Mixed Reference: The Great Reductionist Project

Fol­lowup to: Log­i­cal Pin­point­ing, Causal Reference

Take the uni­verse and grind it down to the finest pow­der and sieve it through the finest sieve and then show me one atom of jus­tice, one molecule of mercy.

- Death, in Hog­father by Terry Pratchett

Med­i­ta­tion: So far we’ve talked about two kinds of mean­ingful­ness and two ways that sen­tences can re­fer; a way of com­par­ing to phys­i­cal things found by fol­low­ing pinned-down causal links, and log­i­cal val­idity by com­par­i­son to mod­els pinned-down by ax­ioms. Is there any­thing else that can be mean­ingfully talked about? Where would you find jus­tice, or mercy?



Sup­pose that I pointed at a cou­ple of piles of ap­ples on a table, a pile of two ap­ples and a pile of three ap­ples.

And lo, I said: “If we took the num­ber of ap­ples in each pile, and mul­ti­plied those num­bers to­gether, we’d get six.”

Nowhere in the phys­i­cal uni­verse is that ‘six’ writ­ten—there’s nowhere in the laws of physics where you’ll find a float­ing six. Even on the table it­self there’s only five ap­ples, and ap­ples aren’t fun­da­men­tal. Or to put it an­other way:

Take the ap­ples and grind them down to the finest pow­der and sieve them through the finest sieve and then show me one atom of six­ness, one molecule of mul­ti­pli­ca­tion.

Nor can the state­ment be true as a mat­ter of pure math, com­par­ing to some Pla­tonic six within a math­e­mat­i­cal model, be­cause we could phys­i­cally take one ap­ple off the table and make the state­ment false, and you can’t do that with math.

This ques­tion doesn’t feel like it should be very hard. And in­deed the an­swer is not very difficult, but it is worth spel­ling out; be­cause cases like “jus­tice” or “mercy” will turn out to pro­ceed in a similar fash­ion.

Nav­i­gat­ing to the six re­quires a mix­ture of phys­i­cal and log­i­cal refer­ence. This case be­gins with a phys­i­cal refer­ence, when we nav­i­gate to the phys­i­cal ap­ples on the table by talk­ing about the cause of our ap­ple-see­ing ex­pe­riences:

Next we have to call the stuff on the table ‘ap­ples’. But how, oh how can we do this, when grind­ing the uni­verse and run­ning it through a sieve will re­veal not a sin­gle par­ti­cle of ap­ple­ness?

This part was cov­ered at some length in the Re­duc­tion­ism se­quence. Stan­dard physics uses the same fun­da­men­tal the­ory to de­scribe the flight of a Boe­ing 747 air­plane, and col­li­sions in the Rel­a­tivis­tic Heavy Ion Col­lider. Nu­clei and air­planes al­ike, ac­cord­ing to our un­der­stand­ing, are obey­ing spe­cial rel­a­tivity, quan­tum me­chan­ics, and chro­mo­dy­nam­ics.

We also use en­tirely differ­ent mod­els to un­der­stand the aero­dy­nam­ics of a 747 and a col­li­sion be­tween gold nu­clei in the RHIC. A com­puter mod­el­ing the aero­dy­nam­ics of a 747 may not con­tain a sin­gle to­ken, a sin­gle bit of RAM, that rep­re­sents a quark. (Or a quan­tum field, re­ally; but you get the idea.)

So is the 747 made of some­thing other than quarks? And is the state­ment “this 747 has wings” mean­ingless or false? No, we’re just mod­el­ing the 747 with rep­re­sen­ta­tional el­e­ments that do not have a one-to-one cor­re­spon­dence with in­di­vi­d­ual quarks.

