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Whence the Force of F = ma?I: Culture Shock

Frank Wilczek

When I was a student, the subjectthat gave me the most troublewas classical mechanics. Thatalways struck me as peculiar,because I had no trouble learningmore advanced subjects, whichwere supposed to be harder. NowI think I've figured it out. It was acase of culture shock. Comingfrom mathematics, I wasexpecting an algorithm. Instead Iencountered something quitedifferent— a sort of culture, infact. Let me explain.

Problems with F = ma

Newton's second law of motion, F = ma, is the soul of classicalmechanics. Like other souls, it is insubstantial. Theright−hand side is the product of two terms with profoundmeanings. Acceleration is a purely kinematical concept,defined in terms of space and time. Mass quite directly reflectsbasic measurable properties of bodies (weights, recoilvelocities). The left−hand side, on the other hand, has noindependent meaning. Yet clearly Newton's second law is fullof meaning, by the highest standard: It proves itself useful indemanding situations. Splendid, unlikely looking bridges, likethe Erasmus Bridge (known as the Swan of Rotterdam), dobear their loads; spacecraft do reach Saturn.

The paradox deepens when we consider force from theperspective of modern physics. In fact, the concept of force isconspicuously absent from our most advanced formulations ofthe basic laws. It doesn't appear in Schrödinger's equation, orin any reasonable formulation of quantum field theory, or inthe foundations of general relativity. Astute observerscommented on this trend to eliminate force even before theemergence of relativity and quantum mechanics.

In his 1895 Dynamics, the prominent physicist Peter G. Tait,who was a close friend and collaborator of Lord Kelvin andJames Clerk Maxwell, wrote

"In all methods and systems which involve the idea of forcethere is a leaven of artificiality. . . . there is no necessity for theintroduction of the word "force" nor of the sense−suggestedideas on which it was originally based."1

Particularly striking, since it is so characteristic and soover−the−top, is what Bertrand Russell had to say in his 1925popularization of relativity for serious intellectuals, The ABCof Relativity:

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"If people were to learn to conceive the world in the new way,without the old notion of "force," it would alter not only theirphysical imagination, but probably also their morals andpolitics. . . . In the Newtonian theory of the solar system, thesun seems like a monarch whose behests the planets have toobey. In the Einsteinian world there is more individualismand less government than in the Newtonian."2

The 14th chapter of Russell's book is entitled "The Abolition ofForce."

If F = ma is formally empty, microscopically obscure, andmaybe even morally suspect, what's the source of itsundeniable power?

The culture of force

To track that source down, let's consider how the formula getsused.

A popular class of problems specifies a force and asks aboutthe motion, or vice versa. These problems look like physics,but they are exercises in differential equations and geometry,thinly disguised. To make contact with physical reality, wehave to make assertions about the forces that actually occur inthe world. All kinds of assumptions get snuck in, often tacitly.

The zeroth law of motion, so basic to classical mechanics thatNewton did not spell it out explicitly, is that mass isconserved. The mass of a body is supposed to be independentof its velocity and of any forces imposed on it; also total massis neither created nor destroyed, but only redistributed, whenbodies interact. Nowadays, of course, we know that none ofthat is quite true.

Newton's third law states that for every action there's an equaland opposite reaction. Also, we generally assume that forcesdo not depend on velocity. Neither of those assumptions isquite true either; for example, they fail for magnetic forcesbetween charged particles.

When most textbooks come to discuss angular momentum,they introduce a fourth law, that forces between bodies aredirected along the line that connects them. It is introduced inorder to "prove" the conservation of angular momentum. Butthis fourth law isn't true at all for molecular forces.

Other assumptions get introduced when we bring in forces ofconstraint, and friction.

I won't belabor the point further. To anyone who reflects on it,it soon becomes clear that F = ma by itself does not providean algorithm for constructing the mechanics of the world. Theequation is more like a common language, in which differentuseful insights about the mechanics of the world can beexpressed. To put it another way, there is a whole cultureinvolved in the interpretation of the symbols. When we learnmechanics, we have to see lots of worked examples to graspproperly what force really means. It is not just a matter ofbuilding up skill by practice; rather, we are imbibing a tacitculture of working assumptions. Failure to appreciate this iswhat got me in trouble.

