sunny y. auyang_mathematics and reality

7/29/2019 Sunny Y. Auyang_Mathematics and Reality

1/14

Philosophy of Science Association

0DWKHPDWLFVDQG5HDOLW\7ZR1RWLRQVRI6SDFHWLPHLQWKH$QDO\WLFDQG&RQVWUXFWLRQLVW9LHZVRI*DXJH)LHOG7KHRULHV$XWKRUV6XQQ\


2/14

Mathematics and Reality:Two Notionsof Spacetime n the Analytic andConstructionistViews of Gauge FieldTheoriesSunny Y. Auyangtt

This paper presents wo interpretations f the fiberbundleformalism hat is applicableto allgaugefieldtheories.The constructionistnterpretation ields a substantival pace-time. The analytic interpretationyields a structuralspacetime, a third option besidesthe familiarsubstantivalismand relationalism.That the same mathematical ormalismcanbe derived n two differentways leading to two differentontological nterpretationsreveals the inadequacyof pureformalarguments.

1. Introduction.Physical theories are mathematical, and the mathematicalstructures of modern physical theories are increasingly complicated. Therelation between mathematics and nature has always been an amazementfor physicists and a puzzle for philosophers. Realists and antirealists lockhorns over it. This paper does not go into the extended debate but focuseson a narrower issue from the realist perspective. Let us assume that mod-ern physical theories are more than sophisticated recipes for predictingexperimentaloutcomes. At least parts of theirmathematics are descriptiveof the microscopic world remote from experience. Assume also that wecan distinguish the objective parts that are ascribed to nature from theinstrumental parts that are not, a difficult task with deep philosophicalimplications. Then we have to interpretthe objective terms: What are theentities and features of the world depicted by the objective part of a physi-cal theory? Usually there are more than one consistent interpretations,tSend requests or reprints o the author, 100MemorialDr., #521B, Cambridge,MA02142;e-mail:[email protected] thank Paul Teller for helpfuldiscussions and John Stachel andAndrewWayneforcriticalreadingsof the first draft.Philosophy of Science, 67 (Proceedings) pp. S482-S494. 0031-8248/2000/67supp-0037$0.00Copyright 2000 by the Philosophy of Science Association. All rights reserved.

S482


3/14

MATHEMATICS AND REALITY S483hence more than one plausible ontologies. Faced with the ambiguity, re-alists can admit defeat and give up the objectivity of physical theories. Orthey can try to narrow down the ambiguity by ferretingout and clarifyingits sources, which are often hidden in presuppositions so familiar no onebut philosophers would worry about. This paper is an exercise in the latterattitude.The particularproblem considered here can be framed as follows. Sup-pose the matured theories for a domain P of the physical world use acomplicated mathematical structureM = {m, m2, . . . mJ, of which sci-entific consensus regards the substructureM' = {ml, . . . mkj, k < X, tobe objective. As terms in a mathematical structure, the mi's are interre-lated. How then do we interpret individual terms? Does a term in theobjective substructure M' necessarily refers to an entity that exists inde-pendently of P, the physical system to which M' as a whole describes?What I call the analytic view answers negatively, the constructionistviewaffirmatively. The two answers lead to two ontologies. In judging theirrelative merits,mathematicalformalism offers no guide. Physical and phil-osophical considerations are mandatory in deciding the relation betweenthe formalism and reality. Much mathematics is abstract construction.However, it is erroneous to prescribe a priori that the construction ofcomplex mathematical structures mirrors the construction of complexphysical structures; that because a mathematical term M3is defined interms of m1 and m2, there must be preexisting entities representedby miand m2 from which M3 s constructed.To be more specific, consider the ontology representedby gauge fieldtheories, especially the status of spacetime in the world of fields. Gaugefield theories-general relativity for the gravitational interaction andquantum field theory for the electromagnetic and nuclear interactions-are presently our most fundamental physical theories with experimentalconfirmation. They posit as the basic ontology of the universe a set ofinteracting fields. What is a physical field? Are there entities in a field?What are the general relations of fields to individual entities on the onehand and spacetime on the other?Gauge field theories contain terms withusual spatiotemporal interpretation. Does this imply that spacetime is anentity existing independently of physical fields? Substantivalism answersyes. From the application of differential geometry to field theories, somesubstantivalists argue that the mathematical notion of differentiablemani-fold must referto spacetimeas an independent primordialsubstance(Field1980, Earman 1989). I will show that their arguments follow a construc-tionist approach, and that an analytic approach yields a different inter-pretation of the same differentialgeometry applied to the same gauge fieldtheories. In the analytic interpretation, spacetime appears as an indis-pensable substructureintegral to the physical field and not a preexisting


