosophy? I’m guessing not, because I never thought to do such a thing myself for a long time.

So now that I’ve made everyone feel guilty, let’s talk about one of the things you’d learn from some basic research: intensional vs extensional definitions.

An **intensional definition** provides some rules about what a word refers to. These rules are called the intension of the word (not to be confused with “intention”). For example, the US president is the head of our executive branch.

An **extensional definition** provides the set of objects that the word refers to. This set is called the extension of the word. For example, we can extensionally define the current US President as Donald Trump.

Now consider the following statement: “The current US President could have been Hillary Clinton.” Clearly what is meant is that Hillary Clinton could have fallen under the *intension* of the current US president. But suppose we substituted the *extension* of the current US president: “Donald Trump could have been Hillary Clinton.” By ignoring the difference between extension and intension, we get nonsense.

Some philosophers argue that a way to deal with the extension/intension distinction is to use the language of possible worlds. The extension of a word is the set of things that the word applies to in the actual world. The intension of a word is the set of things that the word could apply to in all possible worlds. However, I do not think this is entirely satisfactory. Consider, for instance, the following intensional definitions of a triangle:

1. A triangle is a polygon with three sides.

2. A triangle is a polygon with three corners.

Since every polygon with three sides also has three corners, both of these kinds of triangles have the same extension in all possible worlds. Nonetheless, it would be incorrect to say that the two definitions are the same. One way to think about it is that the intension describes an algorithm to determine whether an object is a triangle or not. The two definitions of a triangle correspond to two distinct algorithms, and these algorithms happen to always output the same conclusion, but that does not mean that the algorithms were one and the same.

Question: **Is the extensional or intensional definition more important?**

We often like to believe that the intensional definition is the important one. The intension definition is, after all, what you find in dictionaries. It’s the definition that you try to provide whenever someone asks for a definition.

But there are many reasons to believe that the extensional definition is also important. For example, though I provided two intensional definitions of triangles, most people hardly care which is the “correct” definition. Both definitions specify the correct extension, and that’s pretty much all that matters.

Or consider the definition of “religion”. When a discussion group is asked to define religion, people will think about it and come to different conclusions. But even amidst disagreement, there are at least a few points of agreement. We mostly agree that the *extension* of religion includes Christianity, Judaism, Islam, Buddhism, and Hinduism; it excludes baseball, vegetarianism, and university. And should someone happen to disagree on any of these aspects of the extension, then that might cause us to object to their preferred definition.

What this suggests is that the extension of religion is actually more fundamental than the intension. Or at least, some aspects of the extension are fundamental. We don’t necessarily agree on whether, say, the Sunday Assembly is in the extension of religion. Of course, definitions don’t *need* to be complete and unambiguous, but it’s often nice to be able to classify new things that are brought to our attention. So usually, the approach is to come up with a reasonable intensional definition, such that it matches most or all of our preconceptions about the extension. If the intensional definition doesn’t exactly match the extensional definition, we can choose to either modify the intension, modify the extension, or else be satisfied with multiple definitions.

Why is the extension so fundamental? As I’ve pointed out before, all words must ultimately be defined by pointing to examples. Defining by pointing is called an ostensive definition, and it’s basically a variety of extensional definition.

Extensional definitions also appear to be a fundamental aspect of our psychology. I am especially fond of pointing out the existence of **prototype theory**. The idea is that we think of most concepts by imagining one or more prototypical examples, and then we classify new examples by judging their similarity to the prototypical examples. In other words, for many words, we start with an idea of its extension, and classify new objects according to their similarity to the extension.

Of course, different words behave differently. I avoid claiming that all words operate according to prototype theory. I’m fairly sure, for instance, that our understanding of the US president is based on an intensional definition rather than an extensional one, if only because we don’t like to think about the current extension.

Go forth, and argue about definitions in a more enlightened way. Or don’t, if you prefer.

]]>When I was reading that article about gay loneliness, I followed a reference to “A Longitudinal, Mixed Methods Study of Sexual Position Identity, Behavior, and Fantasies Among Young Sexual Minority Men” by Pachankis et al.

“Sexual position identity” refers to “top” or “bottom” or “versatile”. I would guess that most of my readers are already familiar with these terms, but I don’t want to be presumptive so I’ll just spell it out. The identity terms refer to sex positions in anal sex, with “top” being the insertive position, “bottom” being the receptive position, and “versatile” meaning no strong preference either way.

I will be upfront about my prejudices. These identity labels don’t make much sense to me. If people prefer one sex position over another that’s fine but an identity labels aren’t really useful unless they convey some information that a lot of people need to know. The only people who really need to know are sexual partners, or I suppose potential sexual partners. And we’re talking specifically anal sex, which contrary to stereotypes is not actually the most common sexual practice between men. So, sex position identity labels might make sense if you have a lot of sexual partners, but not otherwise. Given the prevalence of sex position identity labels, I strongly suspect that they are fulfilling some other function, like being a vehicle for stereotypes.

