Mathematics as a lossy compression algorithm gone wild

This is yet an­other half-baked post from my old draft col­lec­tion, but feel free to Crocker away.

There is an old adage from Eu­gene Wigner known as the “Un­rea­son­able Effec­tive­ness of Math­e­mat­ics”. Wikipe­dia:

the math­e­mat­i­cal struc­ture of a phys­i­cal the­ory of­ten points the way to fur­ther ad­vances in that the­ory and even to em­piri­cal pre­dic­tions.

The way I in­ter­pret is that it is pos­si­ble to find an al­gorithm to com­press a set of data points in a way that is also good at pre­dict­ing other data points, not yet ob­served. In yet other words, a good ap­prox­i­ma­tion is, for some rea­son, some­times also a good ex­trap­o­la­tion. The rest of this post elab­o­rates on this anti-Pla­tonic point of view.

Now, this point of view is not ex­actly how most peo­ple see math. They imag­ine it as some near-mag­i­cal thing that tran­scends sci­ence and re­al­ity and, when dis­cov­ered, learned and used prop­erly, gives one limited pow­ers of clair­voy­ance. While only the se­lect few wiz­ard have the power to dis­cover new spells (they are known as sci­en­tists), the rank and file can still use some of the in­can­ta­tions to make oth­er­wise im­pos­si­ble things to hap­pen (they are known as en­g­ineers).

This meta­phys­i­cal view is col­or­fully ex­pressed by Stephen Hawk­ing:

What is it that breathes fire into the equa­tions and makes a uni­verse for them to de­scribe? The usual ap­proach of sci­ence of con­struct­ing a math­e­mat­i­cal model can­not an­swer the ques­tions of why there should be a uni­verse for the model to de­scribe. Why does the uni­verse go to all the bother of ex­ist­ing?

Should one in­ter­pret this as if he pre­sumes here that math, in the form of “the equa­tions” comes first and only then there is a phys­i­cal uni­verse for math to de­scribe, for some val­ues of “first” and “then”, any­way? Pla­ton­ism seems to reach roughly the same con­clu­sions:

Wikipe­dia defines pla­ton­ism as

the philos­o­phy that af­firms the ex­is­tence of ab­stract ob­jects, which are as­serted to “ex­ist” in a “third realm dis­tinct both from the sen­si­ble ex­ter­nal world and from the in­ter­nal world of con­scious­ness, and is the op­po­site of nominalism

In other words, math would have “ex­isted” even if there were no hu­mans around to dis­cover it. In this sense, it is “real”, as op­posed to “imag­ined by hu­mans”. Wikipe­dia on math­e­mat­i­cal re­al­ism:

math­e­mat­i­cal en­tities ex­ist in­de­pen­dently of the hu­man mind. Thus hu­mans do not in­vent math­e­mat­ics, but rather dis­cover it, and any other in­tel­li­gent be­ings in the uni­verse would pre­sum­ably do the same. In this point of view, there is re­ally one sort of math­e­mat­ics that can be dis­cov­ered: tri­an­gles, for ex­am­ple, are real en­tities, not the cre­ations of the hu­man mind.

Of course, the de­bate on whether math­e­mat­ics is “in­vented” or “dis­cov­ered” is very old. Eliezer-2008 chimes in in http://​​less­wrong.com/​​lw/​​mq/​​beau­tiful_math/​​:

To say that hu­man be­ings “in­vented num­bers”—or in­vented the struc­ture im­plicit in num­bers—seems like claiming that Neil Arm­strong hand-crafted the Moon. The uni­verse ex­isted be­fore there were any sen­tient be­ings to ob­serve it, which im­plies that physics pre­ceded physi­cists.

and later:

The amaz­ing thing is that math is a game with­out a de­signer, and yet it is em­i­nently playable.

In the above, I as­sume that what Eliezer means by physics is not the sci­ence of physics (a hu­man en­deavor), but the laws ac­cord­ing to which our uni­verse came into ex­is­tence and evolved. Th­ese laws are not the uni­verse it­self (which would make the state­ment “physics pre­ceded physi­cists” sim­ply “the uni­verse pre­ceded physi­cists”, a vac­u­ous tau­tol­ogy), but some sep­a­rate laws gov­ern­ing it, out there to be dis­cov­ered. If only we knew them all, we could cre­ate a copy of the uni­verse from scratch, if not “for real”, then at least as a faith­ful model. This uni­verse-mak­ing recipe is then what physics (the laws, not sci­ence) is.

And these laws ap­par­ently re­quire math­e­mat­ics to be prop­erly ex­pressed, so math­e­mat­ics must “ex­ist” in or­der for the laws of physics to ex­ist.

Is this the only way to think of math? I don’t think so. Let us sup­pose that the phys­i­cal uni­verse is the only “real” thing, none of those Pla­tonic ab­stract ob­jects. Let is fur­ther sup­pose that this uni­verse is (some­what) pre­dictable. Now, what does it mean for the uni­verse to be pre­dictable to be­gin with? Pre­dictable by whom or by what? Here is one ap­proach to pre­dictabil­ity, based on agency: a small part of the uni­verse (you, the agent) can con­struct/​con­tain a model of some larger part of the uni­verse (say, the earth-sun sys­tem, in­clud­ing you) and op­ti­mize its own ac­tions (to, say, wake up the next morn­ing just as the sun rises).