4/14

S484 SUNNY Y. AUYANGsubstanceon which the field is built (Stachel 1986, Auyang 1995). Opposedto the structural spacetime, the mathematical weaponry deployed by theadvocates of the substantival spacetime against the relational spacetimebecomes impotent, because it cuts both ways. In the end, we are forcedback to philosophical analyses, including an examination of what it meansto be an entity.2. The Analytic and ConstructionistApproaches.A field may be the ele-mentary stratum of the physical world, but this in no way implies that itis trivial in structure. Elementary fields are complex. If they were not,gauge field theories would not be so complicated. Thus interpretations offield theories cut into the general scientificapproaches to complex systems.There are two general approaches to study physical systems of anycomplexity, one proceeds from the top down, the other from the bottomup. The top down approach, which I call synthetic analysis, takes on thetarget system as a whole, then draws distinctions to find the parts appro-priate for explaining the whole's properties. Imagine we analyze a geo-metric whole, say a disc, into two parts by drawing the boundary R todifferentiate the parts A and B. R not merely differentiates but simulta-neously unites the two parts that it delimits. Wheneverwe talk about onepart, we tacitly invoke the other part and the whole via the boundary Rwithout which it is not defined. ThereforeR is an intrinsicrelationbetweenthe two parts. The two parts are intrinsically related, for their mutualrelationship is constitutive of the properties of each. A, B, and their in-trinsicrelation R constitute a gross structureof the whole. Furtheranalysisreveals finer structures of the whole by drawing more distinctions anddelineating more intrinsically related parts.In theoretical science, the "boundaries" that cut up the whole into"parts" are concepts. Metaphorically, we conceptually carve nature at itsjoints. The trick, as Plato said, is to observe the natural joins, not tomangle it up like an unskilled butcher (Phaedrus265-266). The metaphorof the butcher is not quite appropriate, because in conceptual analysis weare careful not to kill our subject. Thus analysis is different from decom-position, which breaks a whole into independent parts. In analysis, weexamine the parts within the whole and study their functions therein. Onthe other hand, analysis is not enslaved to holism. We can make approx-imations that decouple a part from the whole and study it on its own.Nevertheless, even as we talk about entities and treat them independently,we acknowledge our approximations and note their consequences, oftentake into account the effects of the suppressedparts by some parameters.Analyses of complex systems may be long processes. To emphasize thattheir chief goal is to understand the systems as complex wholes, I qualifyanalyses with "synthetic."


5/14

MATHEMATICS AND REALITY S485One consequence of the analytic approach is that we do not assume inadvance any parts and their relations. Depending on the depth and re-finement of analysis, the parts, their properties, and relations can change.Thus we do not posit any given set of constituent entities with absoluteproperties and relations. Elsewhere I have given many examples of syn-thetic analysis from various sciences (Auyang 1998).In contrast; the bottom up approach presupposes a given set of entitieswith intrinsic properties and extrinsic relations among the entities. Imaginea miscellany of blocks, which can be stacked up to form a pyramid or atower. The relations among the blocks are extrinsic to them because itneither depends on their properties nor changes them by its operation.

The constructionist approach sees a complex system as being built fromthe given entities.Both the analytic and constructionistmethods areproductive in science,but neither is appropriate in all circumstances. When we say the gaugefields are the building blocks of the universe, we are assuming a construc-tionist stance. Our question now is: Which approach is more fruitfulwhenwe come to study structure of the gauge field itself? The answersnaturallylead to two different notions of spacetime.3. Fields and Local Fields. A physical field is a dynamical system with aninfinite degrees of freedom. To gain an intuitive idea of what it means, letus start from a system with a finite degree of freedom, a composite systemthat is not a field. Consider the oscillation of N identical beads attachedat various positions to an inelasticweightless thread whose ends are fixed,as depicted in Figure 1. We index the beads by integers n = 1,2, . . , N.The only property of the beads we are interested in is their respectiveposition or displacement. We represent the property of the nth bead attime t by the mathematical function f,(t). The temporal variation of theproperty of the whole system of beads is describedby N coupled equationsof motion for thef(t)'s, taking into account the tension of the thread andthe masses of the beads.