Yes, there are top and bottom stereotypes. Bottoms are supposed to be more submissive and feminine. I don’t understand it.

Unfortunately, I’m not sure that the study by Pachankis really answers any of my questions. But here’s a short overview.

My first reaction is that this study has a really bad case of non-representative sampling from college students. They recruited 93 participants through LGBT student groups at various universities. There is little reason to think any of these numbers are accurate for sexual minority men in general.

The participants were asked to categorize themselves as exclusively top, mostly top, versatile, mostly bottom, or exclusively bottom, never used a label, or not using a label right now. **About 52% of participants changed identities after 2 years**. The authors also tried collapsing into just 4 categories: top, bottom, versatile, or no label. Using the collapsed categories, 39% changed identities after two years.

Here’s a table of the identity labels before and after the two years.

As you can see, the flow of identities is mostly equal in all directions. However, after two years, there are fewer people who don’t use labels, and more people who identify as tops.

The authors asked a subset of participants to give reasons for changing identity. The most commonly cited reasons had to do with increasing self-awareness, or relationships. But there were also reasons relating to practical issues or to stigma on some of the labels.

There were a few quotes that I found striking because they were based on fitting stereotypes:

As I grow up, I can be more confident with people who are larger than me, like more athletic, and because of that I have more opportunities to bottom…

When I was less experienced I wanted to bottom because I felt like I was going to be with more experienced people.

Over [in Southeast Asia], the people who are out are very flamboyant and extreme because of lack of acceptance. Even though here in the U.S. I am more on the feminine bottom side, with all of those men there, I felt more of a top.

Reading these quotes and others, and I’m mystified by the idea that tops are larger, more athletic, and more experienced, while bottoms are more feminine, more submissive, and more gay. Also, tops are supposed to have bigger penises, which is apparently born out by the literature. Incidentally, some of the former bottoms complained that bottoming took more prep time (despite being the position for “inexperienced” people), and that large penises caused medical issues.

The authors also compared sexual position identity to reported behavior and fantasies, but I found their analysis to be uninformative. I’d have like to have learned whether behavior predicted identity change or vice versa. There was also no analysis of which groups had sex more or less frequently.

In the discussion, the authors mention a couple theories. First, there’s a correlation between “top” identities and internalized homophobia, which suggests that men would shift towards bottoming as they had more time to get comfortable with themselves. Second, there’s the theory that gay communities tends to embrace hegemonic masculinity, leading men to identify more as “top” as they interacted more with gay communities. The data appear to confirm the latter, but obviously it’s very simplistic to talk about just one mechanism instead of multiple working in tandem.

]]>*Zelda: Breath of the Wild* has received near universal praise from critics, with Metacritic listing it as one of the best video games of all time. This is an exciting time, as we anticipate the numerous clones that will try (and fail) to capture what makes this game so great.

Like most adventure games, BotW is essentially a power fantasy. What makes the game exciting is the acquisition of power, and the illusion that your power matters. For example, you find better weapons and equipment, which grants you the power to access further game content. If BotW is better than similar games, then it is probably because it maintains a greater illusion of power for a longer period of time.

And indeed, the illusion of power is precisely what most critics praise. BotW is a game that lets you do anything! You can climb anywhere, and paraglide down. You can experience the story in any order, or just skip straight to the final boss immediately, if you so choose.

But as critics praise the extent of power that the game grants you, they are ignoring the other essential characteristic of a power fantasy: the illusion that the power matters.

I say “illusion” because at most you acquire power over the virtual world, and that world doesn’t matter. Yes, you might gain access further story content, but often the truth is you could have gotten better story content from TV and movies. Nonetheless, it is easy enough to maintain the illusion of meaning as long as the power matters *within* the virtual world.

What tends to make the illusion collapse is when it becomes clear that your power doesn’t matter even within the virtual world. At some point, you have so much power, but all you can really do with the power is acquire more power, and more power is useless to you. After becoming lord of all, you glimpse the pointlessness of (virtual) existence and realize that you are lord of nothing. It is not merely that your adventure has become meaningless, but that it had always been meaningless from the start.

Game designers could give you all the power from the start, including the ability to fly, to ignore wall collision, and to destroy enemies instantly. But if they did, then the power would feel pointless just that much faster. A good power fantasy instead rations out power, drip by drip, and tries to make each drip feel meaningful.

What makes BotW a satisfying game? Is it that the player gets so much power? No! The strength of the game is in how little power it gives, while still creating a strong impression of power. And the particular strategy for achieving this is to give you a taste of power, and then quickly take it away.