Similarly with ap­ples. To com­pare a men­tal image of high-level ap­ple-ob­jects to phys­i­cal re­al­ity, for it to be true un­der a cor­re­spon­dence the­ory of truth, doesn’t re­quire that ap­ples be fun­da­men­tal in phys­i­cal law. A sin­gle dis­crete el­e­ment of fun­da­men­tal physics is not the only thing that a state­ment can ever be com­pared-to. We just need truth con­di­tions that cat­e­go­rize the low-level states of the uni­verse, so that differ­ent low-level phys­i­cal states are in­side or out­side the men­tal image of “some ap­ples on the table” or al­ter­na­tively “a kit­ten on the table”.

Now we can draw a cor­re­spon­dence from our image of dis­crete high-level ap­ple ob­jects, to re­al­ity.

Next we need to count the ap­ple-ob­jects in each pile, us­ing some pro­ce­dure along the lines of go­ing from ap­ple to ap­ple, mark­ing those already counted and not count­ing them a sec­ond time, and con­tin­u­ing un­til all the ap­ples in each heap have been counted. And then, hav­ing counted two num­bers, we’ll mul­ti­ply them to­gether. You can imag­ine this as tak­ing the phys­i­cal state of the uni­verse (or a high-level rep­re­sen­ta­tion of it) and run­ning it through a se­ries of func­tions lead­ing to a fi­nal out­put:

And of course op­er­a­tions like “count­ing” and “mul­ti­pli­ca­tion” are pinned down by the num­ber-ax­ioms of Peano Arith­metic:

And we shouldn’t for­get that the image of the table, is be­ing calcu­lated from eyes which are in causal con­tact with the real table-made-of-par­ti­cles out there in phys­i­cal re­al­ity:

And then there’s also the point that the Peano ax­ioms them­selves are be­ing quoted in­side your brain in or­der to pin down the ideal mul­ti­plica­tive re­sult—af­ter all, you can get mul­ti­pli­ca­tions wrong—but I’m not go­ing to draw the image for that one. (We tried, and it came out too crowded.)

So long as the math is pinned down, any table of two ap­ple piles should yield a sin­gle out­put when we run the math over it. Con­strain­ing this out­put con­strains the pos­si­ble states of the origi­nal, phys­i­cal in­put uni­verse:

And thus “The product of the ap­ple num­bers is six” is mean­ingful, con­strain­ing the pos­si­ble wor­lds. It has a truth-con­di­tion, fulfilled by a mix­ture of phys­i­cal re­al­ity and log­i­cal val­idity; and the cor­re­spon­dence is nailed down by a mix­ture of causal refer­ence and ax­io­matic pin­point­ing.

I usu­ally sim­plify this to the idea of “run­ning a log­i­cal func­tion over the phys­i­cal uni­verse”, but of course the small pic­ture doesn’t work un­less the big pic­ture works.


The Great Re­duc­tion­ist Pro­ject can be seen as figur­ing out how to ex­press mean­ingful sen­tences in terms of a com­bi­na­tion of phys­i­cal refer­ences (state­ments whose truth-value is de­ter­mined by a truth-con­di­tion di­rectly cor­re­sp­nd­ing to the real uni­verse we’re em­bed­ded in) and log­i­cal refer­ences (valid im­pli­ca­tions of premises, or el­e­ments of mod­els pinned down by ax­ioms); where both phys­i­cal refer­ences and log­i­cal refer­ences are to be de­scribed ‘effec­tively’ or ‘for­mally’, in com­putable or log­i­cal form. (I haven’t had time to go into this last part but it’s an already-pop­u­lar idea in philos­o­phy of com­pu­ta­tion.)

And the Great Re­duc­tion­ist Th­e­sis can be seen as the propo­si­tion that ev­ery­thing mean­ingful can be ex­pressed this way even­tu­ally.

But it some­times takes a whole bunch of work.

And to no­tice when some­body has sub­tly vi­o­lated the Great Re­duc­tion­ist Th­e­sis—to see when a cur­rent solu­tion is not de­com­pos­able to phys­i­cal and log­i­cal refer­ence—re­quires a fair amount of self-sen­si­ti­za­tion be­fore the trans­gres­sions be­come ob­vi­ous.