'Ai

(a) (b)Figure 1. (a) The integers n index the beads on a thread, whose properties are representedby the function f(t). (b) The real numbers x index the points in a string, whose propertiesare represented by f (x, t).


6/14

S486 SUNNY Y. AUYANGNow imagine that the number of beads N increases but their individualmass m decreases so that the product mN remains constant. As N goes to

infinity, the beads are squashed together, and in the limit we obtain aninelastic string with a continuous and uniform mass distribution. Insteadof using integers n that parameterize the beads, we use real numbers x, 0? x ' L to index the points on a string of length L. Instead of fn(t), theproperty of the string is characterized by f(t, x), called the displacementfield of the string.Generally, a field is a dynamical variable for a continuous system pa-rameterized by one or more continuous variables, which I summarily callx. Unless explicitly specified as in the displacement field, the spatial con-notation of a field is exhausted by the spatiotemporal parameter x, whichhas the same status as the temporal parameter t. With x fixed,f varies butgenerally does not vary spatially. Therefore it is usually called the "inter-nal" property of a point in the field. For a classical field, f may be thestrength of the electric or magnetic field component. For quantum fields,f is a quantum property represented by noncommuting operators, whichare usually not observables.There is a general difference between the displacement field and gaugefields. The property of the displacementfield is represented by an ordinaryfunction. The internal property of a gauge field is represented by a groupof symmetry transformations, for example, the unitary group U(1) for theelectron field. Furthermore, the symmetry group is "gauged" or localizedto each point in the field, hence is often called the gauge group. With thelocal symmetry representation, the property of one point in the gauge fieldcan be described totally independently of those of the other points. Con-sequently it makes explicit the notion that the points in the gauge fieldsare discrete entities, called the localfields. If we want to relate or comparetheir individual properties, then we must introduce explicit interactionmechanisms, which are beyond the scope of this paper. The constitutionof a gauge field by discrete local fields makes its analogy with the beadsystem closer.In short, a gauge field is a continuous system consisting discrete localfields, each with its individual internalproperty and its numerical identity,which is individuated, identified, or indexed by the spatiotemporal vari-able x, the generalizationof the variable x for the displacementfield. Sincex serves the same function as x, it is well to examine the significance ofthe more intuitive x. The continuous variable x for the displacement fieldplays the same role as the discrete variable n for the bead system, differingonly in that x satisfies the mathematical criterion of completeness whereasn does not. A particularvalue of x designates a specific point on the string,just as a value of n designates a specific bead. Sometimesf(t, x) is said tobe the displacement at each space point x. However, x cannot be position


7/14

MATHEMATICS AND REALITY S487in space because it remains fixedwhen the string moves spatially;x denotesnot position in space but position on the string. We can call x a spatialindex but not an index of space points in which the string sits. We cantheoretically abstract the characteristics of the variable x and say it is aone-dimensional continuum. However, this theoretical continuum repre-sents only a structural aspect of the string and not an immaterialsubstancethat can be torn out of the string and let stand on its own. The sameargument applies for the spatiotemporal variable x for the gauge fields.How then do we get the idea of spacetime as a preexisting substratumsupporting the gauge fields?