The historical development of mechanics reflected a similarlearning process. Isaac Newton scored his greatest and mostcomplete success in planetary astronomy, when he discoveredthat a single force of quite a simple form dominates the story.His attempts to describe the mechanics of extended bodies

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and fluids in the second book of The Principia were pathbreaking but not definitive, and he hardly touched the morepractical side of mechanics. Later physicists andmathematicians including notably Jean d'Alembert(constraint and contact forces), Charles Coulomb (friction),and Leonhard Euler (rigid, elastic, and fluid bodies) madefundamental contributions to what we now comprehend in theculture of force.

Physical, psychological origins

Many of the insights embedded in the culture of force, aswe've seen, aren't completely correct. Moreover, what we nowthink are more correct versions of the laws of physics won't fitinto its language easily, if at all. The situation begs for twoprobing questions: How can this culture continue to flourish?Why did it emerge in the first place?

For the behavior of matter, we now have extremely completeand accurate laws that in principle cover the range ofphenomena addressed in classical mechanics and, of course,much more. Quantum electrodynamics (QED) and quantumchromodynamics (QCD) provide the basic laws for building upmaterial bodies and the nongravitational forces between them,and general relativity gives us a magnificent account ofgravity. Looking down from this exalted vantage point, we canget a clear perspective on the territory and boundaries of theculture of force.

Compared to earlier ideas, the modern theory of matter,which really only emerged during the 20th century, is muchmore specific and prescriptive. To put it plainly, you havemuch less freedom in interpreting the symbols. The equationsof QED and QCD form a closed logical system: They informyou what bodies can be produced at the same time as theyprescribe their behavior; they govern your measuring devices— and you, too!— thereby defining what questions are wellposed physically; and they provide answers to such questions— or at least algorithms to arrive at the answers. (I'm wellaware that QED + QCD is not a complete theory of nature,and that, in practice, we can't solve the equations very well.)Paradoxically, there is much less interpretation, less cultureinvolved in the foundations of modern physics than in earlier,less complete syntheses. The equations really do speak forthemselves: They are algorithmic.

By comparison to modern foundational physics, the culture offorce is vaguely defined, limited in scope, and approximate.Nevertheless it survives the competition, and continues toflourish, for one overwhelmingly good reason: It is mucheasier to work with. We really do not want to be picking ourway through a vast Hilbert space, regularizing andrenormalizing ultraviolet divergences as we go, thenanalytically continuing Euclidean Green's functions defined bya limiting procedure, . . . working to discover nuclei that clothethemselves with electrons to make atoms that bind together tomake solids, . . . all to describe the collision of two billiardballs. That would be lunacy similar in spirit to, but worse than,trying to do computer graphics from scratch, in machine code,without the benefit of an operating system. The analogy seemsapt: Force is a flexible construct in a high−level language,which, by shielding us from irrelevant details, allows us to doelaborate applications relatively painlessly.

Why is it possible to encapsulate the complicated deepstructure of matter? The answer is that matter ordinarilyrelaxes to a stable internal state, with high energetic orentropic barriers to excitation of all but a few degrees of

freedom. We can focus our attention on those few effectivedegrees of freedom; the rest just supply the stage for theactors.

While force itself does not appear in the foundationalequations of modern physics, energy and momentum certainlydo, and force is very closely related to them: Roughlyspeaking, it's the space derivative of the former and the timederivative of the latter (and F = ma just states the consistencyof those definitions!). So the concept of force is not quite sofar removed from modern foundations as Tait and Russellinsinuate: It may be gratuitous, but it is not bizarre. Withoutchanging the content of classical mechanics, we can cast it inLagrangian terms, wherein force no longer appears as aprimary concept. But that's really a technicality; the deeperquestions remains: What aspects of fundamentals does theculture of force reflect? What approximations lead to it?

