Natural Laws Are Descriptions, not Rules

Laws as Rules

We speak ca­su­ally of the laws of na­ture de­ter­min­ing the dis­tri­bu­tion of mat­ter and en­ergy, or gov­ern­ing the be­hav­ior of phys­i­cal ob­jects. Im­plicit in this rhetoric is a meta­phys­i­cal pic­ture: the laws are rules that con­strain the tem­po­ral evolu­tion of stuff in the uni­verse. In some im­por­tant sense, the laws are prior to the dis­tri­bu­tion of stuff. The physi­cist Paul Davies ex­presses this idea with a bit more flair: “[W]e have this image of re­ally ex­ist­ing laws of physics en­sconced in a tran­scen­dent aerie, lord­ing it over lowly mat­ter.” The ori­gins of this con­cep­tion can be traced back to the be­gin­nings of the sci­en­tific rev­olu­tion, when Descartes and New­ton es­tab­lished the dis­cov­ery of laws as the cen­tral aim of phys­i­cal in­quiry. In a sci­en­tific cul­ture im­mersed in the­ism, it was un­prob­le­matic, even nat­u­ral, to think of phys­i­cal laws as rules. They are rules laid down by God that drive the de­vel­op­ment of the uni­verse in ac­cord with His di­v­ine plan.

Does this pre­scrip­tive con­cep­tion of law make sense in a sec­u­lar con­text? Per­haps if we re­place the di­v­ine cre­ator of tra­di­tional re­li­gion with a more nat­u­ral­ist-friendly law­giver, such as an ur-simu­la­tor. But what if there is no in­ten­tional agent at the root of it all? Or­di­nar­ily, when I think of a phys­i­cal sys­tem as con­strained by some rule, it is not the rule it­self do­ing the con­strain­ing. The rule is just a piece of lan­guage; it is an ex­pres­sion of a con­straint that is ac­tu­ally en­forced by in­ter­ac­tion with some other phys­i­cal sys­tem—a pro­gram­mer, say, or a phys­i­cal bar­rier, or a po­lice force. In the sort of pic­ture Davies pre­sents, how­ever, it is the rules them­selves that en­force the con­straint. The laws lord it over lowly mat­ter. So on this view, the fact that all elec­trons re­pel one an­other is ex­plained by the ex­is­tence of some ex­ter­nal en­tity, not an or­di­nary phys­i­cal en­tity but a law of na­ture, that some­how forces elec­trons to re­pel one an­other, and this isn’t just short-hand for God or the simu­la­tor forc­ing the be­hav­ior.

I put it to you that this ac­count of nat­u­ral law is ut­terly mys­te­ri­ous and bor­ders on the non­sen­si­cal. How ex­actly are ab­stract, non-phys­i­cal ob­jects—laws of na­ture, liv­ing in their “tran­scen­dent aerie”—sup­posed to in­ter­act with phys­i­cal stuff? What is the mechanism by which the con­straint is ap­plied? Could the laws of na­ture have been differ­ent, so that they forced elec­trons to at­tract one an­other? The view should also be anath­ema to any self-re­spect­ing em­piri­cist, since the laws ap­pear to be idle dan­glers in the meta­phys­i­cal the­ory. What is the differ­ence be­tween a uni­verse where all elec­trons, as a mat­ter of con­tin­gent fact, at­tract one an­other, and a uni­verse where they at­tract one an­other be­cause they are com­pel­led to do so by the re­ally ex­ist­ing laws of physics? Is there any test that could dis­t­in­guish be­tween these states of af­fairs?

Laws as Descriptions

There are those who take the in­co­her­ence of the sec­u­lar pre­scrip­tive con­cep­tion of laws as rea­son to re­ject the whole con­cept of laws of na­ture as an anachro­nis­tic holdover from a be­nighted the­is­tic age. I don’t think the situ­a­tion is that dire. Dis­cov­er­ing laws of na­ture is a hugely im­por­tant ac­tivity in physics. It turns out that the be­hav­ior of large classes of ob­jects can be given a unified com­pact math­e­mat­i­cal de­scrip­tion, and this is cru­cial to our abil­ity to ex­er­cise pre­dic­tive con­trol over our en­vi­ron­ment. The sig­nifi­cant word in the last sen­tence is “de­scrip­tion”. A much more con­ge­nial al­ter­na­tive to the pre­scrip­tive view is available. In­stead of think­ing of laws as rules that have an ex­is­tence above and be­yond the ob­jects they gov­ern, think of them as par­tic­u­larly con­cise and pow­er­ful de­scrip­tions of reg­u­lar be­hav­ior.

