0! = 1 Because I Said So

I remember sitting on the deck of my parents’ house in suburban Philadelphia back in high school an explaining factorials to my friend Paul. I don’t know why I was explaining them to him, since Paul always had way more mathematical chops than me. (As an undergrad he did the Great Books program at St. John’s College in Annapolis, whose math component involved doing proofs about geometric solids using contemporary Euclidean techniques which Paul ate right up but would have given me, a geometrophobe, fits.) “A factorial,” I said, “is a certain kind of multiplication. When you write an exclamation point after a number–an exclamation point because the numbers get really big, really fast!–that means you multiply the number times its predecessor, times its predecessor, and so on down to 1.” Paul grasped the idea immediately. He saw, for instance, that the final multiplication by 1 was a bit of a formality, and asked what the factorial of zero was, because like I said he had mathematical chops and knew the right questions to ask.

“That’s the interesting part. The factorial of zero is one.”

“That’s stupid,” said Paul. “Because zero times any number is zero.”

“Well,” I sputtered. I wasn’t expecting this. “That’s the way it’s defined.”

“Then that definition doesn’t make any sense!”

A bit of background: back in high school the math thing was a sideline, and the main preoccupation of Paul, me, and our circle of friends was reading Lester Bangs’ Psychotic Reactions and Carburetor Dung and using it as an example of how to hold very specific aesthetic opinions very adamantly, so a strong reaction to a thing that didn’t fit wasn’t entirely out of order. The problem was that I couldn’t justify the fact that 0! = 1. I fell back lamely on Appeal To Authority, and hope I didn’t put Paul off factorials for life.

Here, many years later, is the answer I should have given. Of course you could define 0!=0. You can define anything any way you want. Ink is cheap. The limiting factor is the utility of your definition. Factorials, it turns out, are useful when you want to count the number of ways a thing can happen. Pat, Kim, and Sam stand in line for a movie. Maybe Pat is first, then Kim, then Sam. Or maybe Kim is first, then Sam, then Pat. The number of possible configurations is 3! = 3 x 2 x 1 = 6. All manner of sophisticated statistics are built on top of this basic accounting. Now ask, how many ways are there to arrange no things? Granted, this has an annoying Zen koan-like ring to it, but a reasonable answer is: one. “Reasonable” here has two components. First, it’s not unreasonable. When no one shows up to the movie there’s just an empty sidewalk with sad little puddles reflecting blinking marquee lights, and that’s a single thing. Second, when you start using factorials as building blocks for more complex tallies you find yourself writing n! terms, some of which end up in the denominators of fractions, so you either have to define n! so that it doesn’t ever equal zero up front or insert caveats into every equation where that might hurt you. Ink may be cheap, but the mental effort it takes to parse extraneous notation is not.

Mathematics is often a refuge from utility. In theory, once we have specified our initial axioms, the rest is just formal manipulation. Furthermore those axioms usually express such head-slappingly obvious concepts that it’s difficult to imagine things being any other way, and so take on a formal quality of their own. But the definition 0! = 1 does not naturally arise from the business of multiplication, nor does it express some self-evident extra-mathematical concept. Instead it is a finesse we pull to make things easier later on. Mathematics is a real world activity, and demands a certain canniness. As in the rest of life, it helps to be scanning the road ahead.

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A recent Playboy article entitled “Bros, This Is How Your Slut-Shaming Is Backfiring. A Sex Researcher Explains” observes that “When the slut-shaming stops, everyone is likely to get laid more.” This is an obvious point that even very minor bloggers have already made. Still it’s nice to have it echoed in the used-to-be-popular press along with supporting evidence.

Man next to a sleeping woman with a surprised expression on his face

The most interesting thing in the article is a passing reference to “‘precarious masculinities,’ a term…to describe men who perceive manhood as an impermanent state that’s easily lost if they fail to conform to gender norms.” This seems like a real thing to me–proving you’re a man and all that–but also personally very weird. I myself feel not at all macho, but deeply masculine. This is mostly a matter of bearing, but also there’s the matter of my deep voice, broad shoulders, bald head, and thick coat of hair everywhere else. I’m not “comfortable in my masculinity”, an idiotic phrase that makes no more sense than “comfortable in my right-handedness”. It’s a like-it-or-not state of affairs. Nothing precarious about it.

