E pluribus unum: From Complexity, Universality

Terence Tao

© 2012 by the American Academy of Arts & Sciences

TERENCE TAO, a Fellow of theAmerican Academy since 2009, isProfessor of Mathematics at theUniversity of California, Los An-geles. His publications include An Introduction to Measure Theory(2011) and Solving MathematicalProblems: A Personal Perspective(2006). Three of his books, Structureand Randomness (2008), Poincaré’sLegacies (2009), and An Epsilon ofRoom (2010), draw from the math-ematical research blog he hasmaintained since 2007.

Nature is a mutable cloud, which is always and neverthe same.

–Ralph Waldo Emerson, “History” (1841)

Modern mathematics is a powerful tool to modelany number of real-world situations, whether theybe natural–the motion of celestial bodies, for ex-ample, or the physical and chemical properties of amaterial–or man-made: for example, the stockmarket or the voting preferences of an electorate.1In principle, mathematical models can be used tostudy even extremely complicated systems, withmany interacting components. However, in prac-tice, only very simple systems (ones that involveonly two or three interacting agents) can be solvedprecisely. For instance, the mathematical deriva-tion of the spectral lines of hydrogen, with its sin-gle electron orbiting the nucleus, can be given in anundergraduate physics class; but even with the mostpowerful computers, a mathematical derivation ofthe spectral lines of sodium, with eleven electronsinteracting with each other and with the nucleus, isout of reach. (The three-body problem, which asks topredict the motion of three masses with respect toNewton’s law of gravitation, is famously known asthe only problem to have ever given Newton head-aches. Unlike the two-body problem, which has a

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Abstract: In this brief survey, I discuss some examples of the fascinating phenomenon of universality incomplex systems, in which universal macroscopic laws of nature emerge from a variety of differentmicroscopic dynamics. This phenomenon is widely observed empirically, but the rigorous mathematicalfoundation for universality is not yet satisfactory in all cases.

simple mathematical solution, the three-body problem is believed not to have anysimple mathematical expression for itssolution, and can only be solved approxi-mately, via numerical algorithms.) The in-ability to perform feasible computationson a system with many interacting com-ponents is known as the curse of dimen-sionality.

Despite this curse, a remarkable phe-nomenon often occurs once the numberof components becomes large enough:that is, the aggregate properties of thecomplex system can mysteriously becomepredictable again, governed by simplelaws of nature. Even more surprising,these macroscopic laws for the overall sys-tem are often largely independent of theirmicroscopic counterparts that govern theindividual components of that system.One could replace the microscopic com-ponents by completely different types ofobjects and obtain the same governinglaw at the macroscopic level. When thisoccurs, we say that the macroscopic lawis universal. The universality phenomenonhas been observed both empirically andmathematically in many different con-texts, several of which I discuss below. In some cases, the phenomenon is wellunderstood, but in many situations, theunderlying source of universality is mys-terious and remains an active area ofmathematical research.

The U.S. presidential election of No-vember 4, 2008, was a massively compli-cated affair. More than a hundred millionvoters from ½fty states cast their ballots,with each voter’s decision being influencedin countless ways by campaign rhetoric,media coverage, rumors, personal impres-sions of the candidates, or political discus-sions with friends and colleagues. Therewere millions of “swing” voters who werenot ½rmly supporting either of the twomajor candidates; their ½nal decisions

would be unpredictable and perhaps evenrandom in some cases. The same uncer-tainty existed at the state level: whilemany states were considered safe for onecandidate or the other, at least a dozenwere considered “in play” and could havegone either way.

In such a situation, it would seem im-possible to forecast accurately the electionoutcome. Sure, there were electoral polls–hundreds of them–but each poll sur-veyed only a few hundred or a few thou-sand likely voters, which is only a tinyfraction of the entire population. And thepolls often fluctuated wildly and dis-agreed with each other; not all polls wereequally reliable or unbiased, and no twopolling organizations used exactly thesame methodology.

