2-Place and 1-Place Words


I have pre­vi­ously spo­ken of the an­cient, pulp-era mag­a­z­ine cov­ers that showed a bug-eyed mon­ster car­ry­ing off a girl in a torn dress; and about how peo­ple think as if sex­i­ness is an in­her­ent prop­erty of a sexy en­tity, with­out de­pen­dence on the ad­mirer.

“Of course the bug-eyed mon­ster will pre­fer hu­man fe­males to its own kind,” says the artist (who we’ll call Fred); “it can see that hu­man fe­males have soft, pleas­ant skin in­stead of slimy scales. It may be an alien, but it’s not stupid—why are you ex­pect­ing it to make such a ba­sic mis­take about sex­i­ness?”

What is Fred’s er­ror? It is treat­ing a func­tion of 2 ar­gu­ments (“2-place func­tion”):

Sex­i­ness: Ad­mirer, En­tity—> [0, ∞)

As though it were a func­tion of 1 ar­gu­ment (“1-place func­tion”):

Sex­i­ness: En­tity—> [0, ∞)

If Sex­i­ness is treated as a func­tion that ac­cepts only one En­tity as its ar­gu­ment, then of course Sex­i­ness will ap­pear to de­pend only on the En­tity, with noth­ing else be­ing rele­vant.

When you think about a two-place func­tion as though it were a one-place func­tion, you end up with a Vari­able Ques­tion Fal­lacy /​ Mind Pro­jec­tion Fal­lacy. Like try­ing to de­ter­mine whether a build­ing is in­trin­si­cally on the left or on the right side of the road, in­de­pen­dent of any­one’s travel di­rec­tion.

An al­ter­na­tive and equally valid stand­point is that “sex­i­ness” does re­fer to a one-place func­tion—but each speaker uses a differ­ent one-place func­tion to de­cide who to kid­nap and rav­ish. Who says that just be­cause Fred, the artist, and Bloogah, the bug-eyed mon­ster, both use the word “sexy”, they must mean the same thing by it?

If you take this view­point, there is no para­dox in speak­ing of some woman in­trin­si­cally hav­ing 5 units of Fred::Sex­i­ness. All on­look­ers can agree on this fact, once Fred::Sex­i­ness has been speci­fied in terms of curves, skin tex­ture, cloth­ing, sta­tus cues etc. This speci­fi­ca­tion need make no men­tion of Fred, only the woman to be eval­u­ated.

It so hap­pens that Fred, him­self, uses this al­gorithm to se­lect flir­ta­tion tar­gets. But that doesn’t mean the al­gorithm it­self has to men­tion Fred. So Fred’s Sex­i­ness func­tion re­ally is a func­tion of one ob­ject—the woman—on this view. I called it Fred::Sex­i­ness, but re­mem­ber that this name refers to a func­tion that is be­ing de­scribed in­de­pen­dently of Fred. Maybe it would be bet­ter to write:

Fred::Sex­i­ness == Sex­i­ness_20934

It is an em­piri­cal fact about Fred that he uses the func­tion Sex­i­ness_20934 to eval­u­ate po­ten­tial mates. Per­haps John uses ex­actly the same al­gorithm; it doesn’t mat­ter where it comes from once we have it.

And similarly, the same woman has only 0.01 units of Sex­i­ness_72546, whereas a slime mold has 3 units of Sex­i­ness_72546. It hap­pens to be an em­piri­cal fact that Bloogah uses Sex­i­ness_72546 to de­cide who to kid­nap; that is, Bloogah::Sex­i­ness names the fixed Bloogah-in­de­pen­dent math­e­mat­i­cal ob­ject that is the func­tion Sex­i­ness_72546.

Once we say that the woman has 0.01 units of Sex­i­ness_72546 and 5 units of Sex­i­ness_20934, all ob­servers can agree on this with­out para­dox.

And the two 2-place and 1-place views can be unified us­ing the con­cept of “cur­ry­ing”, named af­ter the math­e­mat­i­cian Haskell Curry. Cur­ry­ing is a tech­nique al­lowed in cer­tain pro­gram­ming lan­guage, where e.g. in­stead of writing

x = plus(2, 3) (x = 5)

you can also write

y = plus(2) (y is now a “cur­ried” form of the func­tion plus, which has eaten a 2)
x = y(3) (x = 5)
z = y(7) (z = 9)

So plus is a 2-place func­tion, but cur­ry­ing plus—let­ting it eat only one of its two re­quired ar­gu­ments—turns it into a 1-place func­tion that adds 2 to any in­put. (Similarly, you could start with a 7-place func­tion, feed it 4 ar­gu­ments, and the re­sult would be a 3-place func­tion, etc.)

A true purist would in­sist that all func­tions should be viewed, by defi­ni­tion, as tak­ing ex­actly 1 ar­gu­ment. On this view, plus ac­cepts 1 nu­meric in­put, and out­puts a new func­tion; and this new func­tion has 1 nu­meric in­put and fi­nally out­puts a num­ber. On this view, when we write plus(2, 3) we are re­ally com­put­ing plus(2) to get a func­tion that adds 2 to any in­put, and then ap­ply­ing the re­sult to 3. A pro­gram­mer would write this as:

plus: int—> (int—> int)

This says that plus takes an int as an ar­gu­ment, and re­turns a func­tion of type int—> int.