On this de­scrip­tive con­cep­tion of laws, the laws do not ex­ist in­de­pen­dently in some tran­scen­dent realm. They are not prior to the dis­tri­bu­tion of mat­ter and en­ergy. The laws are just de­scrip­tions of salient pat­terns in that dis­tri­bu­tion. Of course, if this is cor­rect, then our talk of the laws gov­ern­ing mat­ter must be un­der­stood as metaphor­i­cal, but this is a small price to pay for a view that ac­tu­ally makes sense. There may be a con­cern that we are los­ing some im­por­tant ex­plana­tory ground here. After all, on the pre­scrip­tive view the laws of na­ture ex­plain why all elec­trons at­tract one an­other, whereas on the de­scrip­tive view the laws just restate the fact that all elec­trons at­tract one an­other. But con­sider the fol­low­ing di­alogue:

A: Why are these two metal blocks re­pel­ling each other?

B: Be­cause they’re both nega­tively charged, which means they have an ex­cess of elec­trons, and elec­trons re­pel one an­other.

A: But why do elec­trons re­pel one an­other?

B: Be­cause like charges always re­pel.

A: But why is that?

B: Be­cause if you do the path in­te­gral for the elec­tro­mag­netic field (us­ing Maxwell’s La­grangian) with source terms cor­re­spond­ing to two spa­tially sep­a­rated lumps of iden­ti­cal charge den­sity, you will find that the po­ten­tial en­ergy of the field is greater the smaller the spa­tial sep­a­ra­tion be­tween the lumps, and we know the force points in the op­po­site di­rec­tion to the gra­di­ent of the po­ten­tial en­ergy.

A: But why are the dy­nam­ics of the elec­tro­mag­netic field de­rived from Maxwell’s La­grangian rather than some other equa­tion? And why does the path in­te­gral method work at all?


Is the last link in this chain do­ing any ex­plana­tory work at all? Does it give us any fur­ther trac­tion on the prob­lem? B might as well have ended that con­ver­sa­tion by say­ing “Well, that’s just the way things are.” Now, laws of na­ture do have a priv­ileged role in phys­i­cal ex­pla­na­tion, but that priv­ilege is due to their sim­plic­ity and gen­er­al­ity, not to some mys­te­ri­ous quasi-causal power they ex­ert over mat­ter. The fact that a cer­tain gen­er­al­iza­tion is a law of na­ture does not ac­count for the truth and ex­plana­tory power of the gen­er­al­iza­tion, any more than the fact that a sol­dier has won the Medal of Honor ac­counts for his or her courage in com­bat. Law­hood is a recog­ni­tion of the gen­er­al­iza­tion’s truth and ex­plana­tory power. It is an hon­orific; it doesn’t con­fer any fur­ther ex­plana­tory oomph.

The Best Sys­tem Ac­count of Laws

David Lewis offers us a some­what worked out ver­sion of the de­scrip­tive con­cep­tion of law. Con­sider the set of all truths about the world ex­press­ible in a par­tic­u­lar lan­guage. We can con­struct de­duc­tive sys­tems out of this set of propo­si­tions by pick­ing out some of the propo­si­tions as ax­ioms. The log­i­cal con­se­quences of these ax­ioms are the the­o­rems of the de­duc­tive sys­tem. Th­ese de­duc­tive sys­tems com­pete with one an­other along (at least) two di­men­sions: the sim­plic­ity of the ax­ioms, and the strength or in­for­ma­tion con­tent of the sys­tem as a whole. We pre­fer sys­tems that give us more in­for­ma­tion about the world, but this greater strength of­ten comes at the cost of sim­plic­ity. For in­stance, a sys­tem whose ax­ioms com­prised the en­tire set of truths about the world would be max­i­mally strong, but not sim­ple at all. Con­versely, a sys­tem whose only ax­iom is some­thing like “Stuff hap­pens” would be pretty sim­ple, but very un­in­for­ma­tive. What we are look­ing for is the ap­pro­pri­ate bal­ance of sim­plic­ity and strength [1].

Ac­cord­ing to Lewis, the laws of na­ture cor­re­spond to the ax­ioms of the de­duc­tive sys­tem that best bal­ances sim­plic­ity and strength. He does not provide a pre­cise al­gorithm for eval­u­at­ing this bal­ance, and I don’t think his pro­posal should be read as an at­tempt at a tech­ni­cally pre­cise de­ci­sion pro­ce­dure for law­hood any­way. It is more like a heuris­tic pic­ture of what we are do­ing when we look for laws. We are look­ing for sim­ple gen­er­al­iza­tions that can be used to de­duce a large amount of in­for­ma­tion about the world. Laws are highly com­pressed de­scrip­tions of broad classes of phe­nom­ena. This view ev­i­dently differs quite sub­stan­tially from the Davies pic­ture I pre­sented at the be­gin­ning of this post. On Lewis’s view, the col­lec­tion of par­tic­u­lar facts about the world de­ter­mines the laws of na­ture, since the laws are merely com­pact de­scrip­tions of those facts. On Davies’s view, the de­ter­mi­na­tion runs the other way. The laws are in­de­pen­dent en­tities that de­ter­mine the par­tic­u­lar facts about the world. Stuff in the world is ar­ranged the way it is be­cause the laws com­pel­led that ar­range­ment.