Nevertheless, well before election nightwas over, the polls had predicted the out-come of the presidential election (andmost other elections taking place thatnight) quite accurately. Perhaps mostspectacular were the predictions of statis-tician Nate Silver, who used a weightedanalysis of all existing polls to predict cor-rectly the outcome of the presidential elec-tion in forty-nine out of ½fty states, as wellas in all of the thirty-½ve U.S. Senate races.(The lone exception was the presidentialelection in Indiana, which Silver callednarrowly for McCain but which eventu-ally favored Obama by just 0.9 percent.)

The accuracy of polling can be explainedby a mathematical law known as the lawof large numbers. Thanks to this law, weknow that once the size of a random pollis large enough, the probable outcomes ofthat poll will converge to the actual per-centage of voters who would vote for agiven candidate, up to a certain accuracy,known as the margin of error. For instance,in a random poll of a thousand voters, themargin of error is about 3 percent.

A remarkable feature of the law of largenumbers is that it is universal. Does the

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election involve a hundred thousand vot-ers or a hundred million voters? It doesn’tmatter: the margin of error for the poll willremain 3 percent. Is it a state that favorsMcCain to Obama 55 percent to 45 per-cent, or Obama to McCain 60 percent to40 percent? Is the state a homogeneousbloc of, say, affluent white urban voters,or is the state instead a mix of voters of allincomes, races, and backgrounds? Again,it doesn’t matter: the margin of error forthe poll will still be 3 percent. The onlyfactor that makes a signi½cant differenceis the size of the poll; the larger the poll,the smaller the margin of error. The im-mense complexity of a hundred millionvoters trying to decide between presiden-tial candidates collapses to just a handfulof numbers.

The law of large numbers is one of thesimplest and best understood of the uni-versal laws in mathematics and nature, butit is by no means the only one. Over thedecades, many such universal laws havebeen found to govern the behavior of wideclasses of complex systems, regardless ofthe components of a system or how theyinteract with each other.

In the case of the law of large numbers,the mathematical underpinnings of theuniversality phenomenon are well under-stood and are taught routinely in under-graduate courses on probability and sta-tistics. However, for many other universallaws, our mathematical understanding isless complete. The question of why uni-versal laws emerge so often in complexsystems is a highly active direction of re-search in mathematics. In most cases, weare far from a satisfactory answer to thisquestion, but as I discuss below, we havemade some encouraging progress.

After the law of large numbers, perhapsthe next most fundamental example of auniversal law is the central limit theorem.Roughly speaking, this theorem asserts

that if one takes a statistic that is a com-bination of many independent and ran-domly fluctuating components, with noone component having a decisive influ-ence on the whole, then that statistic willbe approximately distributed accordingto a law called the normal distribution (orGaussian distribution) and more popularlyknown as the bell curve. The law is univer-sal because it holds regardless of exactlyhow the individual components fluctuateor how many components there are (al-though the accuracy of the law improveswhen the number of components increas-es). It can be seen in a staggeringly diverserange of statistics, from the incidence rateof accidents; to the variation of height,weight, or other vital statistics among aspecies; to the ½nancial gains or lossescaused by chance; to the velocities of thecomponent particles of a physical system.The size, width, location, and even theunits of measurement of the distributionvary from statistic to statistic, but the bellcurve shape can be discerned in all cases.This convergence arises not because ofany “low level” or “microscopic” connec-tion between such diverse phenomena ascar crashes, human height, trading pro½ts,or stellar velocities, but because in all thesecases the “high level” or “macroscopic”structure is the same: namely, a compoundstatistic formed from a combination of thesmall influences of many independent fac-tors. That the macroscopic behavior of alarge, complex system can be almost total-ly independent of its microscopic struc-ture is the essence of universality.

The universal nature of the central limittheorem is tremendously useful in manyindustries, allowing them to manage whatwould otherwise be an intractably com-plex and chaotic system. With this theo-rem, insurers can manage the risk of, say,their car insurance policies without hav-ing to know all the complicated details ofhow car crashes occur; astronomers can

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measure the size and location of distantgalaxies without having to solve the com-plicated equations of celestial mechanics;electrical engineers can predict the effectof noise and interference on electroniccommunications without having to knowexactly how this noise is generated; and soforth. The central limit theorem, though,is not completely universal; there are im-portant cases when the theorem does notapply, giving statistics with a distributionquite different from the bell curve. (I willreturn to this point later.)