Some kind of approximate, truncated description of thedynamics of matter is both desirable and feasible because it iseasier to use and focuses on the relevant. To explain the roughvalidity and origin of specific concepts and idealizations thatconstitute the culture of force, however, we must considertheir detailed content. A proper answer, like the culture offorce itself, must be both complicated and open−ended. Themolecular explanation of friction is still very much a researchtopic, for example. I'll discuss some of the simpler aspects,addressing the issues raised above, in my next column, beforedrawing some larger conclusions.

Here I conclude with some remarks on the psychologicalquestion, why force was— and usually still is— introduced inthe foundations of mechanics, when from a logical point ofview energy would serve at least equally well, and arguablybetter. The fact that changes in momentum— whichcorrespond, by definition, to forces— are visible, whereaschanges in energy often are not, is certainly a major factor.Another is that, as active participants in statics— for example,when we hold up a weight— we definitely feel we are doingsomething, even though no mechanical work is performed.Force is an abstraction of this sensory experience of exertion.D'Alembert's substitute, the virtual work done in response tosmall displacements, is harder to relate to. (Though ironicallyit is a sort of virtual work, continually made real, that explainsour exertions. When we hold a weight steady, individualmuscle fibers contract in response to feedback signals they getfrom spindles; the spindles sense small displacements, whichmust get compensated before they grow.4) Similar reasonsmay explain why Newton used force. A big part of theexplanation for its continued use is no doubt (intellectual)inertia.

References1. P. G. Tait, Dynamics, Adam & Charles Black, London(1895).2. B. Russell, The ABC of Relativity, 5th rev. ed., Routledge,London (1997).3. I. Newton, The Principia, I. B. Cohen, A. Whitman, trans.,U. of Calif. Press, Berkeley (1999).4. S. Vogel, Prime Mover: A Natural History of Muscle,Norton, New York (2001), p. 79.

Frank Wilczek is the Herman Feshbach Professor ofPhysics at the Massachusetts Institute of Technology inCambridge.

Letters and opinions are encouraged and should be sent toLetters, Physics Today, American Center for Physics, OnePhysics Ellipse, College Park, MD 20740−3842 or by e−mail toptletter@aip.org (using your surname as "Subject"). Pleaseinclude your affiliation, mailing address, and daytime phonenumber. We reserve the right to edit submissions.

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Whence the Force of F = ma? II: Rationalizations

Frank Wilczek

In my previous column(Physics Today, October2004, page 11), I discussedhow assumptions about F andm give substance to the spiritof F = ma. I called this set ofassumptions the culture offorce. I mentioned thatseveral elements of theculture, though oftenpresented as "laws," appearrather strange from theperspective of modernphysics. Here I discuss how,and under whatcircumstances, some of thoseassumptions emerge asconsequences of modernfundamentals—or don't!

Critique of the zeroth law

Ironically, it is the most primitive element of the culture offorce—the zeroth law, conservation of mass—that bears thesubtlest relationship to modern fundamentals.

Is the conservation of mass as used in classical mechanics aconsequence of the conservation of energy in specialrelativity? Superficially, the case might appearstraightforward. In special relativity we learn that the mass ofa body is its energy at rest divided by the speed of lightsquared (m = E/c2); and for slowly moving bodies, it isapproximately that. Since energy is a conserved quantity, thisequation appears to supply an adequate candidate, E/c2, tofill the role of mass in the culture of force.

That reasoning won't withstand scrutiny, however. The gap inits logic becomes evident when we consider how we routinelytreat reactions or decays involving elementary particles.

To determine the possible motions, we must explicitly specifythe mass of each particle coming in and of each particle goingout. Mass is a property of isolated particles, whose masses areintrinsic properties—that is, all protons have one mass, allelectrons have another, and so on. (For experts: "Mass" labelsirreducible representations of the Poincaré group.) There is noseparate principle of mass conservation. Rather, the energiesand momenta of such particles are given in terms of theirmasses and velocities, by well−known formulas, and weconstrain the motion by imposing conservation of energy andmomentum. In general, it is simply not true that the sum of

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the masses of what goes in is the same as the sum of themasses of what goes out.