4. The Constructionist Approach and the Substantival Spacetime. Gaugefield theories are highly mathematical, including a healthy dose of differ-ential geometry. When we look at the mathematics, we find that the mostcommon mode of presentation is abstract construction. Thus a textbookin differential geometry first defines a differentiable manifold by generalcoordinate transformations. Then it introduces various constructions onthe manifold: mathematical objects such as tangent spaces are assigned toeach point in the manifold; structures such as the inner product or coor-dinate frames are introduced on the tangent space; and so on. Mathe-maticians call the collection of a type of object over all points in the mani-fold a "field" because of its smoothly varying nature. Thus the collectionof inner products on the tangent spaces is called the metric field. It isimportant to remember that these mathematical objects and fields are gen-erally not physical objects and fields. They are totally abstract. If theyhappen to represent physical objects and fields in a physical theory, thentheir physical interpretation is an extra step.If we assume that the construction of the physical world mirrors theabstract mathematical construction, then we would interpreteach layer ofmathematic structure as an independent entity. More specifically, wewould automatically identify mathematical fields with physical fields. Un-der the notion that mathematics mirrorsnature, the constructionist stanceleads us to interpretthe differentiable manifold as representativeof a sub-stantival substratum supporting the physical fields represented by themathematical fields. Because the manifold comes first in mathematics andis presupposed by other mathematical constructions, we tend to think ofthe corresponding physical substratum as primitively existing. Conse-quently we have the notion of a substantival space.

The constructionist reasoning is apparent in the writings of leadingmanifold substantivalists. Harty Field wrote, "a field is usually describedas an assignment of some property, or some number or vector or tensor,to each point of space-time; obviously this assumes that there are space-time points" (1980, 35). John Earman called Field's position "manifold


8/14

S488 SUNNY Y. AUYANGsubstantivalism" and rephrased his point: "When relativity theory ban-ished the ether, the space-time manifold M began to function as a kind ofdematerialized ether needed to support the fields." He then proceeded toflesh out the argument by parading the mathematical definitions of thedifferentiable manifold and various mathematical object fields on it, con-cluding: "It is clear that the standard characterization of fields uses thefull manifold structure" (1980, 155, 158-159). The mathematics is clear.What is obscured is their philosophical assumption that no distinctionneed to be made between mathematical and physical fields, which theyidentify under the same word "field." Consequently they marshal the ab-stract construction of mathematical fieldsupon the differentiable manifoldas the direct argument for the construction of physical fields upon a de-materialized ethereal spacetime.Manifold substantivalists count only the bare differential manifold asspacetime; other substantivalists want to include more structures. In allcases, the bottom up constructionist approach plus the notion of mathe-matics-physics mirroring lead to a substantival spacetime as a pizza crustready for toppings. The crust may be thin or thick, but it stands on itsown. The toppings may be plain or as exotic as local quantum fields. Inall cases we have a preexisting spacetime on which material entities willbe placed. Thus an extrinsic relation, supporting, underlying, or occupy-ing, obtains between the spatiotemporal substratum and the fields.I want to argue against this pizza model of physical fields. Even ifabstract construction is the way we learn the mathematics, we have noright to assume that it is mirrored by the construction of the physicalworld. First, there is more than one way to present a mathematical struc-ture, and even more ways to interpretit. Equally important, when appliedin physical theories, many concepts in a mathematical theory are instru-mental and not objective. The pizza model results from the mistake ofindiscriminate objective interpretations of the elements in a mathematicaltheory.

5. The FiberBundle Formulationof Gauge Field Theories.Since spacetimeis a general notion, it would be convenient to have a general frameworkfor discussing field theories featuring it. The mathematics of fiber bundleprovides such a framework. John Stachel (1986) has used the fiberbundleconcept to explain generally covariant field theories and Einstein's viewthat there is no spacetime without the metric field.As seen in Table 1, the fiber bundle formalism applies to most impor-tant dynamical systems. Its generalitymakes it useful for conceptual anal-ysis and comparison. The gauge fields, general relativity and quantumfields, differ from classical mechanics in that their representative fibers


9/14

MATHEMATICS AND REALITY S489TABLE 1 The fiber bundle formulation of gauge field theories. (References canbe found in Auyang, 1995, 60.)

Physical system Physical theory Fiber bundleGravitation interaction General relativity Orthonormal frame bundleElectromagnetic and Quantum field theory Principal fiber bundlenuclear interactionsClassical mechanics Lagrangian formulation Tangent bundleHamiltonian formulation Cotangent bundle

have complex structures characterized by the mathematics of group, thegauge group. I will consider them only.The fiber bundle is a high-level construction in differential geometry.Crudely, a fiber is a complex mathematical object that is associated witha point in the differentiable manifold. For instance, the tangent space withits structures is a fiber. By collecting the tangent spaces over all points ina manifold and bundling them up according to certain mathematical cri-teria, we get a fiber bundle, the tangent bundle. Another kind of mathe-matical object assigned to the points gives another kind of fiber and an-other fiber bundle, but the idea is the same.