One last point about Lewis’s ac­count. Lewis ac­knowl­edges that there is an im­por­tant lan­guage de­pen­dence in his view of laws. If we ig­nore this, we get ab­surd re­sults. For in­stance, con­sider a sys­tem whose only ax­iom is “For all x, x is F” where “F” is defined to be a pred­i­cate that ap­plies to all and only events that oc­cur in this world. This ax­iom is max­i­mally in­for­ma­tive, since it rules out all other pos­si­ble wor­lds, and it seems ex­ceed­ingly sim­ple. Yet we wouldn’t want to de­clare it a law of na­ture. The prob­lem, ob­vi­ously, is that all the com­plex­ity of the ax­iom is hid­den by our choice of lan­guage, with this weird spe­cially rigged pred­i­cate. To rule out this pos­si­bil­ity, Lewis speci­fies that all can­di­date de­duc­tive sys­tems must em­ploy the vo­cab­u­lary of fun­da­men­tal physics.

But we could also re­gard law­hood as a 2-place func­tion which maps a propo­si­tion and vo­cab­u­lary pair to “True” if the propo­si­tion is an ax­iom of the best sys­tem in that vo­cab­u­lary and “False” oth­er­wise. Lewis has cho­sen to curry this func­tion by fix­ing the vo­cab­u­lary vari­able. Leav­ing the func­tion un­cur­ried, how­ever, high­lights that we could have differ­ent laws for differ­ent vo­cab­u­laries and, con­se­quently, for differ­ent lev­els of de­scrip­tion. If I were an economist, I wouldn’t be in­ter­ested (at least not qua economist) in de­duc­tive sys­tems that talked about quarks and lep­tons. I would be in­ter­ested in de­duc­tive sys­tems that talked about prices and de­mand. The best sys­tem for this coarser-grained vo­cab­u­lary will give us the laws of eco­nomics, dis­tinct from the laws of physics.

Law­hood Is in the Map, not in the Territory

There is an­other sig­nifi­cant differ­ence be­tween the de­scrip­tive and pre­scrip­tive ac­counts that I have not yet dis­cussed. On the Davies-style con­cep­tion of laws as rules, law­hood is an el­e­ment of re­al­ity. A law is a dis­tinc­tive beast, an ab­stract en­tity perched in a tran­scen­dent aerie. On the de­scrip­tive ac­count, by com­par­i­son, law­hood is part of our map, not the ter­ri­tory. Note that I am not say­ing that the laws them­selves are a fea­ture of the map and not the ter­ri­tory. Laws are just par­tic­u­larly salient re­dun­dan­cies, ones that per­mit us to con­struct use­ful com­pressed de­scrip­tions of re­al­ity. Th­ese re­dun­dan­cies are, of course, out there in the ter­ri­tory. How­ever, the fact that cer­tain reg­u­lar­i­ties are es­pe­cially use­ful for the or­ga­ni­za­tion of knowl­edge is at least par­tially de­pen­dent on facts about us, since we are the ones do­ing the or­ga­niz­ing in pur­suit of our par­tic­u­lar prac­ti­cal pro­jects. Na­ture does not flag these reg­u­lar­i­ties as laws, we do.

This re­al­iza­tion has con­se­quences for how we eval­u­ate cer­tain forms of re­duc­tion­ism. I should be­gin by not­ing that there is a type of re­duc­tion­ism I ten­ta­tively en­dorse and that I think is un­touched by these spec­u­la­tions. I call this mere­olog­i­cal re­duc­tion­ism [2]; it is the claim that all the stuff in the uni­verse is en­tirely built out of the kinds of things de­scribed by fun­da­men­tal physics. The vague state­ment is in­ten­tional, since fun­da­men­tal physi­cists aren’t yet sure what kinds of things they are de­scribing, but the mo­ti­vat­ing idea be­hind the view is to rule out the ex­is­tence of im­ma­te­rial souls and the like. How­ever, re­duc­tion­ists typ­i­cally em­brace a stronger form of re­duc­tion­ism that one might la­bel nomic re­duc­tion­ism [3]. The view is that the fun­da­men­tal laws of physics are the only re­ally ex­is­tant laws, and that laws in the non-fun­da­men­tal dis­ci­plines are merely con­ve­nient short-cuts that we must em­ploy due to our com­pu­ta­tional limi­ta­tions.