Of course when everything is slowly moving, then mass doesreduce to approximately E/c2. It might therefore appear as ifthe problem, that mass as such is not conserved, can be sweptunder the rug, for only inconspicuous (small and slowlymoving) bulges betray it. The trouble is that as we developmechanics, we want to focus on those bulges. That is, we wantto use conservation of energy again, subtracting off themass−energy exactly (or rather, in practice, ignoring it) andkeeping only the kinetic part E − mc2 ≅ 1/2 mv2. But you can'tsqueeze two conservation laws (for mass and nonrelativisticenergy) out of one (for relativistic energy) honestly. Ascribingconservation of mass to its approximate equality with E/c2

begs an essential question: Why, in a wide variety ofcircumstances, is mass−energy accurately walled off, and notconvertible into other forms of energy?

To illustrate the problem concretely and numerically, considerthe reaction 2H + 3H → 4He + n, which is central for attemptsto achieve controlled fusion. The total mass of the deuteriumplus tritium exceeds that of the alpha plus neutron by 17.6MeV. Suppose that the deuterium and tritium are initially atrest. Then the alpha emerges at .04 c; the neutron at .17 c.

In the (D,T) reaction, mass is not accurately conserved, and(nonrelativistic) kinetic energy has been produced fromscratch, even though no particle is moving at a speed veryclose to the speed of light. Relativistic energy is conserved, ofcourse, but there is no useful way to divide it up into twopieces that are separately conserved. In thought experiments,by adjusting the masses, we could make this problem appearin situations where the motion is arbitrarily slow. Anotherway to keep the motion slow is to allow the liberatedmass−energy to be shared among many bodies.

Recovering the zeroth law

Thus, by licensing the conversion of mass into energy, specialrelativity nullifies the zeroth law, in principle. Why is Natureso circumspect about exploiting this freedom? How didAntoine Lavoisier, in the historic experiments that helpedlaunch modern chemistry, manage to reinforce a centralprinciple (conservation of mass) that isn't really true?

Proper justification of the zeroth law requires appeal tospecific, profound facts about matter.

To explain why most of the energy of ordinary matter isaccurately locked up as mass, we must first appeal to somebasic properties of nuclei, where almost all the mass resides.The crucial properties of nuclei are persistence and dynamicalisolation. The persistence of individual nuclei is a consequenceof baryon number and electric charge conservation, and theproperties of nuclear forces, which result in a spectrum ofquasi−stable isotopes. The physical separation of nuclei andtheir mutual electrostatic repulsion—Coulomb barriers—guarantee their approximate dynamical isolation. Thatapproximate dynamical isolation is rendered completelyeffective by the substantial energy gaps between the groundstate of a nucleus and its excited states. Since the internalenergy of a nucleus cannot change by a little bit, then inresponse to small perturbations, it doesn't change at all.

Because the overwhelming bulk of the mass−energy ofordinary matter is concentrated in nuclei, the isolation andintegrity of nuclei—their persistence and lack of effective

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internal structure—go most of the way toward justifying thezeroth law. But note that to get this far, we needed to appealto quantum theory and special aspects of nuclearphenomenology! For it is quantum theory that makes theconcept of energy gaps available, and it is only particularaspects of nuclear forces that insure substantial gaps abovethe ground state. If it were possible for nuclei to be very muchlarger and less structured—like blobs of liquid or gas—thegaps would be small, and the mass−energy would not belocked up so completely.

Radioactivity is an exception to nuclear integrity, and moregenerally the assumption of dynamical isolation goes out thewindow in extreme conditions, such as we study in nuclearand particle physics. In those circumstances, conservation ofmass simply fails. In the common decay π0 → γγ, for example,a massive π0 particle evolves into photons of zero mass.