I just did a clever double-brag in that last paragraph: reaffirming my conventional male appeal while simultaneously polishing up my liberal feminist bona fides. (It’s the only way I know to be sexy.) As a way of taking myself back down a peg let me point out that there are aspects of machismo I lack–like mechanical aptitude and a capacity for violence–that I feel as true shortcomings and admire in other people. Furthermore, I am a lot closer to the traditional macho guy (whom were are all implicitly looking down on here) than plenty of other men. For example, the chubby, high-voiced, unathletic, geeky, adult role-playing-game enthusiast is nobody’s idea of macho, but still embodies a recognizably male archetype. (Also has a Fetlife profile and gets laid way more than you or I.) Notions of gender are, sigh, fluid.

Only recently have I learned the term “cisgender” as in the opposite of “transgender”. I get a kick out of it because how often do you come across a new prefix, but also because it is a concise way of evoking this sense of inevitability. I have taken to calling myself “cis-as-fuck”. Now this usage has the distinct connotation of trans = good (= oppressed sexual minority) and cis = bad (= frat boy), and it’s a notion of the valorization of one’s born gender more than its certainty that “cis” is getting at, but nevertheless there’s a sense that a man who balances his masculinity precariously on his shoulder is less “cis” than someone who takes it for granted.

Part of learning this delightful new prefix is that I now get to add “cisgendered” to the list of privileges I tick off as part of my ritual white-liberal ablutions. This is justified. Feeling comfortable in one’s own skin is a privilege, and it’s one that the “precariously masculine” frat boy of our imagination enjoys a little less than I do.

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Can You Spot the Differences Between the Pictures?

Girl is wearing a hat

Man’s tie is striped

Flowerpot on balcony

Phone doesn’t work

Boy is lonely

Guy in sweater is Armenian

Not a real blonde

Dog only looks friendly

Couple on bench heading for a divorce

Smells like pickles

Woman on bicycle went to high school with Naomi Judd

It’s daytime

Everything just seems nicer

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Behind Every Mildly Amusing Cartoon…


–That cartoon’s funny, but the attitude it describes is not peculiar to novelists. I’ve spent a lot of time in Silicon Valley, and entrepreneurs have the same romanticized attitude towards an unhealthy obsession with work. The software startup version of this cartoon would change a few of the details–the childhood would be geeky instead of traumatic–but the “Pathologic Ambition”, “Neglected Spouse”, and “Years of Boring Hard Work” panels would remain the same. You’re chasing an illusion. What you’re imagining to be a bohemian rejection of workaday values is really just the Protestant work ethic in disguise. Don’t be a sucker! Abandon your novel and cut out of work early. Spend time with your family and your cat. Live a normal, balanced human life.

–You’re the one chasing an illusion. The illusion is that there is such a thing as a normal, balanced human life. There is not. There are just people doing things, and monomania is one of the things that people do. This jokey celebration of bad habits is healthy in the way that a cigarette after drinks is healthy, as an acknowledgment that there is no answer book against which we’re being judged.

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Lena Dunham is Sexy

I find Lena Dunham sexy because she is talented and funny. The writing on her HBO series Girls is smart, and Dunham’s comic timing is impeccable. Starting with her debut feature Tiny Furniture (a film whose premise practically guarantees headache-inducing preciousness and yet manages to be charming) Dunham has assembled a stable of talented actors so that now in Girls–despite her obvious star presence–she can hang back as part of an ensemble, which bodes well for her current success being the start of a long career instead of just a diverting flash in the pan. To date it seems like Lena Dunham can do no wrong, and this is compelling.

When I say “sexy” I do not mean it as cheeky praise of Dunham’s artistic talent, or a nod to her current status as a hot item in the entertainment world, or some many-layered feminist reappropriation of the concept. What I mean when I say that Lena Dunham is sexy is that I personally would like to have sex with Lena Dunham. Here are some other celebrities I also find sexy: Mila Kunis, Charlize Theron, Halle Berry, and Tia Carrere circa the first Wayne’s World movie.

The difference between these women and Lena Dunham is that I don’t find Lena Dunham physically attractive. Or, no, scratch that. I find Lena Dunham physically attractive. Even when she shows up on Girls in unflattering trying-too-hard outfits, or pasty, naked, and enduring almost-unwatchable humiliation, this is all part of the comic brilliance that is Lena Dunham, which only makes her more attractive in every way. The difference is that I would find any of the other female celebrities mentioned above sexy on the basis of their looks alone, whereas if I had passed Lena Dunham on the street before I’d discovered Tiny Furniture in early 2012 I wouldn’t have given her a second glance. Lena Dunham is not bad looking, but she is frumpy and zaftig, and like most straight men my physical tastes run more in the direction of Mila Kunis et al.

