Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity
Author(s): Alan D. Sokal
Source: Social Text, No. 46/47, Science Wars (Spring - Summer, 1996), pp. 217-252
Published by: Duke University Press
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Transgressingthe Boundaries
TOWARD A TRANSFORMATIVE
HERMENEUTICS
OF QUANTUM
boundaries... [is] a subversiveundertaking
since
disciplinary
Transgressing
it is likelyto violatethe sanctuariesof accepted ways of perceiving.Among
the most fortified
boundarieshave been those betweenthe naturalsciences
and thehumanities.
-Valerie Greenberg,Transgressive
Readings
of ideologyintocriticalscience ... proThe struggleforthetransformation
ceeds on the foundationthatthe critiqueof all presuppositionsof science
and ideologymustbe the onlyabsoluteprincipleof science.
-Stanley Aronowitz,Scienceas Power
There are many natural scientists,and especially physicists,who continue
to reject the notion that the disciplines concerned with social and cultural
criticism can have anything to contribute, except perhaps peripherally,
to their research. Still less are they receptive to the idea that the very
foundations of their worldview must be revised or rebuilt in the light of
such criticism.Rather, they cling to the dogma imposed by the long postEnlightenment hegemony over the Western intellectual outlook, which
can be summarized brieflyas follows: that there exists an external world,
whose properties are independent of any individual human being and
indeed of humanity as a whole; that these properties are encoded in
"eternal" physical laws; and that human beings can obtain reliable, albeit
imperfectand tentative,knowledge of these laws by hewing to the "objective" procedures and epistemological strictures prescribed by the (socalled) scientificmethod.
But deep conceptual shifts within twentieth-centuryscience have
undermined this Cartesian-Newtonian metaphysics (Heisenberg 1958;
Bohr 1963); revisionist studies in the history and philosophy of science
have cast furtherdoubt on its credibility(Kuhn 1970; Feyerabend 1975;
Latour 1987; Aronowitz 1988b; Bloor 1991); and, most recently,feminist
and poststructuralistcritiques have demystifiedthe substantive content of
mainstream Western scientificpractice, revealing the ideology of domination concealed behind the facade of "objectivity" (Merchant 1980; Keller
1985; Harding 1986, 1991; Haraway 1989, 1991; Best 1991). It has thus
become increasingly apparent that physical "reality," no less than social
"reality," is at bottom a social and linguistic construct; that scientific
"knowledge," far from being objective, reflects and encodes the domiSocial Text46/47,Vol. 14, Nos. 1 and 2, Spring/Summer
1996. Copyright? 1996 by Duke
Press.
University
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GRAVITY
Alan D. Sokal
nantideologiesand powerrelationsof the culturethatproducedit; that
thetruthclaimsof scienceare inherently
and self-referential;
theory-laden
and consequently,
thatthediscourseofthescientific
forall its
community,
undeniablevalue, cannot asserta privilegedepistemologicalstatuswith
respect to counterhegemonicnarrativesemanatingfromdissidentor
marginalizedcommunities.These themescan be traced,despite some
differences
of emphasis,in Aronowitz'sanalysisoftheculturalfabricthat
produced quantummechanics(1988b, esp. chaps. 9 and 12); in Ross's
discussion of oppositionaldiscourses in post-quantumscience (1991,
intro,and chap. 1); in Irigaray'sand Hayles'sexegesesofgenderencoding
in fluidmechanics(Irigaray1985; Hayles 1992); and in Harding'scomthenaturalsciences
prehensivecritiqueofthegenderideologyunderlying
in generaland physicsin particular(1986, esp. chaps. 2 and 10; 1991,
esp. chap. 4).
Here myaim is to carrythesedeep analysesone stepfurther,
bytaking
accountof recentdevelopmentsin quantumgravity:theemergingbranch
of physicsin whichHeisenberg'squantummechanicsand Einstein'sgenare at once synthesized
eralrelativity
and superseded.In quantumgravity,
as we shall see, the space-timemanifoldceases to exist as an objective
physicalreality;geometrybecomes relationaland contextual;and the
foundationalconceptualcategoriesof priorscience-among them,existence itself-become problematizedand relativized.This conceptual
revolution,I will argue,has profoundimplicationsforthe contentof a
science.
futurepostmodernand liberatory
My approachwillbe as follows.First,I willreviewverybrieflysome
of thephilosophicaland ideologicalissues raisedby quantummechanics
and by classical generalrelativity.
Next, I will sketchthe outlinesof the
of
emergingtheory quantumgravityand discuss some of theconceptual
issues it raises.Finally,I willcommenton theculturaland politicalimplications of these scientificdevelopments.It should be emphasizedthat
this essay is of necessitytentativeand preliminary;
I do not pretendto
answerall the questionsthatI raise.My aim is, rather,to drawtheattention of readersto theseimportantdevelopmentsin physicalscience and
to sketchas best I can theirphilosophicaland politicalimplications.I
have endeavoredhere to keep mathematicsto a bare minimum;but I
have takencare to providereferenceswhereinterestedreaderscan find
all requisitedetails.