One of the most complained-about mechanics in BotW is weapon durability. Every weapon in the game breaks after a few dozen strikes. A single boss battle can have you cycling through several weapons. There is no way to repair them. You are never in want of weapons, because they are just lying around all over the place, but there is a danger of using up your strongest weapons or weapons that fill a particular niche.

People don’t like this mechanic, because it takes away power. With a little skill and perseverance, you can explore a high level area and find some powerful weapons, but that power is only on loan. In fact, you probably don’t even take advantage of the weapon, instead letting it languish in your inventory in anticipation of tough battle in the hypothetical future. Incidentally, inventory space is extremely limited, and one of the other rewards for exploration is a way to expand your inventory.

It’s understandable that players don’t like weapon durability, because it holds you back. But I believe it’s one of the most important mechanics in the game, because it holds you back *from the abyss*, the same abyss that lurks at the heart of every power fantasy game.

Consider the alternative, where weapons last forever. Instead of exploring to find weapons that grant you power for a finite amount of time, instead you explore to have a *small chance* of finding a weapon better than the one you’re currently using. This is basically skinner box game design, with a random reinforcement schedule. And to prevent you from charging straight into the abyss, the game probably needs areas with different level tiers, forcing the game to be more linear. The much-praised wide open world in BotW is basically made possible by weapon durability.

Of course, the weapon durability mechanic is not without its flaws. Some people intuitively see past the illusion, realizing the pointlessness of it all. I’ve observed that some people tend to hoard, systematically overestimating the value of saving a weapon for the future. Nonetheless, I think this is a case where people are very much impressed by one aspect of the game (a wide open world brimming with rewards) and unimpressed with another (brittle weapons), not realizing that the latter is what makes the former possible.

]]>I grew up in a family that never talked about sex or even really relationships and intimacy. Of course I was still surrounded by sex in media, my peers, etc, but I never got “the talk” or had any discussions about sex within my household. My therapist wanted me to consider if that could have influenced my disinterest in sex and lack of sexual attraction.

–Seen on AVEN

I don’t feel sexual attraction to people but I know my antidepressants repress my sex drive so I don’t know what I feel naturally and what’s been taken away from me if that makes sense.

–A question seen on Asexual Advice

In a world that continually erases Asian (male assigned) sexualities I was coerced into asexuality. It is something I have and will continue to struggle with. My asexuality is a site of racial trauma. I want that sadness, that loss, that anxiety to be a part of asexuality politics. I don’t want to be proud or affirmed […]

There’s a common theme among people questioning whether they’re asexual. What if I’m really this way just because of _____? Replace the blank with “trauma”, “hormones”, “medication”, “my age”, “gender dysphoria”, “abuse”, “anxiety”, “repression”, or “upbringing”.

Even if you’re sure you don’t experience sexual attraction, if the reason you don’t experience it is due to any of the above, your claim to the identity becomes contested. The only universally accepted reason to identify as asexual if it’s “just the way you are”.

Philosophically, this drives me up the wall, because “it’s just the way you are” is not really a reason. It’s an admission that no one knows the reason. Suppose we discovered that 60% of aces are that way because they were subject to larger amounts of a particular brain chemical at the age seven. Would that mean that those 60% are no longer “really” ace? Would that be a case of SCIENCE disproving 60% of asexuality?

Who cares what the reason is? Does it make a difference to your lived experience? Does asexuality-because-hormones feel any different from asexuality-because-genes? If you don’t know whether your asexuality has anything to do with hormones, does that put your experience of sexual attraction into a quantum state?

However, the answer to “who cares?” is you care. And I care. If people just didn’t care, then Alok wouldn’t have written that essay, people wouldn’t ask Asexual Advice for advice, and nobody on AVEN would ever talk about it. Let’s think hard about why people care.

**1. The “real” you**

Many of the “causes” I mentioned appear as external forces, which could push you away from the “real” you. For example, if I’m asexual and taking antidepressants, would the “real” me, who is not on antidepressants, not be asexual? Of course, then the “real” me would also be depressed. The question is not, “Who is the real me?” it’s, “Who do I want to be?”

**2. A product of error**

If one’s asexuality is the product of something like horrible, such as trauma, that feels deeply uncomfortable. How can I celebrate my orientation when it may have been caused by something so terrible?

You are free to celebrate your feelings or not. However, always remember that neutral and good things can come out of bad. For example, I had parents who argued all the time, and that was bad. But I also learned to be good at conflict resolution, and I can still celebrate that consequence.

**3. Unaddressed problems**

If you think you’re asexual because of anxiety, or because of hormones, you might worry that by identifying as asexual, you’re ignoring the real problems in your life, whether those problems are social or medical.

Although, I haven’t heard any cases where an asexual identity caused people to ignore their other problems. If you’re worried about unaddressed problems, an asexual identity doesn’t require you to stop addressing them.