The fiber bundle is a mathematical structure with four major elements:(D, M, Gx, ). M, called the base space, is usually a differentiablemanifold.To illustrate its idea schematically, I have drawn M in two dimensions inFigure. 2c. Each point x in the base space is associated with a mathemat-ical object, a fiber G, which can be a tangent space, a tensor, a differen-tiable manifold, or a group space. The patchwork of the products of thefiber and the manifold, which contains all fibers, is called the total space,D. The manifold and the total space, or each point in the manifold andthe fiber associated with it, are related by the projection map, it. Rigorousdefinitions are readily available in mathematics books, but it is importantin philosophy to catch the main ideas and not be distracted by the com-plicated mathematical details.Applied in physical theories, the total space D is interpretedas the statespace of a gauge field, analogous to all the possible configurations of thedisplacement field of a string. It consists of the distinctive state spaces GXof a set of discrete entities, the local fields, analogous to a point on thestring with its possible displacements. The local field Gx s indexed by thespatiotemporal point x, and M is usually interpretedas spacetime.

As I have presented it, the fiber bundle seems perfect for substantival-ism. M is the substantival spacetime, ft the extrinsic relation of supporting,and Gxa piece of topping on each point in the pizza crust. Nevertheless,mathematicians have shown that the same fiber bundle can be mathemat-ically constructed in another way. This alternative mathematical construc-


10/14

S490 SUNNY Y. AUYANGtion can be physically interpreted as the analysis of a complex physicalsystem instead of the physical construction from prefabricated parts.6. The Analytic Approachand Spacetime as the Principle of Individuation.Suppose the state space of the physical field we aim to understand is some-thing like the Mobius strip in Figure 2a, but we do not know it yet. TheMobius strip shares with the cylinder an arching spatial structure, whichis a circle. How do we find out about it? How do we "extract" the circlefrom the strip?Suppose we observe various qualities u on various locations on theMobius strip. We can say the qualities are properties of the strip as awhole but not the property of any smaller entities, because we have notyet differentiated any entities in the strip. How can we get a finer-graineddescription? How can we analyze the strip into a set of entities? Withoutsuch analysis, the M6bius strip is analogous to a physical field as some-thing amorphous.If we posit that the entities belong to the same type and therefore sharethe same set of possible properties related by a symmetry group G, thenwe have a way of partitioning the whole into entities. A mathematic groupconsists a set of transformations that map one possible quality into an-other. It establishes an equivalence relation among the qualities it con-

< ~ ~ ~ ~ ~~~~~1 DIG(a)(c(b)(c

Figure 2. The analytic view of the fiber bundle formalism. (a) We aim to study an extendedcomplex system with possible properties u on various locations. (b) The qualities are parti-tioned into G-orbits {u': u' u} according to the equivalence relation that is contained inthe symmetry group G. (c) The projection map 7tsends all elements in a G-orbit into a singlep6int x, which serves as the numerical identity of the fiber Gx.The identities x of all fibersGxconstitute the base space M. The fibers constitute the total space D = {Gx x EM}which is a set with an indexing set M.