The mass of an individual electron is a universal constant, asis its charge.Electrons do not support internal excitations, andthe number of electrons is conserved (if we ignore weakinteractions and pair creation). These facts are ultimatelyrooted in quantum field theory. Together, they guarantee theintegrity of electron mass−energy.

In assembling ordinary matter from nuclei and electrons,electrostatics plays the dominant role. We learn in quantumtheory that the active, outer−shell electrons move withvelocities of order αc = e2/4πħ ≈ .007 c. This indicates thatthe energies in play in chemistry are of order me(αc)2/mec2

= α2 ≈ 5 × 10−5 times the electron mass−energy, which in turnis a small fraction of the nuclear mass−energy. So chemicalreactions change the mass−energy only at the level of partsper billion, and Lavoisier rules!

Note that inner−shell electrons of heavy elements, withvelocities of order Zα, can be relativistic. But the inner core ofa heavy atom—nucleus plus inner electron shells—ordinarilyretains its integrity, because it is spatially isolated and has alarge energy gap. So the mass−energy of the core is conserved,though it is not accurately equal to the sum of themass−energy of its component electrons and nucleus.

Putting it all together, we justify Isaac Newton's zeroth law forordinary matter by means of the integrity of nuclei, electrons,and heavy atom cores, together with the slowness of themotion of these building blocks. The principles of quantumtheory, leading to large energy gaps, underlie the integrity; thesmallness of α, the fine−structure constant, underlies the slowmotion.

Newton defined mass as "quantity of matter," and assumed itto be conserved. The connotation of his phrase, whichunderlies his assumption, is that the building blocks of matterare rearranged, but neither created nor destroyed, in physicalprocesses; and that the mass of a body is the sum of themasses of its building blocks. We've now seen, from theperspective of modern foundations, why ordinarily theseassumptions form an excellent approximation, if we take thebuilding blocks to be nuclei, heavy atom cores, and electrons.

It would be wrong to leave the story there, however. For withour next steps in analyzing matter, we depart from thisfamiliar ground: first off a cliff, then into glorious flight. If wetry to use more basic building blocks (protons and neutrons)instead of nuclei, then we discover that the masses don't addaccurately. If we go further, to the level of quarks and gluons,

we can largely derive the mass of nuclei from pure energy, asI've discussed in earlier columns.

Mass and gravity

On the face of it, this complex and approximate justification ofthe mass concept used in classical mechanics poses a paradox:How does this rickety construct manage to support stunninglyprecise and successful predictions in celestial mechanics? Theanswer is that it is bypassed. The forces of celestial mechanicsare gravitational, and so proportional to mass, and m cancelsfrom the two sides of F = ma. This cancellation in theequation for motion in response to gravity becomes afoundational principle in general relativity, where the path isidentified as a geodesic in curved spacetime, with no mentionof mass.

In contrast to a particle's response to gravity, the gravitationalinfluence that the particle exerts is only approximatelyproportional to its mass; the rigorous version of Einstein'sfield equation relates spacetime curvature toenergy−momentum density. As far as gravity is concerned,there is no separate measure of quantity of matter apart fromenergy; that the energy of ordinary matter is dominated bymass−energy is immaterial.

The third and fourth laws

The third and fourth laws are approximate versions ofconservation of momentum and conservation of angularmomentum, respectively. (Recall that the fourth law statedthat all forces are two−body central forces.) In the modernfoundations of physics these great conservation laws reflectthe symmetry of physical laws under translation and rotationsymmetry. Since these conservation laws are more accurateand profound than the assumptions about forces commonlyused to "derive" them, those assumptions have truly becomeanachronisms. I believe that they should, with due honors, beretired.

Newton argued for his third law by observing that a systemwith unbalanced internal forces would begin to acceleratespontaneously, "which is never observed." But this argumentreally motivates the conservation of momentum directly.Similarly, one can "derive" conservation of angularmomentum from the observation that bodies don't spin upspontaneously. Of course, as a matter of pedagogy, one wouldpoint out that action−reaction systems and two−body centralforces provide especially simple ways to satisfy theconservation laws.