**4. Predicting the future**

What if it later turns out I’m wrong? What if it’s due to my gender dysphoria and I stop feeling asexual when I transition? What if it’s due to “repression”, whatever that means, and I stop feeling asexual when I’m no longer repressed? What if it’s due to my age, and I stop feeling the same way in a few years?

The future is scary, and there’s little I can say to make it less scary, since it’s not like I can predict the future. If you’re worried that tomorrow you will stop feeling asexual, you’re welcome to take a day to think it over. If you’re worried that it will happen over the next few years, I can’t tell you what to do with that. You may either give it time, or you can take your experience as it is now.

**5. But other people are saying I can’t be ace…**

They’re not the boss of you. It’s your choice to make.

And although I offer reasons why you may still identify as ace, you may also ultimately decide that an ace identity is too uncomfortable. I respect that, because it is your choice to make.

]]>**Could you remind me what the theorem was?**

The theorem states that a consistent formal system cannot prove its own consistency.

As previously discussed, there are a couple qualifiers. The formal system must include some amount of arithmetic, and must have a computable set of axioms.

**What does consistency mean?**

A system is consistent if it cannot prove any contradictions. A system is inconsistent if it can prove a contradiction.

**Contradictions sound bad. Are they bad?**

Yes. The Explosion Principle states that if you can prove a direct contradiction, then you can prove absolutely any statement.

Here’s how the Explosion Principle works. Suppose A and not-A are both provable. Now consider statement B. “(A implies B) or (not-A implies B)” is a tautology. Since both A and not-A, that means we can prove B. Following the same procedure we can also prove not-B.

**I once heard Gödel’s theorem stated as “A system of logic can prove its consistency if and only if it is inconsistent. Is that right?**

Yeah, that’s another equivalent way to state the theorem. If a system of logic is inconsistent, then by the Explosion Principle, it can prove everything, including its own consistency. If it is consistent, then it cannot prove its own consistency.

**So how do you prove Gödel’s Second Incompleteness Theorem?**

The *First* Incompleteness Theorem says that if a logical system is consistent, then Gödel’s statement (“This statement cannot be proven”) is true.

Now suppose that the logical system can prove its own consistency. Then it can also prove Gödel’s statement. Which also implies that it cannot prove Gödel’s statement, which is a contradiction. Therefore, the logical system cannot prove its own consistency.

**Does this mean we will never prove the consistency of math?**

Well, it’s not quite as bad as it sounds. A consistent system cannot prove its own consistency. But you can prove the consistency of a system by using a second, larger system. For example, the consistency of arithmetic has been proven by set theory. The issue is that now set theory cannot prove its own consistency. You need a yet larger system to prove the consistency of set theory.

**Can I add an axiom asserting that the system is consistent?**

Yes, but then you could prove the system is inconsistent.

**Going back to an earlier point, you said you would discuss what would happen if we added an axiom stating Gödel’s unprovable statement is false.**

Right. So if Gödel’s unprovable statement is false, that means there exists a proof of the unprovable statement. How can that be?

To understand this, we need to understand how “provability” is defined. We have some function Proof(x,y), which means that Statement x is a proof of Statement y. Statement y is “provable” if there exists some x such that Proof(x,y).

So, let g be the Gödel’s number of Gödel’s statement. We know that Proof(0,g) is false, Proof(1,g) is false, Proof(2,g) is false, and so on. And yet, according to our new axiom, there exists some x such that Proof(x,g) is true. This axiom basically asserts that there is a number greater than all other natural numbers.

**Doesn’t that contradict the axiom of induction?**

It depends?

The axiom of induction states that for some predicate P(x), if you can prove P(0) and that P(k) implies P(k+1), then you can prove P(x) for all x. Basically, the axiom of induction says that proof by induction works.

But there are actually multiple forms the axiom of induction. You could say “For all predicates P, proof by induction works” (second-order Peano axioms). Or you could say that “For P, proof by induction works” is an axiom for each predicate P (first-order Peano axioms). It actually makes a difference, because any formal proof is finite in length, and therefore can only refer to a finite number of axioms.

Using the first-order axioms, you can produce non-standard number systems. Using the second-order axioms, my understanding is you can produce a formally consistent logic, but it does not correspond to any number system at all.

**If Gödel’s statement is false, isn’t the system inconsistent?**

The First Incompleteness Theorem says that if the system is consistent, then Gödel’s statement is true. But if we have an axiom saying Gödel’s statement is false, then this appears to require that the system is inconsistent.

And indeed, the non-standard number systems are inconsistent–in a certain sense. There are at least two distinct notions of consistency. Regular consistency means you can’t prove a direct contradiction. ω-consistency means that you also can’t prove a certain class of indirect contradictions. For example, suppose P(0), P(1), P(2), and so on are all false, but there exists some x such that P(x) is true–that is an indirect contradiction.