One ap­peal­ing ar­gu­ment for this form of re­duc­tion­ism is the ap­par­ent su­perfluity of non-fun­da­men­tal laws. Macro­scopic sys­tems are en­tirely built out of parts whose be­hav­ior is de­ter­mined by the laws of physics. It fol­lows that the be­hav­ior of these sys­tems is also fixed by those fun­da­men­tal laws. Ad­di­tional non-fun­da­men­tal laws are otiose; there is noth­ing left for them to do. Barry Loewer puts it like this: “Why would God make [non-fun­da­men­tal laws] the day af­ter he made physics when the world would go on ex­actly as if they were there with­out them?” If these laws play no ex­plana­tory role, Ock­ham’s ra­zor de­mands that we strike them from our on­tolog­i­cal cat­a­log, leav­ing only the fun­da­men­tal laws.

I trust it is ap­par­ent that this ar­gu­ment re­lies on the pre­scrip­tive con­cep­tion of laws. It as­sumes that real laws of na­ture do stuff; they push and pull mat­ter and en­ergy around. It is this im­plicit as­sump­tion that raises the overde­ter­mi­na­tion con­cern. On this as­sump­tion, if the fun­da­men­tal laws of physics are already lord­ing it over all mat­ter, then there is no room for an­other lo­cus of au­thor­ity. How­ever, the ar­gu­ment (and much of the ap­peal of the as­so­ci­ated re­duc­tion­ist view­point) fiz­zles, if we re­gard laws as de­scrip­tive. Em­ploy­ing a Lewisian ac­count, all we have are differ­ent best sys­tems, geared to­wards vo­cab­u­laries at differ­ent re­s­olu­tions, that high­light differ­ent reg­u­lar­i­ties as the ba­sis for a com­pressed de­scrip­tion of a sys­tem. There is noth­ing prob­le­matic with hav­ing differ­ent ways to com­press in­for­ma­tion about a sys­tem. Speci­fi­cally, we are not com­pel­led by wor­ries about overde­ter­mi­na­tion to de­clare one of these meth­ods of com­pres­sion to be more real than an­other. In re­sponse to Loewer’s the­olog­i­cal ques­tion, the pro­po­nent of the de­scrip­tive con­cep­tion could say that God does not get to sep­a­rately spec­ify the non-fun­da­men­tal and fun­da­men­tal laws. By cre­at­ing the pat­tern of events in space-time she im­plic­itly fixes them all.

Nomic re­duc­tion­ism would have us be­lieve that the law­hood of the laws of physics is part of the ter­ri­tory, while the law­hood of the laws of psy­chol­ogy is just part of our map. Once we em­brace the de­scrip­tive con­cep­tion of laws, how­ever, there is no longer this sharp on­tolog­i­cal di­vide be­tween the fun­da­men­tal and non-fun­da­men­tal laws. One rea­son for priv­ileg­ing the laws of physics is re­vealed to be the product of a con­fused meta­phys­i­cal pic­ture. How­ever, one might think there are still other good rea­sons for priv­ileg­ing these laws that en­tail a re­duc­tion­ism more ro­bust than the mere­olog­i­cal va­ri­ety. For in­stance, even if we ac­cept that laws of physics don’t pos­sess a differ­ent on­tolog­i­cal sta­tus, we can still be­lieve that they have a prized po­si­tion in the ex­plana­tory hi­er­ar­chy. This leads to ex­plana­tory re­duc­tion­ism, the view that ex­pla­na­tions couched in the vo­cab­u­lary of fun­da­men­tal physics are always bet­ter be­cause fun­da­men­tal physics pro­vides us with more ac­cu­rate mod­els than the non-fun­da­men­tal sci­ences. Also, even if one de­nies that the laws of physics them­selves are push­ing mat­ter around, one can still be­lieve that all the ac­tual push­ing and pul­ling there is, all the causal ac­tion, is de­scribed by the laws of physics, and that the non-fun­da­men­tal laws do not de­scribe gen­uine causal re­la­tions. We could call this kind of view causal re­duc­tion­ism.

Un­for­tu­nately for the re­duc­tion­ist, ex­plana­tory and causal re­duc­tion­ism don’t fare much bet­ter than nomic re­duc­tion­ism. Stay tuned for the rea­sons why!

[1] Lewis ac­tu­ally adds a third desider­a­tum, fit, that al­lows for the eval­u­a­tion of sys­tems with prob­a­bil­is­tic ax­ioms, but I leave this out for sim­plic­ity of ex­po­si­tion. I have tweaked Lewis’s pre­sen­ta­tion in a cou­ple of other ways as well. For his own ini­tial pre­sen­ta­tion of the view, see Coun­ter­fac­tu­als, pp. 72-77. For a more up-to-date pre­sen­ta­tion, deal­ing es­pe­cially with is­sues in­volv­ing prob­a­bil­is­tic laws, see this pa­per (PDF).

[2] From the Greek meros, mean­ing “part”.

[3] From the Greek nomos, mean­ing “law”.