11/14

MATHEMATICS AND REALITY S491nects. We pick an arbitraryquality u and collect all qualities u' that it canmap into under the transformations of the symmetry group. In this waywe obtain a G-orbit {u': u' u}. We pick another quality w, collect all itscompanions w', and get another G-orbit {w': w' w}. In this way we canexhaustively partition all the qualities on the strip into pairwise disjointG-orbits. Because the grouping is accomplished by an equivalence relation,the resultant G-orbitsdo not share common elements. Two G-orbitseitherhave no element in common or share every element, in which case theyare identical. They make explicit the notion of exclusiveness.A G-orbit is a cluster of possible qualities but not possible properties,for we still lack the notion of what they are properties of. We still lack theconcept of an entity that we can refer to individually and ascribe prop-erties. All G-orbits are identical because they are all cut with the samecookie cutter, the symmetry group G. To make explicit the idea that twoG-orbits are distinct although they share all qualities, we use the idea ofa quotient that results from dividing the whole by a part. We introduce amap i that sends all qualities in a G-orbit into a single point x, i(u) = x,which represents the numerical identity of the G-orbit. The addition of anumerical identity x turns a G-orbit into an entity that we can definitelyrefer to, a fiber Gx.The whole, partitioned into individual fibers, we nowcall the total space D. Furthermore, we find that the points x projectedfrom all fibers Gxin the strip constitute a system, M. In the case of theMobius strip, M has the characteristicof a circle.Here we have another way of arrivingat the fiberbundle, the structure(D, M, Gx, ). Mathematically, it is still abstract construction, as we in-troduce more and more structures.Applied to physical theories, however,the abstract construction appears not as a building from given parts butas an analysis revealing the finer structuresof a large system. We analyzea large system, the Mobius strip, into constituent entities Gx,which arenot given beforehand. Furthermore, by systematically distinguishing thenumerical identities of the entities, we have theoreticallyextracted an arch-ing structure that spans the Mobius strip as a whole, namely, its spatialstructureof a circle, representedby M. M is customarily called spacetime,but this spacetime does not induce the notion of a pregiven substratum.The basic idea behind the analytic view of the fiber bundle is the settheoretic notion of the partition of unity by an equivalence relation, quo-tient space, and a set with an indexing set. These simple set theoretic no-tions bring out an important function of spacetime that is obscured by theglamor of mathematical details. They show that a most primitivefunctionof spatiotemporal concepts in our understanding is the individuationandidentification,the indexing, of physical entities. What we learn from thefiber bundle formulation of field theories is that to represent a system ofphysical entities theoretically, the concept of a simple set is not enough.


12/14

S492 SUNNY Y. AUYANGWe need at least the concept of a set with an indexing set. D is a set withthe indexing set M, D = {Gx x E M}. Spacetime is the indexing set forthe set of local fields in a gauge field, just as certain integers form theindexing set for the system of beads on a string. Unlike substantival sup-porting, the indexing role of spacetime is crucial to physical field theories.Spacetime points do not support local fields in any physical sense. Theymake the local fields distinct.Physically, we still interpret the total space D as the state space of anentire gauge field, a fiber Gxas the state space of a local field, and M asspacetime. However, in this analytic view, spacetime M is neither a sub-stratum nor an entity that can stand on its own. Rather, it is an archingstructureof the physical gauge field as a whole. There is no a priori guar-antee that all dynamical systems have the same spatiotemporal structure,in existing theories they often do not. If we want to posit a universalspacetime, then it must come as an extra constraint on the theories forvarious physical domains. The constraint may serve as a unifying factor,but that is beyond the scope of this paper.An important consequence of analysis is that spacetime and the localfields are intrinsically related. Spacetime is derived by reckoning with theproperties of the local fields. Conversely, the local fields as individual en-tities with definite identities are recognized only within the spatiotemporalsystem. Therefore, to say that spacetime is not an independently existingsubstance does not mean that it is not objective. It is objective as an in-dispensable structure of the physical field. It is absolute in the sense thatwithout it, there would be no individual entities in the field. It rejectsNewton's notion of space as a substance, but agrees to Newton's notionof absolute space: "Space is a disposition of being qua being.... Whenany being is postulated, space is postulated" (1962, 136). When any entityor any local field is posited, spacetime is posited.7. Two Physical Interpretations of the Same Mathematics. To sum up,gauge field theories can be generally represented by the fiber bundle for-malism, and the same mathematical structure can be abstractly con-structed in two different ways. Physically, one way is easily interpretedasthe physical construction from prefabricated parts, the other way as theanalysis of a whole into finer structures. The two interpretations lead totwo different ontologies, especially two notions of spacetime.Spacetime substantivalism, which is based on the bottom up construc-tionist view, assigns physical significanceto all major mathematical termsin the fiber bundle (D, M, Gx, ). It sees the gauge field D as a constructionout of two sets of independent physical entities: local fields Gxand pre-existing spacetime points x, which constitute the substantival spacetimeM. Connecting the two is the extrinsic relation of supporting, represented


13/14

MATHEMATICS AND REALITY S493by i. This is a sumptuous ontology, perhaps too sumptuous. To accountfor all its physical objects and relations, the stock of theoretical conceptsare spread thin, leaving many questions unattended. Each set of entitiesrequiresits criterion of individuation, which is at best left vague. Substan-tivalism does not answer what differentiates one spacetime point fromanother, since all of them are exactly featureless. Furthermore, it saysalmost nothing about the physical relation between spacetime and thefield. Physical theories are most sensitive to causal mechanisms, but thealleged mechanism of spacetime physically supporting the field is notfound in physical field theories, nor is there any evidence for its causalrole.