Most of my sources aren’t clear about which kind of consistency they’re talking about at any given moment. I myself have lots more questions about how this works exactly.

**So let me get this straight. There might be numbers greater than all natural numbers?**

We need to sort out the difference between a “theory” and a “model”. In math, a theory is a set of axioms and inference rules, from which you can prove many theorems. A model is an interpretation of those axioms. In a model, every statement is just true or false, and there’s no notion of a distinction between “axioms” and “theorems”.

Natural numbers are a model, and in this model, we have only the numbers we want to have, no more or less. We wish to come up with a theory that describes that model. Unfortunately, the Incompleteness Theorems say that there is no theory that can completely describe the natural numbers. We run into at least one of the following problems:

1. The theory is inconsistent. The theory proves every true fact about the natural numbers, but also proves every false fact.

2. The theory is incomplete. Some facts can’t be proven, and the theory has non-standard interpretations.

3. The theory is incomputable, aka “ineffective”. There are some things that can be “proven” but for which you can’t show the proof.

Of these three problems, #2 seems to be the least bad. Unfortunately the Second Incompleteness Theorem says you can’t really determine if that’s the problem you have.

]]>The second cause, says Hobbes, is gay culture itself. Well, you get a bunch of people together, all of whom have dealt with minority stress, and it turns out they don’t form a big happy family. Hobbes talks about meanness, often in the form of racism, body policing, and masculinity policing. He laments that for many gay men, hookup apps are the primary way they really interact with other gay people.

I am mostly sympathetic to this article. I’ve long thought the health disparities suffered by gay men (and by other minority groups as well) are an elephant in the room. Instead we talk so much about same-sex marriage, bathroom bills, job and housing discrimination, and bullying. And while these are all important issues, it seems like they were chosen not on the basis of being important, but on the basis of being amenable to public policy changes. Health and economic disparities are tougher to address, because we often don’t know what causes them, much less how to solve them.

But here I will raise a few criticisms of Hobbes’ article, and also discuss other people’s critiques.

My first criticism is that the article focuses a lot on “well-off” gay men who nonetheless suffer from drug addiction or mental health issues. I think this is important to Hobbes thesis, which is that these problems occur even without a clearly identifiable cause. But I think the article would have benefited from more stories from non-white men or bisexual men.

For instance, I think it’s relevant to know that bisexual men are worse off than gay men by most measures, despite being less well connected to any sort of “gay community”. It suggests that insofar as the “gay community” is a problem, it’s not just a problem for people in the middle of it, but also people on the outskirts looking in. I put “gay community” in scare quotes because I think it’s questionable whether there is a single unified community, and whether it is “gay” as opposed to gay+bisexual+queer. But as far as the *image* goes, it looks like there is one gay community. And even if the image is illusory, it can be troubling to look in and say, oh, gay culture is awful and I don’t have any alternatives.

My second criticism is that I am extremely suspicious of the story of the guy who was in a 12-step sex addiction program. Whenever I see “12-step program”, I remind myself that 7 out of 12 steps explicitly invoke God or other religious concepts. I accept that sex addiction could be a problem for some people, but if it’s a real problem and not just a matter of religious values dissonance, then it is also a real problem for atheists, and 12-step programs are unacceptable.

Now, let’s take a look at some articles that have been written in response.

Why Gay ‘Marriage’ Has Not Cured Gay Loneliness – I’ll save you the trouble of reading this one. The answer is that gay people don’t have enough God. Pfft, next!

The Research on Minority Stress and Gay Men Shows “Loneliness”—but Also Resilience – This article is written by Brian Salfas, a researcher in mental health for gay men. Salfas notes that some of the gay men in these studies are doing just fine. This strikes me as a trivial observation. Salfas then criticizes Hobbes for being insufficiently nuanced in the description of the research.

So, this just annoys me as a writer. Hobbes’ article is already pretty long, and it is not possible for an essay to cover all nuances! If Salfas wants to elaborate further on the details of the research, he has his *very own article* where he is free to do so. But instead of actually explaining anything, he wasted the whole space complaining! This is classic clueless academic.

Gay Loneliness Is Real—but “Bitchy, Toxic” Culture Isn’t the Full Story – The author, Ben Miller, appears to be coming from a more radical queer perspective. Miller’s main point is that the problem with gay culture isn’t that people are mean, but rather that the gay men in question are hurt by their own privilege. However, in my reading of Hobbes, his story isn’t that gay culture is too “bitchy” or “toxic” (in fact Hobbes never uses these words at all), but rather that gay culture is extremely judgmental of attractiveness, body size, and race–that is to say, gay men suffer from a many-tiered hierarchy of privilege. I’m not clear who Miller is disagreeing with.

Miller has other specific points, which range from nonsense to distractions.

