# 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.

• This ties in well with the in­tel­li­gence-as-com­pres­sion paradigm: much of math­e­mat­ics can be in­ter­preted as a col­lec­tion of very short pro­grams, and so in a pre­dictable uni­verse with a bias to­wards short pro­grams, it’s un­sur­pris­ing if a lot of them turn out to be use­ful some­where or other.

• Those “very short pro­grams” are use­ful even if the uni­verse has no bias to­wards them. It’s just Oc­cam’s ra­zor. I think it has more to do with the pro­cess of knowl­edge gath­er­ing than with the uni­verse it­self.

• Those “very short pro­grams” are use­ful even if the uni­verse has no bias to­wards them.

They are? How? If they have no priv­ileged sta­tus and phe­nom­ena are due to long pro­grams as likely as short pro­grams (leav­ing aside the is­sue of how they works given that there are so many more long pro­grams than short ones), then they don’t pre­dict well. That doesn’t sound use­ful.

It’s just Oc­cam’s ra­zor.

And what jus­tifies Oc­cam’s ra­zor if the uni­verse has no bias to­wards short pro­grams?

• 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.

Well, yes, and the rea­son isn’t mys­te­ri­ous.

In or­der to com­press a stream of data you need to dis­cover some struc­ture in it. If there is no struc­ture—e.g. if the stream is truly ran­dom—then no com­pres­sion is pos­si­ble. And if the struc­ture you found is “re­ally there” and not an ar­ti­fact of your struc­ture-search­ing tech­niques, then it just as use­ful for ex­trap­o­la­tion and pre­dic­tion.

• 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.

I think when we say that the uni­verse “runs on math,” part of what we mean is that we can use sim­ple math­e­mat­i­cal laws to pre­dict (in prin­ci­ple) all as­pects of the uni­verse. We sus­pect that there is a lossless com­pres­sion al­gorithm, i.e., a the­ory of ev­ery­thing. This is a much stronger state­ment than just claiming that the uni­verse con­tains some pre­dictable reg­u­lar­i­ties, and is part of what makes the Pla­tonic ideas you are ar­gu­ing against seem ap­peal­ing.

We could imag­ine a uni­verse in which physics found lots of ap­prox­i­mate pat­terns that held most of the time and then got stuck, with no hint of any un­der­ly­ing or­der and sim­plic­ity. In such a uni­verse we would prob­a­bly not be so im­pressed with the idea of the uni­verse “run­ning on math” and these Pla­tonic ideas might be less ap­peal­ing.

• We sus­pect that there is a lossless com­pres­sion al­gorithm, i.e., a the­ory of ev­ery­thing.

Yeah, I don’t see this as likely at all. As I re­peat­edly said here, it’s mod­els all the way down.

• Fair enough. I can see the ap­peal of your view if you don’t think there’s a the­ory of ev­ery­thing. But given the suc­cess of fun­da­men­tal physics so far, I find it hard to be­lieve that there isn’t such a the­ory!

• Given that ev­ery time we dis­cover some­thing new we find that there are more ques­tions than an­swers, I find it hard to be­lieve that the pro­cess should con­verge some day.

• ev­ery time we dis­cover some­thing new we find that there are more ques­tions than answers

I don’t think that’s re­ally true though. The ad­vances in physics that have been worth cel­e­brat­ing—New­to­nian me­chan­ics, Maxwellian elec­tro­mag­netism, Ein­stei­nian rel­a­tivity, the elec­troweak the­ory, QCD, etc.--have been those that an­swer lots and lots of ques­tions at once and raise only a few new ques­tions like “why this the­ory?” and “what about higher en­er­gies?”. Now we’re at the point where the Stan­dard Model and GR to­gether an­swer al­most any ques­tion you can ask about how the world works, and there are rel­a­tively few ques­tions re­main­ing, like the prob­lem of quan­tum grav­ity. Think how much more nar­row and neatly-posed this prob­lem is com­pared to the pre-New­to­nian prob­lem of ex­plain­ing all of Na­ture!

• Math­e­mat­ics is gen­er­ally thought of more of a method of pre­dict­ing things. Since pre­dic­tion and com­pres­sion are equiv­a­lent (a good com­pres­sion al­gorithm is pre­cisely one that has shorter state­ments for more likely pre­dic­tions), it’s equiv­a­lent to say that math is a com­pres­sion al­gorithm.

• Can I nom­i­nate for pro­mo­tion to Main/​Front Page?

• Some time ago I sug­gested that (non-link, non-meta) Dis­cus­sion posts should be au­to­mat­i­cally pro­moted to Main if they are up­voted above 20-30 karma. This post is cur­rently well be­low.

• OK, talked to a few reg­u­lars off-line, the opinions are de­cid­edly mixed, so not go­ing to move this post to Main.

• The post us cur­rently on the35, can I vote for the move?