There are distant cousins of the centrallimit theorem that are universal laws forslightly different types of statistics. Oneexample, Benford’s law, is a universal lawfor the ½rst few digits of a statistic of largemagnitude, such as the population of acountry or the size of an account; it givesa number of counterintuitive predictions:for instance, that any given statistic oc-curring in nature is more than six timesas likely to start with the digit 1 than withthe digit 9. Among other things, this law(which can be explained by combiningthe central limit theorem with the math-ematical theory of logarithms) has beenused to detect accounting fraud, becausenumbers that are made up, as opposed tothose that arise naturally, often do notobey Benford’s law (see Figure 1).

In a similar vein, Zipf’s law is a universallaw that governs the largest statistics in agiven category, such as the largest countrypopulations in the world or the most fre-quent words in the English language. Itasserts that the size of a statistic is usual-ly inversely proportional to its ranking;thus, for instance, the tenth largest statis-tic should be about half the size of the½fth largest statistic. (The law tends not towork so well for the top two or three sta-tistics, but becomes more accurate afterthat.) Unlike the central limit theoremand Benford’s law, which are fairly wellunderstood mathematically, Zipf’s law is

primarily an empirical law; it is observedin practice, but mathematicians do nothave a fully satisfactory and convincingexplanation for how the law comes aboutand why it is universal.

So far, I have discussed universal lawsfor individual statistics: complex numer-ical quantities that arise as the combina-tion of many smaller and independent fac-tors. But universal laws have also beenfound for more complicated objects thanmere numerical statistics. Take, for exam-ple, the laws governing the complicatedshapes and structures that arise fromphase transitions in physics and chemistry.As we learn in high school science classes,matter comes in various states, includingthe three classic states of solid, liquid, andgas, but also a number of exotic states suchas plasmas or superfluids. Ferromagneticmaterials, such as iron, also have magne-tized and non-magnetized states; othermaterials become electrical conductorsat some temperatures and insulators atothers. What state a given material is independs on a number of factors, most no-tably the temperature and, in some cases,the pressure. (For some materials, thelevel of impurities is also relevant.) For a½xed value of the pressure, most materi-als tend to be in one state for one range oftemperatures and in another state foranother range. But when the material isat or very close to the temperature divid-ing these two ranges, interesting phasetransitions occur. The material, which isnot fully in one state or the other, tends tosplit into beautifully fractal shapes knownas clusters, each of which embodies oneor the other of the two states.

There are countless materials in exis-tence, each with a different set of keyparameters (such as the boiling point at a given pressure). There are also a largenumber of mathematical models thatphysicists and chemists use to model these

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materials and their phase transitions, inwhich individual atoms or molecules areassumed to be connected to some of theirneighbors by a random number of bonds,assigned according to some probabilisticrule. At the microscopic level, these mod-els can look quite different from eachother. For instance, the ½gures below dis-play the small-scale structure of two typ-ical models: a site percolation model on ahexagonal lattice (Figure 2), in which eachhexagon (or site) is an abstraction of anatom or molecule randomly placed in oneof two states, with the clusters being theconnected regions of a single color; and abond percolation model on a square lattice(Figure 3), in which the edges of the latticeare abstractions of molecular bonds thateach have some probability of being acti-vated, with the clusters being the connect-ed regions given by the active bonds.