First, he complains that Hobbes mostly focuses on “A-gays” (a term I hate already, it basically means privileged gay men). Yes, I also would have liked if Hobbes included more discussion of less privileged groups, and how they are affected by the same issues. But Miller seems to be saying that the *only* way to understand the root of the problem is by talking about stuff like trans murders instead, and I think I’m missing the connection here? This is the literal worst way to use trans people in a rhetorical argument–shrugging aside any problems that have the gall to be less pressing than murder.

Miller’s second complaint is about marriage equality, describing it as a “new and strange celebration of conservative values we’ve constructed as the ultimate goal of gay life”. I don’t agree, and also don’t understand what that has to do with the problems of gay hookup culture.

Miller’s third complaint is that Hobbes relies too much on research, and not enough on Miller’s own anecdotes. Next!

Oh, that’s it. Well, I hope you enjoyed this little cross-section of gay cultural discourse.

]]>4chan: The Skeleton Key to the Rise of Trump – To be honest, the first thing that struck me about this long article, was the name of the author, Dale Beran. Isn’t he the webcomics legend behind A Lesson Is Learned But The Damage Is Irreversible? Yeah, so I’m a webcomics geek. I also remember Dale Beran for this comic–I happen to share his negative opinion of cars.

I’m not sure how far I would vouch for this article. There’s some nice insider perspective on the nihilistic culture of 4chan, and a better explanation of Anonymous than I’ve ever seen from mainstream news outlets. (I remember news outlets saying “anonymous” just refers to people who obscure their identity. But but that’s not what Anonymous is, can you even internet?) It veers a bit much into depicting 4channers as failures who live with their parents. I have lots of friends and relatives who live with their parents–it’s a cheaper way to live in times of economic hardship and I consider this stereotyping to be classist.

I like the bit about models of what men are supposed to strive for in life–either they get a wife and kids, or else they’re supposed to be “players”. Being an ace activist I emphatically reject these models and question whether they’re any good even for non-ace people. Dale Beran suggests that many young men are trapped in these models, and when I question them it’s like I’m saying their problems are in their heads. Something to mull over.

Laurie Penny shared her experience touring with Milo Yiannopolous. This covers some of the same territory as the previous article, but is more compact and focused.

Drug Watch: New Addyi Marketing Campaign, “Find My Spark” – Addyi is the drug recently approved (on weak evidence) to treat Hypoactive Sexual Desire Disorder in women. After disappointing sales, they are trying a marketing campaign. The campaign encourages women who want sex less often than their partners to see this as a medical issue.

BBC’s “Transgender Kids, Who Knows Best?” – HJ Hornbeck and Siobhan team up to counter a recent horrible documentary about transgender kids. I definitely learned a few new things from this series. I had thought that the trans/autism correlation was real but it seems it is not born out by the data.

Luck and Skill in Games – The Game Developers Conference occurred recently, and they released videos of a bunch of old talks. This speaker proposed that luck and skill are not diametrically opposed, and that it is possible to have more of both or less of both. For example, suppose we had a variant on Chess, where at the end of the game, we roll a dice, and if it lands on 1 the loser becomes the winner. This variant of chess requires exactly the same skills to play well, but also has more luck.

Why I support a welfare state – Ozy makes a version of the an argument I would make in favor of universal basic income. So, you know how people often say that the “free market” (for some definition) leads to the most efficient economy, at least in the ideal case? But even in the ideal case, that’s not true when people start out with unequal resources. (Also it’s not actually true in monopolistic competition, but that’s a separate issue.)

]]>To some extent, Hilbert’s Program was successful. We now have Zermelo-Fraenkel set theory, which is a solid foundation for the vast majority of mathematics. But there are two problems. First, set theory isn’t complete. Second, we can’t prove it’s consistent. And Gödel showed that these problems have no solutions.

Gödel’s First Incompleteness Theorem: No consistent formal system is complete.

Gödel’s Second Incompleteness Theorem: No consistent formal system can prove its own consistency.

(Both of these theorems have additional qualifiers that I’ll get to later.)

Here I will explain the proof for the First Incompleteness Theorem, and a few of its implications. In a later post, I will talk about the Second Incompleteness Theorem.

**What does completeness mean?**

A formal mathematical system starts with a set of axioms, and from those can prove many statements. The system is “complete” if every statement can be proven either true or false.

**In a complete mathematical system, could I prove the sky is blue?**

No. Only statements that are well-defined within the mathematical language count. Gödel isn’t trolling us.

**So if math is incomplete, what does the unprovable statement look like?**

Are you familiar with the liar paradox? “This statement is a lie.”

Informally, the unprovable statement (called Gödel’s statement) says “This statement cannot be proven.”

**But that’s not allowed, is it?**

You’re right. “This statement” is too vague and is not allowable within the mathematical language. In order to be clear, you must substitute “This statement” for the specific statement in question. Unfortunately this leads to an infinite recursion, which is also not allowed:

“‘”‘”‘”[…]” is a lie.’ is a lie.” is a lie.’ is a lie.” is a lie.’ is a lie.”