• Thanks. If any of the ad­mins find it worth it, I don’t mind if they move it.

• Pos­si­bly re­lated:

The Rea­son­able Ineffec­tive­ness Of Math­e­mat­ics (warn­ing: PDF)

• Neat. While this ar­ti­cle does not go far enough for my tastes, I am quite happy that it con­firms my “strong non-Pla­ton­ism” in­tu­ition (I called it anti-Pla­tonic.)

• Not hav­ing read the whole thing, it seems to make much of the failure of clas­si­cal mod­els to de­scribe and pre­dict elec­tronic cir­cuits. But if the cor­rect model is quan­tum, this isn’t nec­es­sar­ily sur­pris­ing.

It seems pos­si­ble that a set of phys­i­cal laws will be dis­cov­ered that al­lows lossless com­pres­sion. Maybe QM as now un­der­stood will be in­cluded in that set. If so, it seems that those laws, and the math­e­mat­ics in­volved therein, de­scribe some­thing real.

But that’s not Pla­ton­ism by the Wikipe­dia defi­ni­tion.

Say that math­e­mat­ics is about gen­er­at­ing com­pressed mod­els of the world. How do we gen­er­ate these mod­els? Surely we will want to study (com­press) our most pow­er­ful com­pres­sion heurestics. Is that not what pure math is?

• I agree I was too brief there. The origi­nal mo­ti­va­tion for math was to help figure out the phys­i­cal world. At some point (mul­ti­ple points, re­ally, start­ing with Eu­clid), perfect­ing the tools for their own sake be­came just as much of a mo­ti­va­tion. This is not a judge­ment but an ob­ser­va­tion. Yes, some­times “pure math” yields un­ex­pected benefits, but this is more of a co­in­ci­dence then the rea­son peo­ple do it (de­spite what the grant ap­pli­ca­tions might say).

Say that math­e­mat­ics is about gen­er­at­ing com­pressed mod­els of the world. How do we gen­er­ate these mod­els? Surely we will want to study (com­press) our most pow­er­ful com­pres­sion heuris­tics. Is that not what pure math is?

The main rea­son is pure cu­ri­os­ity with­out any care for even­tual ap­pli­ca­bil­ity to un­der­stand­ing the phys­i­cal world. Pre­tend­ing oth­er­wise would be dis­in­gen­u­ous.

The­o­ret­i­cal physics is not very differ­ent in that re­gard. To quote Feyn­man

Physics is like sex: sure, it may give some prac­ti­cal re­sults, but that’s not why we do it.

• I guess I always took the phrase “un­rea­son­able effec­tive­ness” to re­fer to the “co­in­ci­dence” you men­tion in your re­ply. I’m not re­ally sure you’ve gone far to­ward ex­plain­ing this co­in­ci­dence in your ar­ti­cle. Just what is it that you think math­e­mat­i­ci­ans have “pure cu­ri­ousity” about? What does it mean to “perfect a tool for its own sake” and why do those perfec­tions some­times wind up hav­ing prac­ti­cal fur­ther use. As a pure math­e­mat­i­cian, I never think about ap­ply­ing a tool to the real world, but I do think I’m work­ing to­wards a very com­pressed un­der­stand­ing of tool mak­ing.

• I guess I always took the phrase “un­rea­son­able effec­tive­ness” to re­fer to the “co­in­ci­dence” you men­tion in your re­ply.

Un­rea­son­able effec­tive­ness tends to re­fer to the ob­ser­va­tion that the same math­e­mat­i­cal tools, like, say, math­e­mat­i­cal anal­y­sis, end up use­ful for mod­el­ing very dis­parate phe­nom­ena. In a more ba­sic form it is “why do math­e­mat­i­cal ideas help us un­der­stand the world so well?”. The an­swer sug­gested in the OP is that the ques­tion is a tau­tol­ogy: math is a meta-model build by hu­man minds, not a col­lec­tion of some ab­stract ob­jects which hu­mans dis­cover in their pur­suit of bet­ter mod­els of the world. The JPEG anal­ogy is that ask­ing why math is un­rea­son­ably effec­tive in con­struct­ing dis­parate (lossy) mod­els is like ask­ing why the JPEG al­gorithm is un­rea­son­ably effec­tive in lossy com­pres­sion of dis­parate images.

The “co­in­ci­dence” part referred to some­thing else: that pur­su­ing math re­search for its own sake may oc­ca­sion­ally work out ot be use­ful for mod­el­ing the phys­i­cal world, num­ber the­ory and en­cryp­tion be­ing the stan­dard ex­am­ple.

As a pure math­e­mat­i­cian, I never think about ap­ply­ing a tool to the real world, but I do think I’m work­ing to­wards a very com­pressed un­der­stand­ing of tool mak­ing.