Jack Black lifting up his shirt to reveal writing on his chest.

The criteria for sexiness in contemporary western society are roughly that you should be young, thin, and discernibly masculine or feminine as the case may be. These criteria apply to civilians as well as celebrities, though for celebrities the volume is turned up quite a bit. The same criteria also apply to both men and women, but with a crucial difference: a man who is not physically attractive in a conventional sense may compensate by being strong-willed, talented, or successful. This is not just a case of the markers of male power making potential mates willing to look past a general trollishness (though that happens too). No, the fireplug-shaped alpha male who commands every room he enters, the fleshy oaf with an angel’s voice, and the scrawny geek who leaves you helpless with laughter–these are sexy men. Their looks are part of the package. For women, though, the baseline is set much higher, and the fungibility of other charms is steeply discounted. So really all I’m saying about Lena Dunham is that for me she plays by a man’s rules.

From time to time in discussions of society’s standards of beauty (which are inevitably discussions of female beauty, since women are the ones compelled to play for the highest stakes in this arena) you run across a sentiment like the following: “Young-old, skinny-fat, tall-short–why can’t everyone be beautiful?” Though well-intentioned, this is hopelessly naive and can be dispatched with a concise answer: because that’s not what beauty is. By its nature, beauty is exclusive. Beautiful people only quicken our pulse because they are visibly different than us. (Or because we number among them, in which case we are visibly different from others.) The same goes for the rich, powerful, and talented. When Garrison Keillor describes his fictional town of Lake Wobegon as the place where “all the children are above average” this gets a laugh because it is, by definition, impossible. (Garrison Keillor: another person who is funnier than you or me.) In a perfect world there would still be a tiny cadre of women whose beauty makes the rest of us sick with desire and envy. It’s just that more of them would look like Lena Dunham.

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Samantha!. We’re confident we can build the whole show around her. She’s funny, sexy, and larger than life! The women will want to be her and the men will want to bed her. She’s also campy, so the gay men will want to be her too.
–The concept…
–…is unbeatable!
–…a single woman…
–…and her comic misadventures in the big city. Friends, romance, success. Having it all!
–…in her forties.
–Yes. Somewhere in her mid-forties. It’s not plausible for a younger woman to own her own public relations firm.
–You promised HBO it would be sexy. The development prospectus we agreed upon specifically said–here it is right here: “Sexy”.
–And indeed it will be. Part of the delight of the show will be experiencing the weekly vicarious thrill of sex with our blonde bombshell lead.
–Who is in her forties.
–Who is played by Kim Cattrall. Miss Honeywell the gym teacher from Porky’s. I can guarantee you no small enthusiasm on the vicarious-having-sex-with front. Besides which, romance is only half the show. The other half is a workplace comedy about Samantha running a publi-
–That sounds great but let’s get back to the. Uh. There’ll also be…younger women, right?
–Sure, her group of close gal pals are all younger than her. Samantha’s crabby lesbian best friend will be played by Cynthia Nixon, who’s a hoot. For the part of the naive young writer Samantha takes under her wing we’ve lined up Sarah Jessica Parker.
–Sarah Jessica…?
–One of the girls from Square Pegs.
–The one with the braces or the horse-faced one?
–Horse-face. But she’s matured into a lovely young woman.
–Oh, she’s cute…
–Well that’s wonderful, but here at HBO we need assurances…
–…that you will take full advantage of the unique story-telling opportunities that cable television offers.
–…I had such a crush on her. [Sings] Square Pegs! One size does not fit all…
–I don’t follow.
–That you will deliver an artistic product finely attuned to the medium for which it is created.
–I still don’t follow.
–He means will we get to see tits?
–Oh. Yes. Lots.
–Will we get to see Sarah Jessica Parker’s tits?
–Unfortunately miss Parker does not do nudity. But that actually works out, because her character is terrified of sex. On the other hand, as I understand it doffing her top is just Kim Cattrall’s way of saying, “Nice to meet ya.”
–Okay. So–Samantha has a boyfriend?
–Samantha has lots of boyfriends!
–Which one does she marry?
–She doesn’t marry any of them. [General consternation] Every few weeks we have a new guest star who is Kim Cattrall’s Mr. Right Now. We figure we’ll alternate between hunks and funny character actors. That’s how we’ll keep things fresh.
–Okay, but which one does she secretly want to marry?
–She doesn’t want to marry any of them. Samantha doesn’t want to pick just one man. Besides she’s having too much fun with her friends and her public relations firm.
–I’m lost. I was following you up to the tits part, but now I’m lost. Who gets married then?
No one gets married. Well, maybe the naive young writer…
–…Sarah Jessica Parker?…
–…comes close. There’s a story arc we’re kicking around, a romance between her and a character we haven’t fully worked out yet–we’re just calling him Mr. Big for now. He’s a wealthy, older, emotionally abusive sociopath who she falls for because she’s so insecure. She almost marries him, but at the last minute Samantha talks some sense into her, so everything works out okay.
–Alright. Well, I didn’t get where I am by not taking chances. I think you’ve got something here.
–Thank you, thank you, thank you.
–Just one thing.
–It can’t be set in Duluth, Minnesota.
–But-but! We’ve got this whole recurring Lutefisk motif…
–Duluth isn’t sexy.
–…and the Coen brothers have already signed on to direct an episode!
–No. No flyover territory. The show is set in New York City. Trust me. I know what I’m talking about. Sometimes you have to compromise. New York City.
–…and maybe a bigger role for Sarah Jessica Parker…
–[Deflating acquiescence] New York City.
–Listen, you’re on to something here. You’ve almost got it. Come back in a week. Give me sex, give me the city, and you’ve got a deal!