Trans­lat­ing the metaphor back into the hu­man use of words, we could imag­ine that “sex­i­ness” starts by eat­ing an Ad­mirer, and spits out the fixed math­e­mat­i­cal ob­ject that de­scribes how the Ad­mirer cur­rently eval­u­ates pul­chritude. It is an em­piri­cal fact about the Ad­mirer that their in­tu­itions of de­sir­a­bil­ity are com­puted in a way that is iso­mor­phic to this math­e­mat­i­cal func­tion.

Then the math­e­mat­i­cal ob­ject spit out by cur­ry­ing Sex­i­ness(Ad­mirer) can be ap­plied to the Wo­man. If the Ad­mirer was origi­nally Fred, Sex­i­ness(Fred) will first re­turn Sex­i­ness_20934. We can then say it is an em­piri­cal fact about the Wo­man, in­de­pen­dently of Fred, that Sex­i­ness_20934(Wo­man) = 5.

In Hilary Put­nam’s “Twin Earth” thought ex­per­i­ment, there was a tremen­dous philo­soph­i­cal brouhaha over whether it makes sense to pos­tu­late a Twin Earth which is just like our own, ex­cept that in­stead of wa­ter be­ing H20, wa­ter is a differ­ent trans­par­ent flow­ing sub­stance, XYZ. And fur­ther­more, set the time of the thought ex­per­i­ment a few cen­turies ago, so in nei­ther our Earth nor the Twin Earth does any­one know how to test the al­ter­na­tive hy­pothe­ses of H20 vs. XYZ. Does the word “wa­ter” mean the same thing in that world, as in this one?

Some said, “Yes, be­cause when an Earth per­son and a Twin Earth per­son ut­ter the word ‘wa­ter’, they have the same sen­sory test in mind.”

Some said, “No, be­cause ‘wa­ter’ in our Earth means H20 and ‘wa­ter’ in the Twin Earth means XYZ.”

If you think of “wa­ter” as a con­cept that be­gins by eat­ing a world to find out the em­piri­cal true na­ture of that trans­par­ent flow­ing stuff, and re­turns a new fixed con­cept Water_42 or H20, then this world-eat­ing con­cept is the same in our Earth and the Twin Earth; it just re­turns differ­ent an­swers in differ­ent places.

If you think of “wa­ter” as mean­ing H20 then the con­cept does noth­ing differ­ent when we trans­port it be­tween wor­lds, and the Twin Earth con­tains no H20.

And of course there is no point in ar­gu­ing over what the sound of the syl­la­bles “wa-ter” re­ally means.

So should you pick one defi­ni­tion and use it con­sis­tently? But it’s not that easy to save your­self from con­fu­sion. You have to train your­self to be de­liber­ately aware of the dis­tinc­tion be­tween the cur­ried and un­cur­ried forms of con­cepts.

When you take the un­cur­ried wa­ter con­cept and ap­ply it in a differ­ent world, it is the same con­cept but it refers to a differ­ent thing; that is, we are ap­ply­ing a con­stant world-eat­ing func­tion to a differ­ent world and ob­tain­ing a differ­ent re­turn value. In the Twin Earth, XYZ is “wa­ter” and H20 is not; in our Earth, H20 is “wa­ter” and XYZ is not.

On the other hand, if you take “wa­ter” to re­fer to what the prior thinker would call “the re­sult of ap­ply­ing ‘wa­ter’ to our Earth”, then in the Twin Earth, XYZ is not wa­ter and H20 is.

The whole con­fus­ing­ness of the sub­se­quent philo­soph­i­cal de­bate, rested on a ten­dency to in­stinc­tively curry con­cepts or in­stinc­tively un­curry them.

Similarly it takes an ex­tra step for Fred to re­al­ize that other agents, like the Bug-Eyed-Mon­ster agent, will choose kid­nappees for rav­ish­ing based on Sex­i­nessBEM(Wo­man), not Sex­i­nessFred(Wo­man). To do this, Fred must con­sciously re-en­vi­sion Sex­i­ness as a func­tion with two ar­gu­ments. All Fred’s brain does by in­stinct is eval­u­ate Wo­man.sex­i­ness—that is, Sex­i­nessFred(Wo­man); but it’s sim­ply la­beled Wo­man.sex­i­ness.

The fixed math­e­mat­i­cal func­tion Sex­i­ness_20934 makes no men­tion of Fred or the BEM, only women, so Fred does not in­stinc­tively see why the BEM would eval­u­ate “sex­i­ness” any differ­ently. And in­deed the BEM would not eval­u­ate Sex­i­ness_20934 any differ­ently, if for some odd rea­son it cared about the re­sult of that par­tic­u­lar func­tion; but it is an em­piri­cal fact about the BEM that it uses a differ­ent func­tion to de­cide who to kid­nap.

If you’re won­der­ing as to the point of this anal­y­sis, we shall need it later in or­der to Ta­boo such con­fus­ing words as “ob­jec­tive”, “sub­jec­tive”, and “ar­bi­trary”.