When I talk to some­one who works in pure math, they usu­ally de­scribe the mo­ti­va­tion for what they do in al­most artis­tic terms, not car­ing whether what they do can be use­ful for any­thing, so “tool mak­ing” does not seem like the right term.

• Great ar­ti­cle, though I’ve always been a bit more of a math­e­mat­i­cal re­al­ist my­self.

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

The part that still fas­ci­nates me is how tak­ing a cou­ple of differ­ent math­e­mat­i­cal de­scrip­tions of cer­tain phe­nom­ena and work­ing solely with the num­bers un­der the “laws” of math­e­mat­ics can lead to math­e­mat­i­cal the­o­ries and pre­dic­tions of seem­ingly un­re­lated phe­nom­ena.

For ex­am­ple, Ein­stein de­vel­oped Spe­cial Rel­a­tivity to ac­count for the in­con­sis­ten­cies be­tween clas­si­cal me­chan­ics and Maxwell’s equa­tions us­ing pri­mar­ily the ob­ser­va­tion that the speed of light is ab­solute re­gard­less of the mo­tion of the light source and the pos­tu­late that the laws of physics are the same in all in­er­tial refer­ence frames. He just worked with the num­bers un­der the rules of math­e­mat­ics to (in­de­pen­dently) de­velop the Lorentz trans­for­ma­tions which lead to his Spe­cial Rel­a­tivity.

The key fea­ture here is that Ein­stein did not perform ex­per­i­ments. He knew form the null-re­sult of the Michel­son-Mor­ley ex­per­i­ment that the speed of light is con­stant, but be­sides that, the vast amount of the work done was what Richard Ham­ming called “Scholas­tic” in its ap­proach. I’ve even heard it said that as far back as New­ton, the idea of non-lo­cal­ity was con­sid­ered pre­pos­ter­ous and that in it­self gives the idea of a uni­ver­sal speed limit which might have been enough, along with Gal­ileo’s Prin­ci­ple of rel­a­tivity to get very close to Spe­cial Rel­a­tivity us­ing only the tools of math­e­mat­ics he had available to him and the cur­rent the­o­ries of mo­tion and elec­tro­mag­netism.

Now ob­vi­ously a lot of work went into that, but so many strange pre­dic­tions fell out of it that it re­ally is amaz­ing, some might say “un­rea­son­able”. For ex­am­ple, time di­la­tion, mass/​en­ergy equiv­alence, length con­trac­tion, and even­tu­ally (af­ter GR) black holes, rel­a­tivis­tic cos­mol­ogy, grav­i­ta­tional lens­ing, and the ex­is­tence of dark mat­ter.

That we can de­rive knowl­edge about how the uni­verse works be­cause of in­con­sis­ten­cies be­tween sim­pler math­e­mat­i­cal de­scrip­tions of var­i­ous phe­nom­ena re­ally does seem to sug­gest that the uni­verse “runs on math”. Now I don’t mean to sug­gest that the equa­tions are writ­ten some­where in the sky and that some en­tity “breathes fire” into them; Just that the struc­ture of the uni­verse is iso­mor­phic to the ideal math that we could use to ex­plain and pre­dict it. I would not be at all sur­prised to find out that some­how, “they are the same thing,” what­ever that might mean.

• the struc­ture of the uni­verse is iso­mor­phic to the ideal math that we could use to ex­plain and pre­dict it. I would not be at all sur­prised to find out that some­how, “they are the same thing,” what­ever that might mean.

Teg­mark’s Math­e­mat­i­cal uni­verse hy­poth­e­sis is one an­swer to what that might mean.

• For ob­servers to ex­ist some parts of the uni­verse must fol­low pat­terns, but not nec­es­sar­ily all of it. Could the “Un­rea­son­able Effec­tive­ness of Math­e­mat­ics” re­late to why all of the uni­verse can prob­a­bly be mod­eled with math?

• That seems re­lated to the an­thropic ques­tions of why the uni­verse is much more or­dered than the min­i­mum re­quired for Boltz­mann brains, or much older than the min­i­mum re­quired for life to de­velop.

• I think this is in­ter­est­ing even if I don’t fully but into it’s ar­gu­ment. May I ask what your math­e­mat­i­cal back­ground is? I have a men­tal pre­dic­tion based on the post that I’d like to test.

• He shouldn’t tell it to you, but in se­cret to a third party. Other­wise the ex­per­i­ment would be ru­ined.

• I think we can trust shminux not to go get ad­di­tional de­grees in re­sponse to WQ’s an­swer.

• Never un­der­es­ti­mate the race for sta­tus.

• He has sent me his pre­dic­tion in a pri­vate mes­sage.

• Well?

• The pre­dic­tion was:

My pre­dic­tion is that the writer has a strong back­ground in calcu­lus in­clud­ing multi vari­ate but has not taken more the­o­ret­i­cal proof courses like set the­ory /​ logic 2 /​ or num­ber the­ory.