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Couldda, Wouldda, Shouldda

Recently my wife was talking about something that didn’t strike her as important. “I could’ve cared less,” she said, then corrected herself: “I couldn’t have cared less.” If you stop to think about it, the latter is the literally correct expression. If you don’t care about something your level of caring is already at a minimum, so it would not be possible to care less.1  But really both are correct. Responding to the former with, “I think you mean you could not have cared less” would go beyond obnoxious into nonsensical.

“I could’ve cared less” is one of those prescriptively incorrect expressions that is nevertheless the way people actually talk. Linguists have many of examples of these supposed solecisms at their fingertips–flammable/inflammable, can/may, “That is a rule up with which I will not put”–and delight in explaining why they are not ungrammatical. The “error” when you say “Can I use the restroom?” when you “mean” “May I use the restroom?” is actually a demonstration of your linguistic competence. Part of being a native English speaker lies in possessing three pieces of knowledge: 1)  that there is a subtle difference in the meaning of “can” and “may” that hinges on the latter being a request 2) outside deontically ambiguous contexts this distinction collapses and “can” can (may?) be used everywhere 3) nevertheless there is a convention of correcting someone else’s use of “can” in a strictly-speaking “may” situation, usually as a way of making fun of the sort of person who would do such a thing.

Could-have/couldn’t-have flounders at step (1). There isn’t a well-understood distinction between the two phrases that native speakers choose to ignore because it’s a difficult phrase to understand. There is a subtle three-way interaction between the irrealis of “could”, the negation of “n’t”, and the relational semantics of “less”–and that’s before you throw “have” into the auxiliary verb pileup. The aspiring martinet must be willing to solve a logic puzzle before being able to get on someone’s case about their supposedly incorrect usage. In its focus on the details of negation, could-have/couldn’t-have resembles the prescriptive criticism of negative concord in English–e.g. “If you say ‘I ain’t got no money’ doesn’t that really mean you have some money?”2 But there’s a difference in that English negative concord is highly stigmatized, whereas could-have/couldn’t-have is mostly overlooked.3

My wife didn’t even say “couldn’t have cared less”. In fluid speech it ran together, sounding more like “couldn’t’ve have cared less”. You have to be listening closely to even hear the distinction. So given all this may I now propose a new English word: couldda. It can appear in any context that admits “could have/couldn’t have” and means the same thing, except it also signals to the listener “I am aware of the subtle logical distinction that may arise at this point and emphatically don’t care because we both know what I mean.”

Question: does this word already exist?

1 The former works if you’re being sarcastic–“I could have cared less (…but it’s not likely)”–but this requires an extra twist of prosody.

2 In both cases the smug prescriptivist ignores the fact that it is English being spoken, not propositional calculus. If upon being asked whether I may dump of pot of hot coffee into his lap he responds “No, no, no, no!” I do not think “Aha!–you said ‘no’ four times, and -1 raised to an even power equals +1” and then proceed to scald him.

3 By the way, my wife disagrees with this judgement and thinks that “could care less” is just plain wrong.

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