If one zooms out to look at the large-scale structure of clusters while at or nearthe critical value of parameters (such astemperature), the differences in micro-scopic structure fade away, and one beginsto see a number of universal laws emerg-ing. While the clusters have a random sizeand shape, they almost always have a frac-tal structure; thus, if one zooms in on anyportion of the cluster, the resulting imagemore or less resembles the cluster as awhole. Basic statistics such as the numberof clusters, the average size of the clusters,or the frequency with which a cluster con-nects two given regions of space appear toobey some speci½c universal laws, knownas power laws (which are somewhat similar,though not quite the same, as Zipf’s law).These laws arise in almost every mathe-matical model that has been put forwardto explain (continuous) phase transitions

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Figure 1Histogram of the First Digits of the Populations of the 237 Countries of the World in 2010

The black dots indicate the Benford’s law prediction. Source: Wikipedia, http://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world%27s_countries_population.png; used here under the Creative Commons Attribution-Share Alike license.

and have been observed many times innature. As with other universal laws, theprecise microscopic structure of the mod-el or the material may affect some basicparameters, such as the phase transitiontemperature, but the underlying structureof the law is the same across all modelsand materials.

In contrast to more classical universallaws such as the central limit theorem, ourunderstanding of the universal laws ofphase transition is incomplete. Physicistshave put forth some compelling heuristicarguments that explain or support many

of these laws (based on a powerful, but notfully rigorous, tool known as the renormal-ization group method), but a completely rig-orous proof of these laws has not yet beenobtained in all cases. This is a very activearea of research; for instance, in August2010, the Fields medal, one of the mostprestigious prizes in mathematics, wasawarded to Stanislav Smirnov for hisbreakthroughs in rigorously establishingthe validity of these universal laws forsome key models, such as percolationmodels on a triangular lattice.

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Figure 2Site Percolation Model on a Hexagonal Lattice at the Critical Threshold

Source: Michael Kozdron, http://stat.math.uregina.ca/~kozdron/Simulations/Percolation/percolation.html;used here with permission from Michael Kozdron.

As we near the end of our tour of uni-versal laws, I want to consider an exampleof this phenomenon that is closer to myown area of research. Here, the object ofstudy is not a single numerical statistic (asin the case of the central limit theorem)or a shape (as with phase transitions), buta discrete spectrum: a sequence of points(or numbers, or frequencies, or energylevels) spread along a line.

Perhaps the most familiar example of adiscrete spectrum is the radio frequen-cies emitted by local radio stations; thisis a sequence of frequencies in the radioportion of the electromagnetic spectrum,

which one can access by turning a radiodial. These frequencies are not evenlyspaced, but usually some effort is made tokeep any two station frequencies separat-ed from each other, to reduce interference.

Another familiar example of a discretespectrum is the spectral lines of an atom-ic element that come from the frequen-cies that the electrons in the atomic shellscan absorb and emit, according to the lawsof quantum mechanics. When these fre-quencies lie in the visible portion of theelectromagnetic spectrum, they give indi-vidual elements their distinctive colors,from the blue light of argon gas (which,

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Figure 3Bond Percolation Model on a Square Lattice at the Critical Threshold

Note the presence of both very small clusters and extremely large clusters. Source: Wikipedia, http://en.wikipedia.org/wiki/File:Bond_percolation_p_51.png; used here under the Creative Commons Attribution-Share Alikelicense.

confusingly, is often used in neon lamps,as pure neon emits orange-red light) tothe yellow light of sodium. For simpleelements, such as hydrogen, the equationsof quantum mechanics can be solved rel-atively easily, and the spectral lines fol-low a regular pattern; but for heavier ele-ments, the spectral lines become quitecomplicated and not easy to work out justfrom ½rst principles.

An analogous, but less familiar, exam-ple of spectra comes from the scatteringof neutrons off of atomic nuclei, such asthe Uranium-238 nucleus. The electro-magnetic and nuclear forces of a nucleus,when combined with the laws of quantummechanics, predict that a neutron will passthrough a nucleus virtually unimpeded forsome energies but will bounce off that nu-cleus at other energies, known as scatter-ing resonances. The internal structures ofsuch large nuclei are so complex that it hasnot been possible to compute these reso-nances either theoretically or numerical-ly, leaving experimental data as the onlyoption.