Tacit simplicities

Some tacit assumptions about the simplicity of F are so deeplyembedded that we easily take them for granted. But they haveprofound roots.

In calculating the force, we take into account only nearbybodies. Why can we get away with that? Locality in quantumfield theory, which deeply embodies basic requirements ofspecial relativity and quantum mechanics, gives usexpressions for energy and momentum at a point—andthereby for force—that depend only on the position of bodiesnear that point. Even so−called long−range electric andgravitational forces (actually 1/r2—still falling rapidly withdistance) reflect the special properties of locally coupled gaugefields and their associated covariant derivatives.

Similarly, the absence of significant multibody forces is

connected to the fact that sensible (renormalizable) quantumfield theories can't support them.

In this column I've stressed, and maybe strained, therelationship between the culture of force and modernfundamentals. In the final column of this series, I'll discuss itsimportance both as a continuing, expanding endeavor and asa philosophical model.

Frank Wilczek is the Herman Feshbach Professor ofPhysics at the Massachusetts Institute of Technology inCambridge.

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The concept of force, as we haveseen, defines a culture. In the pre-

vious columns of this series (PHYSICSTODAY, October 2004, page 11, andDecember 2004, page 10) I’ve indi-cated how F ⊂ ma acquires meaningthrough interpretation of—that is, ad-ditional assumptions about—F. Thisbody of interpretation is a sort of folk-lore. It contains both approximationsthat we can derive, under appropriateconditions, from modern foundations,and also rough generalizations (suchas “laws” of friction and of elastic be-havior) abstracted from experience.

In the course of that discussion itbecame clear that there is also asmaller, but nontrivial, culturearound m. Indeed, the conservation ofm for ordinary matter provides an ex-cellent, instructive example of anemergent law. It captures in a simplestatement an important consequenceof broad regularities whose basis inmodern fundamentals is robust butcomplicated. In modern physics, theidea that mass is conserved is drasti-cally false. A great triumph of modernquantum chromodynamics (QCD) isto build protons and neutrons, whichcontribute more than 99% of the massof ordinary matter, from gluons thathave exactly zero mass, and from uand d quarks that have very smallmasses. To explain from a modernperspective why conservation of massis often a valid approximation, weneed to invoke specific, deep proper-ties of QCD and quantum electrody-namics (QED), including the dynami-cal emergence of large energy gaps inQCD and the smallness of the finestructure constant in QED.

Isaac Newton and AntoineLavoisier knew nothing of all this, ofcourse. They took conservation of massas a fundamental principle. And they

were right to do so, because by adopt-ing that principle they were able tomake brilliant progress in the analysisof motion and of chemical change. De-spite its radical falsity, their principlewas, and still is, an adequate basis formany quantitative applications. To dis-card it is unthinkable. It is an invalu-able cultural artifact and a basic in-sight into the way the world worksdespite—indeed, in part, because of—its emergent character.

The culture of aWhat about a? There’s a culture at-tached to acceleration, as well. To ob-tain a, we are instructed to considerthe change of the position of a body inspace as a function of time, and totake the second derivative. This pre-scription, from a modern perspective,has severe problems.

In quantum mechanics, bodies don’thave definite positions. In quantumfield theory, they pop in and out of ex-istence. In quantum gravity, space isfluctuating and time is hard to define.So evidently serious assumptions andapproximations are involved even inmaking sense of a’s definition.

Nevertheless, we know very wellwhere we’re going to end up. We’regoing to have an emergent, approxi-mate concept of what a body is. Phys-ical space is going to be modeledmathematically as the Euclideanthree-dimensional space R3 that sup-ports Euclidean geometry. Thistremendously successful model ofspace has been in continuous use formillennia, with applications in sur-veying and civil engineering that evenpredate Euclid’s formalization.

Time is going to be modeled as theone-dimensional continuum R1 of realnumbers. This model of time, at a topo-logical level, goes into our primitive in-tuitions that divide the world into pastand future. I believe that the metricstructure of time—that is, the ideathat time can be not only ordered butdivided into intervals with definite nu-merical magnitude—is a much more

recent innovation. That idea emergedclearly only with Galileo’s use of pen-dulum clocks (and his pulse!).