Ex­am­ple: Coun­ter­fac­tu­als.

Con­sider the fol­low­ing pair of sen­tences, widely used to in­tro­duce the idea of “coun­ter­fac­tual con­di­tion­ing”:

  • (A) If Lee Har­vey Oswald didn’t shoot John F. Kennedy, some­one else did.

  • (B) If Lee Har­vey Oswald hadn’t shot John F. Kennedy, some­one else would’ve.

The first sen­tence seems agree­able—John F. Kennedy definitely was shot, his­tor­i­cally speak­ing, so if it wasn’t Lee Har­vey Oswald it was some­one. On the other hand, un­less you be­lieve the Illu­mi­nati planned it all, it doesn’t seem par­tic­u­larly likely that if Lee Har­vey Oswald had been re­moved from the equa­tion, some­body else would’ve shot Kennedy in­stead.

Which is to say that sen­tence (A) ap­pears true, and sen­tence (B) ap­pears false.

One of the his­tor­i­cal ques­tions about the mean­ing of causal mod­els—in fact, of causal as­ser­tions in gen­eral—is, “How does this so-called ‘causal’ model of yours, differ from as­sert­ing a bunch of statis­ti­cal re­la­tions? Okay, sure, these statis­ti­cal de­pen­den­cies have a nice neigh­bor­hood-struc­ture, but why not just call them cor­re­la­tions with a nice neigh­bor­hood-struc­ture; why use fancy terms like ‘cause and effect’?”

And one of the most widely en­dorsed an­swers, in­clud­ing nowa­days, is that causal mod­els carry an ex­tra mean­ing be­cause they tell us about coun­ter­fac­tual out­comes, which or­di­nary statis­ti­cal mod­els don’t. For ex­am­ple, sup­pose this is our causal model of how John F. Kennedy got shot:

Kennedy causes Oswald

Roughly this is in­tended to con­vey the idea that there are no Illu­mi­nati: Kennedy causes Oswald to shoot him, does not cause any­body else to shoot him, and causes the Moon land­ing; but once you know that Kennedy was elected, there’s no cor­re­la­tion be­tween his prob­a­bil­ity of caus­ing Oswald to shoot him and his prob­a­bil­ity of caus­ing any­one else to shoot him. In par­tic­u­lar, there’s no Illu­mi­nati who mon­i­tor Oswald and send an­other shooter if Oswald fails.

In any case, this di­a­gram also im­plies that if Oswald hadn’t shot Kennedy, no­body else would’ve, which is mod­ified by a coun­ter­fac­tual surgery a.k.a. the do(.) op­er­a­tor, in which a node is sev­ered from its former par­ents, set to a par­tic­u­lar value, and its de­scen­dants then re­com­puted:

do Oswald=N

And so it was claimed that the mean­ing of the first di­a­gram is em­bod­ied in its im­plicit claim (as made ex­plicit in the sec­ond di­a­gram) that “if Oswald hadn’t shot Kennedy, no­body else would’ve”. This state­ment is true, and if all the other im­plicit coun­ter­fac­tual state­ments are also true, the first causal model as a whole is a true causal model.

What’s wrong with this pic­ture?

Well… if you’re strict about that whole com­bi­na­tion-of-physics-and-logic busi­ness… the prob­lem is that there are no coun­ter­fac­tual uni­verses for a coun­ter­fac­tual state­ment to cor­re­spond-to. “There’s ap­ples on the table” can be true when the par­ti­cles in the uni­verse are ar­ranged into a con­figu­ra­tion where there’s some clumps of or­ganic molecules on the table. What ar­range­ment of the par­ti­cles in this uni­verse could di­rectly make true the state­ment “If Oswald hadn’t shot Kennedy, no­body else would’ve”? In this uni­verse, Oswald did shoot Kennedy and Kennedy did end up shot.