A grad­u­ate physics de­gree no doubt re­quires calcu­lus. What about the rest?

• That’s about right. I did learn el­e­men­tary set the­ory and logic, and had to prove the­o­rems in the course of my re­search, but I was not in­ter­ested in, say, com­putabil­ity, ad­vanced logic, for­mal sys­tems or the num­ber the­ory. I guess Pla­ton­ism is more per­va­sive among pure math­e­mat­i­ci­ans.

• You’ve stud­ied ab­stract alge­bra too, no? At min­i­mum, you’ve dealt with groups.

• Oh, for sure, had to learn some, like finite groups, Lie groups/​alge­bras.

• How would it be ru­ined? My “math­e­mat­i­cal back­ground” does not de­pend on his pre­dic­tion and I can­not change it af­ter learn­ing what the pre­dic­tion is.

• But his pre­dic­tion could change sub­tly af­ter learn­ing what your back­ground is.

“Ruined” is too strong; I should have said “in­fluenced”.

• Right, that’s why I asked what it is be­fore re­veal­ing the an­swer.

• Your way of for­mu­lat­ing the an­swer may have been in­fluenced by his the­ory?

I’m quib­bling. I recom­mended the “full pro­ce­dure” out of habit. It’s wasn’t re­ally nec­es­sary in this par­tic­u­lar case.

• Any­way, I have about 3 years of col­lege math (geared to­ward physics, not math stu­dents), plus some el­e­ments of grad-level math re­quired for my grad de­gree, such as differ­en­tial ge­om­e­try and alge­braic topol­ogy.

• While I love your anal­ogy and agree that maths is simplifying

It’s also, not. I wish Wikipe­dia ed­i­tors and ju­rors were more sym­pa­thetic to the idea that the vast ma­jor­ity of Wikipe­dia’s au­di­ence doesn’t have have ca­pac­ity to over­come the cog­ni­tive com­plex­ity of math­e­mat­i­cal for­mal­isms in many tech­ni­cal ar­ti­cles and would ap­pre­ci­ate an en­glish in in­stead as well. Sim­ple en­glish Wikipe­dia doesn’t have its share of tech­ni­cal ar­ti­cles if that’s where you’d rather the en­glish went.

Mathsy peo­ple, to help you put your­self in our shoes, con­sider this Wiki ar­ti­cle on logic. Imag­ine that for­eign lan­guage but all the sym­bols are highly com­pressed in an area, like one side of a nu­mer­a­tor, with all kinds of re­la­tions be­tween them, rep­re­sent­ing differ­ent things. I sim­ply don’t have the work­ing mem­ory to make any sense of it by the time I’ve looked up what a par­tic­u­lar thing is and it’s re­la­tion to a few oth­ers.

• What is a uni­verse with­out hu­mans?

We are limited to sub­jec­tive ob­ser­va­tions, and can not con­firm what ob­jec­tive ob­ser­va­tions of the uni­verse would be.

I’d like to ar­gue in this com­ment that math­e­mat­ics is an im­plied prop­erty of the uni­verse. We might “mis­take” to think that math­e­mat­ics are gov­ern­ing the uni­verse, but rather the way the uni­verse works can be de­scribed from our sub­jec­tive per­spec­tive with the seem­ingly ab­stract en­tity of math­e­mat­ics. The uni­verse con­tains math­e­mat­ics in the way it ex­ists.

Claiming that math­e­mat­ics ex­ist in some other di­men­sion, is just about as rea­son­able as claiming that ap­ples ex­ist in an­other di­men­sion.

If there’s some­one who can proove that they’ve bro­ken the laws of physics while perform­ing math­e­mat­ics, then they have a valid ba­sis for math­e­mat­i­cal re­al­ism.

• Would a differ­ent uni­verse have differ­ent maths?

• What d’ya mean by “differ­ent” and by “maths”? Sure they would use differ­ent no­ta­tion but it’d de­note math­e­mat­i­cal struc­tures iso­mor­phic to our own.

• I was try­ing to fol­low through 395′s premise. If maths is im­plied by the uni­verse, then other uni­verses would im­ply other maths, n’est ce pas.

• We are limited to sub­jec­tive ob­ser­va­tions, and can not con­firm what ob­jec­tive ob­ser­va­tions of the uni­verse would be.

What does “ob­jec­tive ob­ser­va­tions of the uni­verse” mean?

• Hav­ing knowl­edge that would not be limited by be­ing an ob­server, a small part of the uni­verse.

What is time if you’re not a crea­ture ex­ist­ing in time?

• That is not the usual ob­jec­tive/​sub­jec­tive dis­tinc­tion.

• Hav­ing knowl­edge that would not be limited by be­ing an observer

Who/​what is the sub­ject of “hav­ing knowl­edge”?