The mathematical structures in-volved are so familiar and fully devel-oped that they can be, and are, usedroutinely in computer programs. Thisis not to say they are trivial. Theymost definitely aren’t. The classicalGreeks agonized over the concept of acontinuum. Zeno’s famous paradoxesreflect these struggles. Indeed, Greekmathematics never won through tocomfortable algebraic treatment ofreal numbers. Continuum quantitieswere always represented as geometricintervals, even though that represen-tation involved rather awkward con-structions to implement simple alge-braic operations.

The founders of modern analysis(René Descartes, Newton, GottfriedWilhelm Leibniz, Leonhard Euler,and others) were on the whole muchmore freewheeling, trusting their in-tuition while manipulating infinitesi-mals that lacked any rigorous defini-tion. (In his Principia, Newton didoperate geometrically, in the style ofthe Greeks. That is what makes thePrincipia so difficult for us to readtoday. The Principia also contains asophisticated discussion of deriva-tives as limits. From that discussion Iinfer that Newton and possibly otherearly analysts had a pretty good ideaabout what it would take to make atleast the simpler parts of their workrigorous, but they didn’t want to slowdown to do it.) Reasonable rigor, at the level commonly taught in mathe-matics courses today—the much-bemoaned epsilons and deltas—entered into the subject in the 19thcentury.

“Unreasonable” rigor entered in theearly 20th century, when the funda-mental notions from which real num-bers and geometry are constructedwere traced to the level of set theoryand ultimately symbolic logic. In theirPrincipia Mathematica Bertrand Rus-

Whence the Force of F ⊂⊂ ma? III: Cultural DiversityFrank Wilczek

Frank Wilczek is the Herman Fesh-bach Professor of Physics at the Mass-achusetts Institute of Technology inCambridge.

10 July 2005 Physics Today © 2005 American Institute of Physics, S-0031-9228-0507-210-8

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sell and Alfred Whitehead develop375 pages of dense mathematics be-fore proving 1 ⊕ 1 ⊂ 2. To be fair, theirtreatment could be slimmed down con-siderably if attaining that particularresult were the ultimate goal. But inany case, an adequate definition ofreal numbers from symbolic logic in-volves some hard, complicated work.Having the integers in hand, you thenhave to define rational numbers andtheir ordering. Then you must com-plete them by filling in the holes sothat any bounded increasing sequencehas a limit. Then finally—this is thehardest part—you must demonstratethat the resulting system supportsalgebra and is consistent.

Perhaps all that complexity is ahint that the real-number model ofspace and time is an emergent conceptthat some day will be derived fromphysically motivated primitives thatare logically simpler. Also, scrutiny ofthe construction of real numbers sug-gests natural variants, notably JohnConway’s surreal numbers, which in-clude infinitesimals (smaller than anyrational number!) as legitimate quan-tities.1 Might such quantities, whoseformal properties seem no less natu-ral and elegant than those of ordinaryreal numbers, help us to describe na-ture? Time will tell.

Even the unreasonable rigor ofsymbolic logic does not reach idealstrictness. Kurt Gödel demonstratedthat this ideal is unattainable: No rea-sonably complex, consistent ax-iomatic system can be used to demon-strate its own consistency.

But all the esoteric shortcomingsin defining and justifying the cultureof a clearly arise on an entirely differ-ent level from the comparatively mun-dane, immediate difficulties we havein doing justice to the culture of F. Wecan translate the culture of a, withoutserious loss, into C or FORTRAN.That completeness and precision giveus an inspiring benchmark.