But it’s a sub­tle sort of thing, to no­tice when you’re try­ing to es­tab­lish the truth-con­di­tion of a sen­tence by com­par­i­son to coun­ter­fac­tual uni­verses that are not mea­surable, are never ob­served, and do not in fact ac­tu­ally ex­ist.

Be­cause our own brains carry out the same sort of ‘coun­ter­fac­tual surgery’ au­to­mat­i­cally and na­tively—so na­tively that it’s em­bed­ded in the syn­tax of lan­guage. We don’t say, “What if we perform coun­ter­fac­tual surgery on our mod­els to set ‘Oswald shoots Kennedy’ to false?” We say, “What if Oswald hadn’t shot Kennedy?” So there’s this coun­ter­fac­tual-sup­po­si­tion op­er­a­tion which our brain does very quickly and in­visi­bly to imag­ine a hy­po­thet­i­cal non-ex­is­tent uni­verse where Oswald doesn’t shoot Kennedy, and our brain very rapidly re­turns the sup­po­si­tion that Kennedy doesn’t get shot, and this seems to be a fact like any other fact; and so why couldn’t you just com­pare the causal model to this fact like any other fact?

And in one sense, “If Oswald hadn’t shot Kennedy, no­body else would’ve” is a fact; it’s a mixed refer­ence that starts with the causal model of the ac­tual uni­verse where there are ac­tu­ally no Illu­mi­nati, and pro­ceeds from there to the log­i­cal op­er­a­tion of coun­ter­fac­tual surgery to yield an an­swer which, like ‘six’ for the product of ap­ples on the table, is not ac­tu­ally pre­sent any­where in the uni­verse. But you can’t say that the causal model is true be­cause the coun­ter­fac­tu­als are true. The truth of the coun­ter­fac­tu­als has to be calcu­lated from the truth of the causal model, fol­lowed by the im­pli­ca­tions of the coun­ter­fac­tual-surgery ax­ioms. If the causal model couldn’t be ‘true’ or ‘false’ on its own, by di­rect com­par­i­son to the ac­tual real uni­verse, there’d be no way for the coun­ter­fac­tu­als to be true or false ei­ther, since no ac­tual coun­ter­fac­tual uni­verses ex­ist.


So that busi­ness of coun­ter­fac­tu­als may sound like a rel­a­tively ob­scure ex­am­ple (though it’s go­ing to play a large role in de­ci­sion the­ory later on, and I ex­pect to re­visit it then) but it sets up some even larger points.

For ex­am­ple, the Born prob­a­bil­ities in quan­tum me­chan­ics seem to talk about a ‘de­gree of re­al­ness’ that differ­ent parts of the con­figu­ra­tion space have (pro­por­tional to the in­te­gral over squared mod­u­lus of that ‘world’).

Could the Born prob­a­bil­ities be ba­sic—could there just be a ba­sic law of physics which just says di­rectly that to find out how likely you are to be in any quan­tum world, the in­te­gral over squared mod­u­lus gives you the an­swer? And the same law could’ve just as eas­ily have said that you’re likely to find your­self in a world that goes over the in­te­gral of mod­u­lus to the power 1.99999?

But then we would have ‘mixed refer­ences’ that mixed to­gether three kinds of stuff—the Schrod­inger Equa­tion, a de­ter­minis­tic causal equa­tion re­lat­ing com­plex am­pli­tudes in­side a con­figu­ra­tion space; log­i­cal val­idi­ties and mod­els; and a law which as­signed fun­da­men­tal-de­gree-of-re­al­ness a.k.a. mag­i­cal-re­al­ity-fluid. Mean­ingful state­ments would talk about some mix­ture of phys­i­cal laws over par­ti­cle fields in our own uni­verse, log­i­cal val­idi­ties, and de­gree-of-re­al­ness.

This is just the same sort of prob­lem if you say that causal mod­els are mean­ingful and true rel­a­tive to a mix­ture of three kinds of stuff, ac­tual wor­lds, log­i­cal val­idi­ties, and coun­ter­fac­tu­als, and log­i­cal val­idi­ties. You’re only sup­posed to have two kinds of stuff.