• In my opinion that is ir­rele­vant as long as the in­for­ma­tion is not limited by the na­ture of the ob­server. How­ever I don’t in­tend to say that “hav­ing knowl­edge” could have a mean­ing out­side some sort of a sub­jec­tive thing hav­ing it. So I’d like to sep­a­rate the idea of some kind of truth from a sub­jec­tive ex­pe­rience of hav­ing it. If there is any truth like that. If there was, how could we know, if we don’t cur­rently? Does it make sense to con­tem­plate on the pos­si­bil­ity of there be­ing knowl­edge we can’t have? We are limited and we can’t re­ally think out­side the box. Know­ing that we can’t think out­side the box, does not provide the ca­pac­ity to sug­gest ev­ery­thing that could be out­side the box.

• Depends on the value of can’t. We can’t have the knowl­edge in the library of Alexan­dria..but coun­ter­fac­tu­ally we could have had.

• So I’d like to sep­a­rate the idea of some kind of truth from a sub­jec­tive ex­pe­rience of hav­ing it. If there is any truth like that. If there was, how could we know, if we don’t cur­rently?

That’s a pretty well-trod­den philos­o­phy topic :-)

• All analo­gies are sus­pect, but if I had to choose one I’d say physics’ the­o­ries—at best—are if any­thing like code that re­turns the Fibonacci se­quence through a speci­fied range. The the­o­ries give us a for­mula we can use to make cer­tain pre­dic­tions, in some cases with ar­bi­trary pre­ci­sion. Video, losslessly- or lossy-com­pressed, is still video. Whereas

fib n = take n fib­list where fib­list = 0:1:(zipWith (+) fib­list (tail fib­list))

is not a bag hol­ing the en­tire Fibonacci se­quence, wait­ing for us to com­press it so we can look at a slightly more pix­e­lated ver­sion of the ac­tual piece.

Also, I don’t think it makes sense to say math is part of na­ture (ex­cept in the sense ev­ery­thing is part of na­ture), though it may be that math is a psy­cholog­i­cal anal­ogy to some fea­ture of na­ture—like vi­sion is an anal­ogy to part of the EM spec­trum. It would be a strange co­in­ci­dence oth­er­wise, con­sid­er­ing how use­ful math is helping us make cer­tain pre­dic­tions. At the same time, many fea­tures of na­ture are ut­terly un­pre­dictable, so ei­ther math only images se­lect parts—we can’t see x-ray—or we haven’t fully un­der­stood how to use math­e­mat­ics yet.

In­ci­den­tally, I think it is true that math—all our sec­u­lar, ma­te­ri­al­ist pre­tense aside—is still widely felt to have mag­i­cal prop­er­ties. In this re­gard Descartes’ god is still with us.

• Shouldn’t this be physics as a lossy com­pres­sion al­gorithm? A lot of math has noth­ing to do with any­thing in the real world. I guess I agree, the math­e­mat­i­cal na­ture of phys­i­cal laws is sim­ply an ex­pres­sion of the pre­dictabil­ity of the uni­verse.

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

While num­bers might not be use­ful in some uni­verses, they would nev­er­the­less be cor­rect. 1 + 1 would still equal 2 be­cause they are defined that way. Whether the uni­verse con­tains an en­tity ca­pa­ble of ex­press­ing that fact doesn’t mat­ter.

• Shouldn’t this be physics as a lossy com­pres­sion al­gorithm?

Think of math as the JPEG com­pres­sion al­gorithm, and physics as cat pic­tures (not “real cats”). And some other sci­ence as, say, pic­tures of the sun­set. JPEG works on both kinds of pic­tures, even if there is not much in com­mon be­tween cats and sun­sets.

While num­bers might not be use­ful in some uni­verses, they would nev­er­the­less be cor­rect. 1 + 1 would still equal 2 be­cause they are defined that way. Whether the uni­verse con­tains an en­tity ca­pa­ble of ex­press­ing that fact doesn’t mat­ter.

That’s the Pla­tonic view. I am sug­gest­ing that in a uni­verse with­out cats and pic­tures no one would in­vent JPEG, just like in some other (or even this) uni­verse no one would in­vent hu­man moral­ity, even though one can state that “moral­ity ex­ists” as an ab­stract con­cept.

• I’m not sure that this is a use­ful dis­agree­ment. But any­way: the JPEG com­par­i­son is in­ter­est­ing be­cause JPEG is an al­gorithm, it doesn’t “ex­ist” but it isn’t con­tin­gent on any­thing phys­i­cal. Its just an al­gorithm. Does Ham­let “ex­ist” in an al­ter­nate uni­verse where Shake­speare was never born. The ques­tion doesn’t re­ally mean any­thing.