The computational imperativeBefore they tried to do it, most com-puter scientists anticipated that toteach a computer to play chess like agrand master would be much morechallenging than to teach one to domundane tasks like drive a car safely.Notoriously, experience has provedotherwise. A big reason for that sur-prise is that chess is algorithmic,whereas driving a car is not. In chessthe rules are completely explicit; weknow very concretely and unambigu-ously what the degrees of freedom areand how they behave. Car driving isquite different: Essential conceptslike “other driver’s expectations” and

“pedestrian,” when you start to ana-lyze them, quickly burgeon into cul-tures. I wouldn’t trust a computerdriver in Boston’s streets because itwouldn’t know how to interpret themixture of intimidation and deferencethat human drivers convey by ges-tures, maneuvers, and eye contact.

The problem with teaching a com-puter classical mechanics is, of course,of more than academic interest: We’dlike robots to get around and manip-ulate things; computer gamesterswant realistic graphics; engineers andastronomers would welcome smartsilicon collaborators—up to a point, Isuppose.

The great logician and philosopherRudolf Carnap made brave, pioneeringattempts to make axiomatic systemsfor elementary mechanics, amongmany other things.2 Patrick Hayes is-sued an influential paper, “NaivePhysics Manifesto,” challenging artifi-cial-intelligence researchers to codifyintuitions about materials and forcesin an explicit way.3 Physics-based com-puter graphics is a lively, rapidly ad-vancing endeavor, as are several vari-eties of computer-assisted design. MyMIT colleagues Gerald Sussman andJack Wisdom have developed an in-tensely computational approach to me-chanics,4 supported every step of theway with explicit programs. The timemay be ripe for a powerful synthesis,incorporating empirical properties ofspecific materials, successful knowndesigns of useful mechanisms, andgeneral laws of mechanical behaviorinto a fully realized computational cul-ture of F ⊂ ma. Functioning robotsmight not need to know a lot of me-chanics explicitly, any more than mosthuman soccer players do; but design-ing a functioning robotic soccer playermay be a job that can best be accom-plished by a very smart and knowl-edgeable man-machine team.

Blur and focusAn overarching theme of this serieshas been that the law F ⊂ ma, whichis sometimes presented as the epit-ome of an algorithm describing na-ture, is actually not an algorithm thatcan be applied mechanically (pun in-tended). It is more like a language inwhich we can easily express impor-tant facts about the world. That’s notto imply it is without content. Thecontent is supplied, first of all, bysome powerful general statements inthat language—such as the zerothlaw, the momentum conservationlaws, the gravitational force law, thenecessary association of forces withnearby sources—and then by the wayin which phenomenological observa-

tions, including many (though not all)of the laws of material science can beexpressed in it easily.

Another theme has been thatF ⊂ ma is not in any sense an ulti-mate truth. We can understand, frommodern foundational physics, how itarises as an approximation underwide but limited circumstances.Again, that does not prevent it frombeing extraordinarily useful; indeed,one of its primary virtues is to shieldus from the unnecessary complexity ofirrelevant accuracy!

Viewed this way, the law of physicsF ⊂ ma comes to appear a little softerthan is commonly considered. It reallydoes bear a family resemblance toother kinds of laws, like the laws of ju-risprudence or of morality, whereinthe meaning of the terms takes shapethrough their use. In those domains,claims of ultimate truth are wiselyviewed with great suspicion; yetnonetheless we should actively aspireto the highest achievable level of co-herence and explicitness. Our physicsculture of force, properly understood,has this profoundly modest but prac-tically ambitious character. And onceit is no longer statuized, put on apedestal, and seen in splendid isola-tion, it comes to appear as an inspir-ing model for intellectual endeavormore generally.

References1. D. Knuth, Surreal Numbers, Addison-

Wesley, Reading, MA (1974).2. R. Carnap, Introduction to Symbolic

Logic and Its Applications, Dover, NewYork (1958).

3. P. Hayes, in Expert Systems in the Mi-croelectronic Age, D. Michie, ed., Edin-burgh U. Press, Edinburgh, UK (1979).

4. G. Sussman, J. Wisdom, Structure andInterpretation of Classical Mechanics,MIT Press, Cambridge, MA (2001). �

http://www.physicstoday.org July 2005 Physics Today 11

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