These resonances have an interestingdistribution; they are not independent ofeach other, but instead seem to obey a pre-cise repulsion law that makes it unlikelythat two adjacent resonances are too closeto each other–somewhat analogous tohow radio station frequencies tend to avoidbeing too close together, except that theformer phenomenon arises from the lawsof nature rather than from governmentregulation of the spectrum. In the 1950s,the renowned physicist and Nobel laure-ate Eugene Wigner investigated these res-onance statistics and proposed a remark-able mathematical model to explain them,an example of what we now call a randommatrix model. The precise mathematicaldetails of these models are too technicalto describe here, but in general, one canview such models as a large collection ofmasses, all connected to each other by

springs of various randomly selectedstrengths. Such a mechanical system willoscillate (or resonate) at a certain set offrequencies; and the Wigner hypothesisasserts that the resonances of a largeatomic nucleus should resemble that ofthe resonances of a random matrix model.In particular, they should experience thesame repulsion phenomenon. Because itis possible to rigorously prove repulsionof the frequencies of a random matrixmodel, a heuristic explanation can be giv-en for the same phenomenon that is ex-perimentally observed for nuclei.

Of course, an atomic nucleus does notactually resemble a large system of massesand springs (among other things, it is gov-erned by the laws of quantum mechan-ics rather than of classical mechanics).Instead, as we have since discovered, Wig-ner’s hypothesis is a manifestation of auniversal law that governs many types ofspectral lines, including those that osten-sibly have little in common with atomicnuclei or random matrix models. Forinstance, the same spacing distributionwas famously found in the waiting timesbetween buses arriving at a bus stop inCuernavaca, Mexico (without, as yet, acompelling explanation for why this dis-tribution emerges in this case).

Perhaps the most unexpected demon-stration of the universality of these lawscame from the wholly unrelated area ofnumber theory, and in particular the distri-bution of the prime numbers 2, 3, 5, 7, 11,and so on–the natural numbers greaterthan 1 that cannot be factored into small-er natural numbers. The prime numbersare distributed in an irregular fashionthrough the integers; but if one performsa spectral analysis of this distribution,one can discern certain long-term oscilla-tions in the distribution (sometimesknown as the music of the primes), thefrequencies of which are described by asequence of complex numbers known as

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the (non-trivial) zeroes of the Riemannzeta function, ½rst studied by BernhardRiemann in 1859. (For this discussion, itis not important to know exactly whatthe Riemann zeta function is.) In princi-ple, these numbers tell us everything wewould wish to know about the primes.One of the most famous and importantproblems in number theory is the Rie-mann hypothesis, which asserts that thesenumbers all lie on a single line in the com-plex plane. It has many consequences innumber theory and, in particular, givesmany important consequences about theprime numbers. However, even the pow-erful Riemann hypothesis does not settleeverything on this subject, in part becauseit does not directly say much about howthe zeroes are distributed on this line. Butthere is extremely strong numerical evi-dence that these zeroes obey the sameprecise law that is observed in neutronscattering and in other systems; in par-ticular, the zeroes seem to “repel” eachother in a manner that matches the pre-dictions of random matrix theory withuncanny accuracy. The formal descriptionof this law is known as the Gaussian Uni-tary Ensemble (gue) hypothesis. (Thegue is a fundamental example of a ran-dom matrix model.) Like the Riemannhypothesis, it is currently unproven, butit has powerful consequences for the dis-tribution of the prime numbers.

The discovery of the gue hypothesis,connecting the music of the primes andthe energy levels of nuclei, occurred atthe Institute for Advanced Study in 1972,and the story is legendary in mathemati-cal circles. It concerns a chance meetingbetween the mathematician Hugh Mont-gomery, who had been working on thedistribution of zeroes of the zeta function(and more speci½cally, on a certain statis-tic relating to that distribution known asthe pair correlation function), and the re-nowned physicist Freeman Dyson. In his

book Stalking the Riemann Hypothesis,mathematician and computer scientistDan Rockmore describes that meeting:

As Dyson recalls it, he and Montgomeryhad crossed paths from time to time at theInstitute nursery when picking up anddropping off their children. Nevertheless,they had not been formally introduced. Inspite of Dyson’s fame, Montgomery hadn’tseen any purpose in meeting him. “Whatwill we talk about?” is what Montgomerypurportedly said when brought to tea. Nev-ertheless, Montgomery relented and uponbeing introduced, the amiable physicistasked the young number theorist about hiswork. Montgomery began to explain hisrecent results on the pair correlation, andDyson stopped him short–“Did you getthis?” he asked, writing down a particularmathematical formula. Montgomery almostfell over in surprise: Dyson had writtendown the sinc-infused pair correlation func-tion. . . . Whereas Montgomery had traveleda number theorist’s road to a “prime pic-ture” of the pair correlation, Dyson hadarrived at this formula through the study ofthese energy levels in the mathematics ofmatrices.2

The chance discovery by Montgomeryand Dyson–that the same universal lawthat governs random matrices and atomicspectra also applies to the zeta function–was given substantial numerical supportby the computational work of AndrewOdlyzko beginning in the 1980s (see Fig-ure 4). But this discovery does not meanthat the primes are somehow nuclear-powered or that atomic physics is some-how driven by the prime numbers; in-stead, it is evidence that a single law forspectra is so universal that it is the naturalend product of any number of differentprocesses, whether from nuclear physics,random matrix models, or number theory.

The precise mechanism underlyingthis law has not yet been fully unearthed;

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in particular, we still do not have a com-pelling explanation, let alone a rigorousproof, of why the zeroes of the zeta func-tion are subject to the gue hypothesis.However, there is now a substantial bodyof rigorous work (including some of myown work, and including some substan-tial breakthroughs in just the last fewyears) that gives support to the universal-ity of this hypothesis, by showing that awide variety of random matrix models(not just the most famous model of thegue) are all governed by essentially thesame law for their spacings. At present,these demonstrations of universality havenot extended to the number theoretic orphysical settings, but they do give indirectsupport to the law being applicable inthose cases.

The arguments used in this recent workare too technical to give here, but I willmention one of the key ideas, which mycolleague Van Vu and I borrowed from anold proof of the central limit theorem byJarl Lindeberg from 1922. In terms of themechanical analogy of a system of massesand springs (mentioned above), the keystrategy was to replace just one of thesprings by another, randomly selectedspring and to show that the distributionof the frequencies of this system did notchange signi½cantly when doing so. Apply-ing this replacement operation to eachspring in turn, one can eventually replacea given random matrix model with acompletely different model while keepingthe distribution mostly unchanged–which can be used to show that large class-

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Figure 4Spacing Distribution for a Billion Zeroes of the Riemann Zeta Function, with the CorrespondingPrediction from Random Matrix Theory

Source: Andrew M. Odlyzko, “The 1022-nd Zero of the Riemann Zeta Function,” in Dynamical, Spectral, and Arith-metic Zeta Functions, Contemporary Mathematics Series, no. 290, ed. Machiel van Frankenhuysen and Michel L.Lapidus (American Mathematical Society, 2001), 139–144, http://www.dtc.umn.edu/~odlyzko/doc/zeta.10to22.pdf; used here with permission from Andrew Odlyzko.

es of random matrix models have essen-tially the same distribution.

This is a very active area of research; forinstance, simultaneously with Van Vu’sand my work from last year, László Erdös,Benjamin Schlein, and Horng-Tzer Yaualso gave a number of other demonstra-tions of universality for random matrixmodels, based on ideas from mathemati-cal physics. The ½eld is moving quickly,and in a few years we may have manymore insights into the nature of this mys-terious universal law.

There are many other universal laws ofmathematics and nature; the examples Ihave given are only a small fraction ofthose that have been discovered over theyears, from such diverse subjects as dy-namical systems and quantum ½eld theo-ry. For instance, many of the macroscopiclaws of physics, such as the laws of thermo-dynamics or the equations of fluid mo-tion, are quite universal in nature, makingthe microscopic structure of the materialor fluid being studied almost irrelevant,other than via some key parameters suchas viscosity, compressibility, or entropy.