I’m not a Pla­ton­ist, if any­thing I would lean to to­wards for­mal­ism but I am very wary of tak­ing a firm po­si­tion on a philo­soph­i­cal is­sue that I am not in­ti­mately fa­mil­iar with. And I think that For­mal­ists and Pla­ton­ists would an­swer yes to the ques­tion of whether 1+ 1 = 2 in an al­ter­nate chaotic uni­verse (The logi­cists and in­tu­ition­ists are rare). You can read about this here.

The Pla­ton­ist view is that 1 + 1 = 2 ex­presses some truth about the en­tities “1”, “+”, “2” and “=”

The for­mal­ist view is that 1 + 1 = 2 fol­lows from the ax­ioms, its isn’t a truth but rather the pred­i­cate of a sys­tem of rules. If “math­e­mat­i­cal ax­ioms are granted” then “1 + 1 = 2″

But pretty much no philoso­pher thinks that math­e­mat­i­cal state­ments are con­tin­gent on some­thing. And I’m not sure if you’re say­ing that. Math­e­mat­i­cal state­ments would not be gen­er­ated in some uni­verses yes, but that’s not rele­vant to whether the state­ments them­selves are true. The state­ments are not con­tin­gent on the na­ture of the uni­verse.

If you think that they are con­tin­gent, how far does this go? Is the law of non-con­tra­dic­tion con­tin­gent be­cause in a suffi­ciently weird uni­verse, no one would dis­cover it?What about the state­ment “If X then X”?

• Math­e­mat­i­cal state­ments would not be gen­er­ated in some uni­verses yes, but that’s not rele­vant to whether the state­ments them­selves are true.

I guess I failed to pre­sent my view clearly enough. See if this works bet­ter:

Is the law of non-con­tra­dic­tion con­tin­gent be­cause in a suffi­ciently weird uni­verse, no one would dis­cover it?

Is the virtue of mercy con­tin­gent be­cause in a suffi­ciently weird uni­verse, no one would dis­cover it?

• Is the virtue of mercy con­tin­gent be­cause in a suffi­ciently weird uni­verse, no one would dis­cover it?

I don’t know if I’m a moral re­al­ist. But as­sum­ing I were for a sec­ond: mercy es­sen­tially means “be nice to peo­ple even if they are bad/​ do bad things”. Its not a state­ment about the world. Its not a truth claim. Its not a defi­ni­tion. Its not an al­gorithm. Its an im­per­a­tive, an in­struc­tion. As such it is not sub­ject to be­ing con­tin­gent or nec­es­sary. Is the state­ment “Eat your veg­eta­bles” true? Well no it doesn’t have a truth value. Like­wise with eth­i­cal state­ments.

Which isn’t re­motely similar to math­e­mat­i­cal state­ments which most philoso­phers ei­ther think ex­press truths (Pla­ton­ists) or for­mal­isms—se­quences of sym­bols that are gen­er­a­ble from sim­ple rules. But no philoso­phers that I know of think that math­e­mat­i­cal state­ments re­quire refer­ents in the real world. There are some very strange math­e­mat­i­cal fields that have noth­ing to do with this uni­verse. My take on this is that it doesn’t re­ally mat­ter all that mat­ters is that math­e­mat­ics is use­ful.

• 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.

Can you ac­tu­ally imag­ine or de­scribe one? I in­tel­lec­tu­ally can ac­cept that they might ex­ist, but I don’t know that my mind is ca­pa­ble of imag­in­ing a uni­verse which could not be simu­lated on a Tur­ing Ma­chine.

The way that I define Teg­mark’s Ul­ti­mate Ensem­ble is as the set of all wor­lds that can be simu­lated by a Tur­ing Ma­chine. Is it pos­si­ble to imag­ine in any con­crete way a uni­verse which doesn’t fall un­der that defi­ni­tion? Is there an even more Ul­ti­mate Ensem­ble that we can’t con­ceive of be­cause we’re crea­tures of a Tur­ing uni­verse?

• The set of all Tur­ing Machines is merely countable. In­stead, imag­ine an en­sem­ble of uni­verses cor­re­spond­ing to the real num­bers, some of which aren’t even com­putable.

For ex­am­ple, uni­verses run­ning on clas­si­cal me­chan­ics, where val­ues can be mea­sured with in­finite pre­ci­sion, and which also have con­tin­u­ous space and time to rep­re­sent those in­finitely pre­cise val­ues. In other words, a set of uni­verses where two uni­verses can be differ­ent in the po­si­tion of a par­ti­cle, po­si­tion be­ing defined as a real num­ber.

This seems easy to imag­ine—it’s a pretty stan­dard New­to­nian world.

• Even in a New­ton world, you can make pre­dic­tions. You just won’t be able to make them to in­finite pre­ci­sion.

I, too, am hav­ing trou­ble imag­in­ing a perfectly un­pre­dictable uni­verse with agents in it.