Does wak­ing up on time count as do­ing math? Cer­tainly not by the con­ven­tional defi­ni­tion of math. Do mi­gra­tory birds do math when they mi­grate thou­sands of miles twice a year, suc­cess­fully pre­dict­ing that there would be food sources and warm weather once they get to their des­ti­na­tion? Cer­tainly not by the con­ven­tional defi­ni­tion of math. Now, sup­pose a ship cap­tain lays a course to fol­low the birds, us­ing maps and ta­bles and calcu­la­tions? Does this count as do­ing math? Why, cer­tainly the cap­tain would say so, even if the math in ques­tion is rel­a­tively sim­ple. Some­times the in­puts both the birds and the hu­mans are us­ing are the same: sun and star po­si­tions at var­i­ous times of the day and night, the mag­netic field di­rec­tion, the shape of the ter­rain.

What is the differ­ence be­tween what the birds are do­ing and what hu­mans are do­ing? Cer­tainly both make pre­dic­tions about the uni­verse and act on them. Only birds do this in­stinc­tively and hu­mans con­sciously, by “ap­ply­ing math”. But this is a state­ment about the differ­ences in cog­ni­tion, not about some Pla­tonic math­e­mat­i­cal ob­jects. One can even say that birds perform the rele­vant math in­stinc­tively. But this is a rather slip­pery slope. By this defi­ni­tion amoe­bas solve the diffu­sion equa­tion when they move along the sugar gra­di­ent to­ward a food source. While this view has mer­its, the math­e­mat­i­ci­ans an­a­lyz­ing cer­tain as­pects of the Navier-Stokes equa­tion might not take kindly be­ing com­pared to a pro­to­zoa.

So, like JPEG is a lossy image com­pres­sion al­gorithm of the part of the uni­verse which cre­ates an image on our retina when we look at a pic­ture, the col­lec­tion of the New­ton’s laws is a lossy com­pres­sion al­gorithm which de­scribes how a thrown rock falls to the ground, or how planets go around the Sun. in both cases we, a tiny part of the uni­verse, are able to model and pre­dict a much larger part, albeit with some loss of ac­cu­racy.

What would it mean then for a Uni­verse to not “run on math”? In this ap­proach it means that in such a uni­verse no sub­sys­tem can con­tain a model, no mat­ter how coarse, of a larger sys­tem. In other words, such a uni­verse is com­pletely un­pre­dictable from the in­side. Such a uni­verse can­not con­tain agents, in­tel­li­gence or even the sim­plest life forms.

Now, to the “gone wild” part of the ti­tle. This is where the tra­di­tional ap­plied math, like count­ing sheep, or calcu­lat­ing how many can­nons you can arm a ship with be­fore it sinks, or how to pre­dict/​cause/​ex­ploit the stock mar­ket fluc­tu­a­tions, be­comes “pure math”, or math for math’s sake, be it prov­ing the Pythagorean the­o­rem or solv­ing a Millen­nium Prize prob­lem. At this point the math­e­mat­i­cian is no longer in­ter­ested in mod­el­ing a larger part of the uni­verse (ex­cept in­so­far as she pre­dicts that it would be a fun thing to do for her, which is prob­a­bly not very math­e­mat­i­cal).

Now, there is at least one se­ri­ous ob­jec­tion to this “math is jpg” episte­mol­ogy. It goes as fol­lows: “in any uni­verse, no mat­ter how con­voluted, 1+1=2, so clearly math­e­mat­ics tran­scends the spe­cific struc­ture of a sin­gle uni­verse”. I am skep­ti­cal of this logic, since to me 1,+,= and 2 are semi-in­tu­itive mod­els run­ning in our minds, which evolved to model the uni­verse we live in. I can cer­tainly imag­ine a uni­verse where none of these con­cepts would be use­ful in pre­dict­ing any­thing, and so they would never evolve in the “mind” of what­ever en­tity in­hab­its it. To me math­e­mat­i­cal con­cepts are no more uni­ver­sal than moral con­cepts: some­times they crys­tal­lize into use­ful mod­els, and some­times they do not. Like the hu­man con­cept of honor would not be use­ful to spi­ders, the con­cept of num­bers (which prob­a­bly is use­ful to spi­ders) would not be use­ful in a uni­verse where size is not a well-defined con­cept (like some­thing based on a Con­for­mal Field The­ory).

So the “Un­rea­son­able Effec­tive­ness of Math­e­mat­ics” is not at all un­rea­son­able: it re­flects the pre­dictabil­ity of our uni­verse. Noth­ing “breathes fire into the equa­tions and makes a uni­verse for them to de­scribe”, the equa­tions are but one way a small part of the uni­verse pre­dicts the salient fea­tures of a larger part of it. Rather, an in­ter­est­ing ques­tion is what fea­tures of a pre­dictable uni­verse en­able agents to ap­pear in it, and how com­plex and pow­er­ful can these agents get.