Peo­ple who think qualia are fun­da­men­tal are also try­ing to build refer­ences out of at least three differ­ent kinds of stuff: phys­i­cal laws, logic, and ex­pe­riences.

An­thropic prob­lems similarly re­volve around a mys­te­ri­ous de­gree-of-re­al­ness, since pre­sum­ably when you make more copies of peo­ple, you make their ex­pe­riences more an­ti­ci­pate-able some­how. But this doesn’t say that an­thropic ques­tions are mean­ingless or in­co­her­ent. It says that since we can only talk about an­thropic prob­lems us­ing three kinds of stuff, we haven’t finished Do­ing Re­duc­tion­ism to it yet. (I have not yet en­coun­tered a claim to have finished Re­duc­ing an­throp­ics which (a) ends up with only two kinds of stuff and (b) does not seem to im­ply that I should ex­pect my ex­pe­riences to dis­solve into Boltz­mann-brain chaos in the next in­stant, given that if all this talk of ‘de­gree of re­al­ness’ is non­sense, there is no way to say that phys­i­cally-lawful copies of me are more com­mon than Boltz­mann brain copies of me.)

Or to take it down a notch, naive the­o­ries of free will can be seen as ob­vi­ously not-com­pleted Re­duc­tions when you con­sider that they now con­tain physics, logic, and this third sort of thingy called ‘choices’.

And—alas—mod­ern philos­o­phy is full of ‘new sorts of stuff’; we have modal re­al­ism that makes pos­si­bil­ity a real sort of thing, and then other philoso­phers ap­peal to the truth of state­ments about con­ceiv­abil­ity with­out any at­tempt to re­duce con­ceiv­abil­ity into some mix­ture of the ac­tu­ally-phys­i­cally-real-in-our-uni­verse and log­i­cal ax­ioms; and so on, and so on.

But lest you be tempted to think that the cor­rect course is always to just en­vi­sion a sim­pler uni­verse with­out the ex­tra stuff, con­sider that we do not live in the ‘naive un-free uni­verse’ in which all our choices are con­strained by the malev­olent out­side hand of physics, leav­ing us as slaves—re­duc­ing choices to physics is not the same as tak­ing a naive model with three kinds of stuff, and delet­ing all the ‘choices’ from it. This is con­fus­ing the pro­ject of get­ting the gnomes out of the haunted mine, with try­ing to un­make the rain­bow. Coun­ter­fac­tual surgery was even­tu­ally given a for­mal and log­i­cal defi­ni­tion, but it was a lot of work to get that far—causal mod­els had to be in­vented first, and be­fore then, peo­ple could only wave their hands fran­ti­cally in the air when asked what it meant for some­thing to be a ‘cause’. The over­all moral I’m try­ing con­vey is that the Great Re­duc­tion­ist Pro­ject is difficult; it’s not a mat­ter of just pro­claiming that there’s no gnomes in the mine, or that rain­bows couldn’t pos­si­bly be ‘su­per­nat­u­ral’. There are all sorts of state­ment that were not origi­nally, or are presently not ob­vi­ously de­com­pos­able into phys­i­cal law plus logic; but that doesn’t mean you just give up im­me­di­ately. The Great Re­duc­tion­ist Th­e­sis is that re­duc­tion is always pos­si­ble even­tu­ally. It is nowhere writ­ten that it is easy, or that your prior efforts were enough to find a solu­tion if one ex­isted.

Con­tinued next time with jus­tice and mercy (or rather, fair­ness and good­ness). Be­cause clearly, if we end up with mean­ingful moral state­ments, they’re not go­ing to cor­re­spond to a com­bi­na­tion of physics and logic plus moral­ity.


Main­stream sta­tus.

Part of the se­quence Highly Ad­vanced Episte­mol­ogy 101 for Beginners

Next post: “By Which It May Be Judged

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