**So Gödel’s proof is infinitely bogus?**

No. Gödel found a trick to get around the infinite recursion. The general idea is that each statement can be assigned a number (called its Gödel number). This is easy and can be done in any number of ways. For instance, you could simply convert the statement to ascii code. And now instead of saying “this statement” we can refer to statements by number.

All that remains is to find some number n such that Statement n is equivalent to “Statement n cannot be proven.”

**I don’t think it’s possible for a statement to include the ascii code of itself.**

Right, it seems ridiculous. For illustration purposes, suppose we have the following:

Statement 11: “Statement 1 cannot be proven.”

Statement 111: “Statement 11 cannot be proven.”

Statement 1123: “Statement 123 cannot be proven.”

Statement 18953: “Statement 8953 cannot be proven.”

Hopefully you recognize the pattern, and can see that here we will *never* find n such that Statement n is “Statement n cannot be proven.” Fortunately, that’s not what Gödel needs. He needs *two* statements fulfilling the following conditions:

Statement n is logically equivalent to statement m.

Statement m is “Statement n cannot be proven.”

**So what is Gödel’s statement?**

I’ll get to that, but first I need to talk about substitution. Consider the statement “x is odd”. This statement is not well-defined (ie we can’t say whether it is true until the value of x is declared), but it does have a Gödel number. Let’s say its number is 3. Now suppose we substitute x with a specific number, let’s say the number 5. Now we have “5 is odd,” which is well-defined, and has a Gödel’s number of its own.

So let’s define sub(j,k) to be Gödel’s number of the statement j, where every instance of “x” is replaced the number k. For instance, sub(3,5) is the Gödel’s number of “5 is odd.”

Now, let d be the Gödel’s number of the statement “Statement sub(x,x) is well-defined and cannot be proven”. Gödel’s statement is

Statement sub(d,d) is well-defined and cannot be proven.

When we expand out the sub function we find that Gödel’s statement is equivalent to

“Statement sub(d,d) is well-defined and cannot be proven” is well-defined and cannot be proven.

Which of course simply means “Gödel’s statement is well-defined and cannot be proven.”

**Wow, how did he come up with that?**

Gödel used what we call Gödel’s Diagonal Lemma. Basically, if you have any function G(x), then the Diagonal Lemma finds some number n such that statement n is logically equivalent to G(n).

**What’s diagonal about the Diagonal Lemma?**

There’s some similarity between Gödel’s Diagonal Lemma and Cantor’s Diagonal Argument, the latter which was used to prove that real numbers are uncountable.

To prove the Diagonal Lemma, we draw out a table of sub(j,k).

We’re particularly interested in the diagonal of this table. We ask, is G(sub(1,1)) true? is G(sub(2,2)) true? And so on. Construct the function D(x) which basically says “G(sub(x,x))”. Of course, statement D(x) has its own Gödel number d, which corresponds to a row on the table. So we can construct D(d), whose Gödel number is sub(d,d), and which is logically equivalent to G(sub(d,d)). And that is the statement we wanted.

**Can I use the Diagonal Lemma to generate a liar paradox?**

No. In order to prove Gödel’s First Incompleteness Theorem, we needed to formally construct the function “Statement x cannot be proven”. This is possible. But in order to create the liar paradox, we need to formally construct a function “Statement x is false”. This is not possible. This was proven by Tarski’s Undefinability Theorem.

Tarski’s Undefinability Theorem is easy to prove. Basically, if you could construct “Statement x is false”, then by the Diagonal Lemma we could create a liar paradox, which is a contradiction.

Incidentally, this is related to Cantor’s Diagonal Argument. Cantor showed that real numbers are uncountable. Likewise, the set of all functions are uncountable. Thus, there are some functions that are impossible to assign a Gödel’s number. Which is to say, some functions cannot be formally constructed. “Statement x is false” is one of those functions.

**What if I create an exotic math system where none of this argument applies?**

You can do that! *First qualifier to Gödel’s Incompleteness Theorems*: They only apply when the mathematical system includes some basic arithmetic.

**What if we just add an axiom asserting that Gödel’s statement is true?**

That… would be fine! Keep in mind, “provability” is always relative to a particular set of axioms. If we have axioms S, then we can construct Gödel’s statement, which is unprovable under S. Now we create a new set of axioms S’, adding an axiom asserting the truth of Gödel’s statement. Gödel’s statement still cannot be proven under S, but can be proven under S’. Of course, S’ is still incomplete, because we can construct a second Gödel’s statement from S’.