Quantum Mechanics: Uncertainty,Complementarity,
Discontinuity,and Interconnectedness
It is not my intention to enter here into the extensive debate on the conceptual foundations of quantum mechanics.1 Suffice it to say that anyone
who has seriously studied the equations of quantum mechanics will assent
2J8
Alan D. Sokal
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to Heisenberg's measured (pardon the pun) summary of his celebrated
uncertaintyprinciple:
ofthe
We can no longerspeak of thebehaviourof theparticleindependently
a
the
natural
of
As
final
laws
formulated
observation.
consequence,
process
in quantumtheoryno longerdeal withthe elementaryparmathematically
ticlesthemselves
butwithourknowledgeofthem.Nor is itanylongerpossible
to ask whetheror not theseparticlesexistin space and timeobjectively...
When we speak of the pictureof naturein the exact science of our age, we
with
do not mean a pictureof natureso much as a pictureofourrelationships
Science no longerconfrontsnatureas an objectiveobserver,but
nature.
...
sees itselfas an actorin thisinterplaybetweenman [sic]and nature.The scihas become conscious
entificmethodof analysing,explainingand classifying
science
whicharise out of the factthatby its intervention
of its limitations,
In otherwords,methodand
altersand refashionsthe objectof investigation.
object can no longerbe separated. (Heisenberg 1958, 28-29; emphasis in
original)2
Along the same lines, Niels Bohr (1928; cited in Pais 1991, 314) wrote:
"An independent realityin the ordinary physical sense can ... neitherbe
ascribed to the phenomena nor to the agencies of observation." Stanley
Aronowitz (1988b, 251-56) has convincinglytraced this worldview to the
crisis of liberal hegemony in Central Europe in the years prior and subsequent to World War I.3
A second important aspect of quantum mechanics is its principle of
or dialecticism.Is lighta particle or a wave? Complemencomplementarity,
tarity "is the realization that particle and wave behavior are mutually
exclusive, yet that both are necessary for a complete description of all
phenomena" (Pais 1991, 23).4More generally,notes Heisenberg,
intuitivepictureswhich we use to describe atomic systems,
the different
althoughfullyadequate for given experiments,are neverthelessmutually
exclusive.Thus, forinstance,the Bohr atom can be describedas a smallscale planetarysystem,having a centralatomic nucleus about which the
externalelectronsrevolve.For otherexperiments,
however,it mightbe more
convenientto imaginethatthe atomicnucleus is surroundedby a systemof
waveswhose frequencyis characteristic
of theradiationemanating
stationary
fromthe atom. Finally,we can considerthe atom chemically. . . Each picture is legitimatewhen used in the rightplace, but the different
pictures
are contradictoryand thereforewe call them mutuallycomplementary.
(1958, 40-41)
And once again Bohr (1934; cited in Jammer 1974, 102): "A complete
elucidation of one and the same object may require diverse points of
view which defy a unique description. Indeed, strictlyspeaking, the conscious analysis of any concept stands in a relation of exclusion to its
the Boundaries
Transgressing
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219
immediateapplication."5This foreshadowingof postmodernistepistemologyis by no means coincidental.The profoundconnectionsbetween
and deconstructionhave recentlybeen elucidatedby
complementarity
Froula (1985) and Honner (1994), and, in great depth, by Plotnitsky
(1994).6,7
A thirdaspectof quantumphysicsis discontinuity,
or rupture:
as Bohr
(1928; citedin Jammer1974, 90) explained,[the]essence [ofthe quantumtheory]maybe expressedin the so-calledquantumpostulate,which
attributes
to any atomicprocessan essentialdiscontinuity,
or ratherindividuality,completelyforeignto the classical theoriesand symbolizedby
Planck'squantumof action."A halfcenturylater,the expression"quantumleap" has so enteredour everydayvocabularythatwe are likelyto use
it withoutanyconsciousnessof its originsin physicaltheory.
showthatan act
Finally,Bell'stheoremsand itsrecentgeneralizations9
of observationhere and now can affectnot only the object being
observed-as Heisenbergtold us-but also an object arbitrarily
faraway
Andromeda
This
on
Einstein
termed
(say,
galaxy).
phenomenon-which
"spooky"-imposes a radical reevaluationof the traditionalmechanistic
concepts of space, object, and causality,10and suggestsan alternative
worldviewin whichthe universeis characterizedby interconnectedness
and (w)holism:whatphysicistDavid Bohm (1980) has called "implicate
New Age interpretations
of theseinsightsfromquantumphysics
order.""11
have oftengone overboardin unwarrantedspeculation,but the general
soundness of the argumentis undeniable.12 In Bohr's words,"Planck's
discovery of the elementaryquantum of action . .
revealed a feature of
inherentin atomicphysics,goingfarbeyondtheancientidea of
wholeness
thelimiteddivisibility
of matter"(Bohr 1963, 2; emphasisin original).