The top down analytic view sees the whole gauge field as an integralsystem, complex but primitive. It analyzes the whole D into a set of in-trinsicallyrelated physical entities, the local fields Gx, he individuation ofwhich depends on a structure spanning the whole, the spatiotemporalstructure M. This is a parsimonious ontology with a single set of physicalentities. That it engages the same stock of theoretical concepts as substan-tivalism reveals the conceptual complexity requiredto characterizephysi-cal entitiesproperly.Conversely, it revealsthe conditions a physical systemmust satisfy if it is to be analyzed into a set of individual entities. Analysisaddresses the problem of individuation explicitly, showing that to refer toand describe particular entities individually, we need certain spatiotem-poral concepts. In this interpretation, the mapping i represents not aphysical mechanism but the conceptual relation between identification andpredication.Which physical interpretation of the mathematical formalism is better?The mathematics itself is neutral, as it should be, to this point. Here I willnot examine the pros and cons for the two notions of spacetime, havingconsidered it elsewhere (Auyang 1995). In laying out alternativeinterpre-tations of the same mathematics, this paper aims to emphasize the insuf-ficiency of formal and technical considerations in philosophical matters.The mathematical structures of physical theories are important, butequally indispensable are careful consideration of the physics and analysisof general concepts such as entity and substance. This adage is more read-ily acknowledged than practiced. General concepts are intuitive but dif-ficult to analyze and articulate, although the conclusions of philosophicalarguments are framed in terms of them. The mathematical structures ofphysical theories are complicated but susceptible to clear articulation.Thus the mathematics sometimes turns into a "techniquetrap"wherephi-losophers are snowed by formalism and blinded to its physical meaning.An example of the trap's effects is the jump from a rehearsal of differentialgeometric definitions to the conclusion that substantival spacetime pointsare the basic objects of reference and of predication. The jump ignores


14/14

S494 SUNNY Y. AUYANGimportant physical and philosophical factors and invokes general conceptsmost carelessly. What does it mean to be the basic object of predication?How do we refer to individual objects? These are central questions incontemporary analytic philosophy. They cannot be taken lightly by phi-losophers of physics. It is precisely in the world of fields where individualobjects are not obvious at all that these questions become more poignantand demand close attention. By ignoring them and concentrating on math-ematical details, philosophers of physics can be oblivious of the philo-sophical assumptions hidden in their arguments, such as physical con-struction mirrors mathematical construction. It is well to remember SaulKripke's admonition: "There is no mathematical substitution for philos-ophy" (1976, 416).

REFERENCESAuyang, Sunny (1995), How Is QuantumField Theory Possible? New York: Oxford Univer-sity Press.. (1998), Foundations of Complex System Theories in Economics, Evolutionary Biol-ogy, and Statistical Physics. New York: Cambridge University Press.Earman, John (1989), WorldEnoughand Spacetime. Cambridge, MA: MIT Press.Field, Harty (1980), Science WithoutNumber. Princeton: Princeton University Press.Kripke, Saul (1976), "Is There a Problem about Substitutional Quantification?", n E. Evansand J. McDowell (eds.), Truth and Meaning. New York: Oxford University Press.Newton, Isaac (1962), "On the Gravity and Equilibriumof Fluids", in UnpublishedScientificPapers of Isaac Newton, selected and translated by A. R. Hall and M. B. Hall. NewYork: Cambridge University Press.Stachel, John (1986), "What a Physicist Can Learn from the Discovery of General Rela-tivity", in R. Ruffini (ed.), Proceedings of the Fourth Marcel GrossmannMeeting onGeneralRelativity. Uppsala: Elsevier Science Publishers, 1857-1862.

sunny y. auyang_mathematics and reality

Documents