**What if we add an infinite number of axioms stating that all of Gödel’s statements are true?**

Depending on how you did it, you’d run into one of the following problems:

1. The axioms are still incomplete. That is, there exist yet more unprovable statements.

2. The axioms are complete but incomputable. That is, there is no effective method to determine whether a given statement is one of the axioms or not. Here the notion of “provability” is questionable, since in general you can’t actually show a proof. *Second qualifier to Gödel’s Incompleteness Theorems*: they only apply to computable sets of axioms.

**Isn’t Gödel’s First Incompleteness Theorem itself a proof of the unprovable statement?**

No, not unless you can use the axioms to prove their own consistency. Gödel’s Second Incompleteness Theorem says you can’t do that.

**Can we add an axiom asserting that Gödel’s statement is false?**

Yes. Remind me to come back to this later.

**Where do you get your information?**

Wikipedia and the Stanford Encyclopedia of Philosophy are good resources, although at times very technical. My goal is to translate such resources, while preserving more detail than the typical popular explanation.

**Update**: Part 2 has been posted.

Here’s a little mystery that my readers can help me out with: **What are the cultural differences between queer men’s and queer women’s spaces?**

The differences are directly relevant to my life. I am gay, and I have hung out in many spaces for queer men. However, I am also active in online ace communities, which are predominantly made up of women. Occasionally, this causes a disconnect between the cultures I see online, and the cultures I see offline. For example, ace communities experience a lot of gatekeeping, wherein people try to say aces aren’t queer, or else reject the word “queer”. To me this has always felt like absurd internet nonsense, because my impression is queer men don’t engage in the same variety of gatekeeping at all. But the ability to dismiss gatekeeping as absurd is a kind of privilege. I want to understand the differences rather than dismissing them.

Obviously, one of the major differences is the difference between offline and online. But recently, I came to recognize gender as an important factor. I wanted to investigate this further by seeing what other people say, but all I found was a silly Buzzfeed article. Clearly this warrants more serious discussion.

So, here are a few observations and impressions:

- The first item in the Buzzfeed article is about the scarcity of lesbian nights, and that sounds about right. In San Francisco, there are gay bars and clubs all over the place, but hardly any for queer women. This appears to cause queer women to police their spaces rather heavily, since their spaces are in short supply. In contrast, the attitude of many queer men is that the more people who are gay, the more space there is.
- I think queer men do have gatekeeping, but of a different variety. They have a way of pushing out people who aren’t sufficiently attractive, or who aren’t white, or who otherwise disrupt whatever culture they think is worth preserving. I have, in the past criticized rape culture in queer men’s spaces, and the most common negative response is of the form, “If you don’t like it, then leave.” Anti-ace gatekeeping also looks like that.
- Queer men and women face different kinds of hostility from society. Homophobia towards men often comes in the form of disgust: people recognize what we are, and don’t like it. Women, on the other hand, are often seen as performing queerness to get attention. See, for instance, the stereotype that bisexual men are really just gay, while bisexual women are really just straight. I don’t know so much about women, but among men there’s always the tension between men who own the stereotypes, and men who reject them.
- In general, queer women seem to be more social-justice-conscious than queer men. This is most dramatically demonstrated by the political fringes. Queer women have TERFs, queer men have an MRA/libertarian cluster. I mean, look at Milo Yiannopolous–when I found out he was gay, I was not the least bit surprised.
- As noted at the top, not all of the women and men in question identify as queer. But, the reasons are different. Usually when men don’t identify as queer, it’s because they’ve grown up seeing it used as an insult, and have a negative visceral reaction to it. Also, they say “gay” works just fine thank you very much. When women don’t identify as queer (at least on Tumblr), they say they are trying to protect
*other people*who may be triggered. I find this ironic for a number of reasons that could fill out a separate post.

What do you think? Are any of my impressions incorrect? Any other ideas to add?

]]>**Arrow Illusion**, my design.

The arrow illusion was inspired by a much more impressive optical illusion, the Ambiguous Cylinder Illusion. Video below the fold.

[Description: a series of illusions using objects that look like cylinders, but whose mirror reflections look like squares. Whenever the objects are rotated 180 degrees, they look like squares, but their reflections look like cylinders.]

YouTube suggested a few responses to this video, with people recreating the illusion with 3D printers and explaining how it works. Although, I’m not sure why they bothered, when the author of the illusion, Kokichi Sugihara, literally published an academic paper on the subject! (Incidentally, both of those video responses got the math wrong.)

Anyway, I obviously don’t have a 3D printer, so I wanted to design an origami version of the illusion. At first I wanted to make an ambiguous cylinder, but curved origami is hard, and it was a bit too much math even for me. So instead I went with the first example in Sugihara’s paper, an arrow.

Also! I have origami diagrams, which I sketched for myself. This was a relatively recent model, so by this point I was familiar with origami diagramming conventions. These… may not be entirely self-explanatory, but I’m available to elaborate on anything.

(Please do not share these diagrams without permission.)

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