However, the principle of universalitydoes have de½nite limitations. Take, forinstance, the central limit theorem, whichgives a bell curve distribution to anyquantity that arises from a combinationof many small and independent factors.This theorem can fail when the requiredhypotheses are not met. The distributionof, say, the heights of all human adults(male and female) does not obey a bellcurve distribution because one single fac-tor–gender–has so large an impact onheight that it is not averaged out by all theother environmental and genetic factorsthat influence this statistic.

Another very important way in whichthe central limit fails is when the individ-ual factors that make up a quantity do notfluctuate independently of each other, but

are instead correlated, so that they tendto rise or fall in unison. In such cases, “fattails” (also known colloquially as “blackswans”) can develop, in which the quan-tity moves much further from its averagevalue than the central limit theoremwould predict. This phenomenon is par-ticularly important in ½nancial modeling,especially when dealing with complex½nancial instruments such as the collat-eralized debt obligations (cdos) thatwere formed by aggregating mortgages.As long as the mortgages behaved inde-pendently of each other, the central limittheorem could be used to model the riskof these instruments; but in the recent½nancial crisis (a textbook example of ablack swan), this independence hypothe-sis broke down spectacularly, leading tosigni½cant ½nancial losses for manyholders of these obligations (and for theirinsurers). A mathematical model is onlyas strong as the assumptions behind it.

A third way in which a universal law canbreak down is if the system does not haveenough degrees of freedom for the law totake effect. For instance, cosmologists canuse universal laws of fluid mechanics todescribe the motion of entire galaxies, butthe motion of a single satellite under theinfluence of just three gravitational bod-ies can be far more complicated (being,quite literally, rocket science).

Another instance where the universallaws of fluid mechanics break down is atthe mesoscopic scale: that is, larger than themicroscopic scale of individual molecules,but smaller than the macroscopic scalesfor which universality applies. An impor-tant example of a mesoscopic fluid is theblood flowing through blood vessels; theblood cells that make up this liquid are solarge that they cannot be treated merelyas an ensemble of microscopic molecules,but rather as mesoscopic agents with com-plex behavior. Other examples of materi-als with interesting mesoscopic behavior

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include colloidal fluids (such as mud), cer-tain types of nanomaterials, and quantumdots; it is a continuing challenge to math-ematically model such materials properly.

There are also many macroscopic situ-ations in which no universal law is knownto exist, particularly in cases where thesystem contains human agents. The stockmarket is a good example: despite ex-tremely intensive efforts, no satisfactoryuniversal laws to describe the movementof stock prices have been discovered.(The central limit theorem, for instance,does not seem to be a good model, as dis-cussed earlier.) One reason for this short-coming is that any regularity discoveredin the market is likely to be exploited byarbitrageurs until it disappears. For simi-lar reasons, ½nding universal laws formacroeconomics appears to be a movingtarget; according to Goodhart’s law, if anobserved statistical regularity in economicdata is exploited for policy purposes, ittends to collapse. (Ironically, Goodhart’slaw itself is arguably an example of a uni-versal law.)

Even when universal laws do exist, itstill may be practically impossible to usethem to make predictions. For instance,we have universal laws for the motion offluids, such as the Navier-Stokes equa-tions, and these are used all the time insuch tasks as weather prediction. But theseequations are so complex and unstablethat even with the most powerful com-puters, we are still unable to accuratelypredict the weather more than a week ortwo into the future. (By unstable, I meanthat even small errors in one’s measure-ment data, or in one’s numerical compu-tations, can lead to large fluctuations inthe predicted solution of the equations.)

Hence, between the vast, macroscopicsystems for which universal laws hold swayand the simple systems that can be ana-lyzed using the fundamental laws of na-ture, there is a substantial middle groundof systems that are too complex for fun-damental analysis but too simple to beuniversal–plenty of room, in short, forall the complexities of life as we know it.

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endnotes1 This essay bene½ted from the feedback of many readers of my blog. They commented on a

draft version that (together with additional ½gures and links) can be read at http://terrytao.wordpress.com/2010/09/14/a-second-draft-of-a-non-technical-article-on-universality/.

2 Dan Rockmore, Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of PrimeNumbers (New York: Pantheon Books, 2005).