• I can’t ei­ther. But I can imag­ine a par­tially pre­dictable world, which can­not be perfectly simu­lated by any Tur­ing ma­chine (that’s what Gavin’s ques­tion was about), and yet is com­pletely de­ter­minis­tic and has a sim­ple math­e­mat­i­cal de­scrip­tion. For in­stance, a con­tin­u­ous, de­ter­minis­tic world where a fun­da­men­tal phys­i­cal con­stant is an un­com­putable num­ber.

• I see—you’re go­ing af­ter a much weaker crite­rion than the one de­scribed in the OP.

I’d note that my ‘finite pre­ci­sion’ com­ment cov­ers un­com­putable con­stants. Un­less no com­pu­ta­tion can even con­verge on them...

• I see—you’re go­ing af­ter a much weaker crite­rion than the one de­scribed in the OP.

I was re­spond­ing to Gavin’s com­ment and its crite­rion was “a uni­verse which could not be simu­lated on a Tur­ing Ma­chine”.

I’d note that my ‘finite pre­ci­sion’ com­ment cov­ers un­com­putable con­stants. Un­less no com­pu­ta­tion can even con­verge on them...

The set of all con­stants on which some com­pu­ta­tion con­verges is still only countable (each Tur­ing ma­chine’s out­put con­verges on at most one con­stant).

A uni­verse with un­com­putable con­stants in the laws of na­ture might still be pre­dictable in prac­tice. We don’t know if the con­stants in our own uni­verse are com­putable or not, be­cause we can’t even mea­sure them be­yond a few hun­dreds bits each. And as long as we keep mea­sur­ing them in­stead of de­riv­ing them from some equa­tion, our knowl­edge will pre­sum­ably re­main finite, so com­putabil­ity does not mat­ter for us in prac­tice.

On the other hand, full simu­la­tion does de­pend on the pre­cise val­ues. (If it didn’t, the con­stants wouldn’t have those pre­cise val­ues in any mean­ingful sense.)

• The set of all con­stants on which some com­pu­ta­tion con­verges is still only countable (each Tur­ing ma­chine’s out­put con­verges on at most one con­stant).

I meant to say ‘no com­pu­ta­tion can even con­strain them to a use­ful de­gree’. Strict con­ver­gence is not nec­es­sary.

• That seems to be the same as ‘no com­pu­ta­tion can out­put a pre­fix that is suffi­ciently long to be use­ful in pre­dic­tions’?

This would de­pend on how long a pre­fix you need. If there ex­ists a con­stant K such that K-length pre­fixes of all uni­ver­sal con­stants are enough, then cer­tainly all pos­si­ble bit­strings of length K can be com­puted.

The ques­tion be­comes: what does a uni­verse look like where you need to know the ex­act value of some con­stants to make use­ful pre­dic­tions? (Per­haps you can make some pre­dic­tions us­ing ap­prox­i­ma­tions, but some other pos­si­ble sce­nar­ios re­main un­pre­dictable with­out pre­cise knowl­edge.)

This re­quires know­ing an in­finite amount of in­for­ma­tion: some kind of phys­i­cal reifi­ca­tion of an un­com­putable real num­ber. (I’m only us­ing real num­bers as a fa­mil­iar ex­am­ple; larger in­fini­ties could be used too.) Not just to know the con­stant, but prob­a­bly also to know the pre­cise ini­tial state of the sys­tem whose evolu­tion you want to pre­dict.

One pos­si­ble reifi­ca­tion is the pre­cise po­si­tion, mass, mo­men­tum, etc. of an el­e­ment in the simu­la­tion—as­sum­ing these prop­er­ties can also have any real value, and that you can con­trol them to that de­gree.

I feel I could still imag­ine such a uni­verse. But ei­ther way that’s a fact about me, not about uni­verses. I don’t know how to pre­dict the mea­sure of such uni­verses in any mul­ti­verse the­ory.

• I would think those would all be rep­re­sentable by a Tur­ing Ma­chine, but I could be wrong about that. Cer­tainly, my un­der­stand­ing of the Ul­ti­mate Ensem­ble is that it would in­clude uni­verses that are con­tin­u­ous or in­clude ir­ra­tional num­bers, etc.

• Tur­ing Machines have dis­crete, not con­ti­nous states. There is a countable in­finity of Tur­ing Machines.

• I don’t know that my mind is ca­pa­ble of imag­in­ing a uni­verse which could not be simu­lated on a Tur­ing Ma­chine.

I never said it could not be, just that the Tur­ing Ma­chine would not be a con­cept that is likely to evolve there.

Imag­ine a uni­verse where there are no dis­crete en­tities, so num­bers/​ad­di­tion is not a use­ful model. What­ever in­hab­its such a uni­verse, if any­thing, would not de­velop the ab­strac­tion of count­ing. This uni­verse could still be Tur­ing-simu­lated (Tur­ing Ma­chine is an ab­strac­tion from our uni­verse),

This is the es­sen­tial point I am try­ing to make. Math­e­mat­ics is de­ter­mined by the struc­ture of the uni­verse and is not an in­de­pen­dent ab­stract en­tity. I feel like I failed, though.

• if you could figure out phys­i­cal laws from apri­ori maths, that re­ally would be un­rea­son­ably effec­tive. As it is, physi­cists have to care­fully se­lect maths that works phys­i­cally from the much larger amount that doesn’t. That this is pos­si­ble is about as sur­pris­ing as find­ing a true his­tory in the library of Ba­bel. The un­rea­son­able effec­tive­ness of maths is the rea­son­able effec­tive­ness of physics.

• For for­mal­ists (anti Pla­ton­ists), 1+1=2 is true in all uni­verses, pro­vid­ing you adopt ax­ioms from which it can be de­rived. It isn’t a truth about the uni­verse, for them, but it is con­tin­gent on choice of ax­ioms. This seems to lead to a vi­sion mul­ti­ple con­flict­ing math­e­mat­i­cal truths, which will seem coun­ter­in­tu­itive to some. Form­lists can but­tress their case by by ap­peal­ing to two ur -ax­ioms: maths is about min­imis­ing con­tra­dic­tion whilst max­imis­ing rich­ness of struc­ture,; and spec­u­lat­ing that these provide con­sid­er­able con­straints to ax­ioms choice.

• Start­ing from the premise that maths talks us about the world, you are faced with a vi­sion num­ber of puz­zles; whether maths is as real as the world, whether the world is as ab­stract as maths, why physi­cists and math­e­mat­i­ci­ans are differ­ent peo­ple do­ing differ­ent things, why some maths is in­cor­rect physics… Start­ing from the premise that ap­ple’s are or­anges, you are like­wise faced with puz­zles about why they look and taste differ­ent… Physics is the sci­ence of ex­plain­ing the world with maths. That maths can be used to ex­plain the world has the samem­con­tent as say­ing that physics works. It is not state­ment about maths per se. His­to­ri­ans suc­ceed in us­ing words to de­scribe events. That isn’t a fact about lex­i­cog­ra­phy.

• I be­lieve I’m in ba­sic agree­ment. Definitely in the nom­i­nal­ist camp.

Math is an evolved con­cep­tual struc­ture. Why does the math we use work? About the same rea­son the ham­mers we use work. Things that work, get used. We make changes, see which ones work bet­ter, and use those.

How is it that math can work? Well, how is it that the con­cep­tual struc­tures we use work? We try to use the ones that do, and move on from those that don’t. Noth­ing to see here folks, move along.

There’s an in­finite space of con­cep­tual struc­tures. Most of them suck. Some don’t. Math doesn’t suck. Huck­le­berry Finn doesn’t ei­ther. Were both “dis­cov­ered” out of that in­finite space of struc­tures? I guess you could say so, but that seems quite a pe­cu­liar way of look­ing at it.

To say that hu­man be­ings “in­vented num­bers”—or in­vented the struc­ture im­plicit in num­bers—seems like claiming that Neil Arm­strong hand-crafted the Moon.

Doesn’t seem that way to me. The moon is not a con­cep­tual struc­ture.

• Work at what? For whom? Math­e­mat­i­ci­ans are happy with maths that is of no use to physi­cists.

• And that’s fine. In fact, that’s great. If peo­ple want to en­joy the aes­thet­ics of con­cep­tual struc­ture, I hope they call me over for the fun.

But the “what” in the “work at what” I was speak­ing, is “pre­dict­ing other data points, not yet ob­served”, per the OP.

• Not ac­tu­ally the job of maths...”this ham­mer doesn’t work, you can’t drive in screws with it”

• Math doesn’t have a “job”. It’s use by peo­ple to fulfill their ends. For most peo­ple, those ends are pre­dic­tion.

• Grab an ar­bi­trary piece of maths and it won’t pre­dict any­thing. There is a tech­nique and spe­cial­ity and skill of find­ing the right pieceof maths to match the ter­ri­tory, and that is called physics.

Mean­while...no pro­fes­sional math­e­mat­i­cian gets sacked for failing to pre­dict or oth­er­wise be­ing em­piri­cally cor­rect. It’s not their job.

• I be­lieve I’m in ba­sic agree­ment. Definitely in the nom­i­nal­ist camp.

Ac­tu­ally, I am not a nom­i­nal­ist, I only adopted a some­what-nom­i­nal­ist po­si­tion for this post to ex­press what I think about math. In ac­tu­al­ity I slide all the way down the slip­pery slope and con­sider the term “ex­ist” mean­ingless ex­cept in the origi­nal sense of “per­ceived”. But that would be a sub­ject for a differ­ent post.

• Not even “po­ten­tially re­ceiv­able”?