Hermeneuticsof Classical General Relativity
In theNewtonianmechanisticworldview,
space and timeare distinctand
absolute.13In Einstein'sspecialtheoryof relativity
(1905), thedistinction
betweenspace and timedissolves:thereis onlya new unity,four-dimensional space-time,and the observer'sperceptionof "space" and "time"
dependson herstateof motion.14In HermannMinkowski'sfamouswords
(1908): "Henceforthspace by itself,and time by itself,are doomed to
fade away into mere shadows,and only a kindof union of the two will
preservean independentreality"(translatedin Lorentzet al. 1952, 75).
Nevertheless,the underlyinggeometryof Minkowskian space-time
remainsabsolute.15
It is in Einstein's general theory of relativity(1915) that the radical
conceptual break occurs: the space-time geometry becomes contingent
220
Alan D. Sokal
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and dynamical, encoding in itselfthe gravitationalfield. Mathematically,
Einstein breaks with the traditiondating back to Euclid (which is inflicted
on high-school students even today!), and employs instead the nonEuclidean geometry developed by Riemann. Einstein's equations are
highly nonlinear, which is why traditionallytrained mathematicians find
them so difficultto solve.16Newton's gravitationaltheory corresponds to
the crude (and conceptually misleading) truncationof Einstein's equations
in which the nonlinearity is simply ignored. Einstein's general relativity
thereforesubsumes all the putative successes of Newton's theory, while
going beyond Newton to predict radically new phenomena that arise
directly from the nonlinearity: the bending of starlightby the sun, the
precession of the perihelion of Mercury, and the gravitationalcollapse of
stars into black holes.
General relativityis so weird that some of its consequences-deduced
by impeccable mathematics, and increasingly confirmed by astrophysical
observation-read like science fiction.Black holes are by now well-known,
and wormholes are beginning to make the charts. Perhaps less familiaris
G6del's construction of an Einstein space-time admitting closed timelike
curves: that is, a universe in which it is possible to travel into one's own
past!17
Thus, general relativityforces upon us radically new and counterintuitivenotions of space, time, and causality;18so it is not surprisingthat it
has had a profound impact not only on the natural sciences but also on
philosophy, literarycriticism,and the human sciences. For example, in a
celebrated symposium three decades ago on Les Langages critiqueset les sciencesde l'homme,Jean Hyppolite raised an incisive question about Jacques
Derrida's theory of structureand sign in scientificdiscourse:
When I take,forexample,the structureof certainalgebraicconstructions
[ensembles],where is the center?Is the centerthe knowledgeof general
rules which,aftera fashion,allow us to understandthe interplayof the elements?Or is the centercertainelementswhichenjoy a particularprivilege
withinthe ensemble?. . . With Einstein,forexample,we see the end of a
kindof privilegeof empiricevidence.And in thatconnectionwe see a constantappear, a constantwhichis a combinationof space-time,which does
who live the experience,but which,
not belongto any of the experimenters
in a way,dominatesthewhole construct;and thisnotionof the constant-is
thisthe center?19
Derrida's perceptive reply went to the heart of classical general relativity:
The Einsteinianconstantis not a constant,is not a center.It is theveryconthe conceptof thegame. In otherwords,it is
cept of variability-itis, finally,
not the concept of something-ofa centerstartingfromwhichan observer
could masterthe field-but theveryconceptof thegame.20
Transgressingthe Boundaries
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221
In mathematicalterms,Derrida's observationrelatesto the invarianceof
theEinsteinfieldequationG, = 8&GT , undernonlinearspace-timediffeomorphisms(self-mappingsof the space-timemanifoldthatare infibutnotnecessarilyanalytic).The keypointis thatthis
nitelydifferentiable
thismeansthatanyspace-timepoint,
invariancegroup"acts transitively":
intoanyother.In thiswaytheinfiniteifit existsat all,can be transformed
betweenobserverand
dimensionalinvariancegrouperodesthedistinction
observed;the7rof Euclid and the G of Newton,formerly
thoughtto be
constantand universal,are now perceivedin theirineluctablehistoricity;
disconnectedfrom
and theputativeobserverbecomesfatallyde-centered,
any epistemiclinkto a space-timepointthatcan no longerbe definedby
geometryalone.
Quantum Gravity:String,Weave, or MorphogeneticField?
whileadequate withinclassicalgeneralrelaHowever,thisinterpretation,
tivity,becomes incompletewithinthe emergingpostmodernview of
quantum gravity.When even the gravitationalfield-geometryincarnate-becomes a noncommuting(and hence nonlinear)operator,how
of Gv as a geometricentitybe sustained?
can the classicalinterpretation
becomes
Now not onlythe observer,but the veryconceptof geometry,
relationaland contextual.
is thus the
The synthesisof quantum theoryand generalrelativity
no one todaycan precentralunsolvedproblemof theoreticalphysics;21
dict withconfidencewhatwill be the language and ontology,much less
the content,of this synthesis,when and if it comes. It is, nevertheless,
usefulto examinehistoricallythe metaphorsand imagerythattheoretical physicistshave employedin theirattemptsto understandquantum
gravity.
The earliestattempts,datingback to the early 1960s, to visualize
geometryon the Planck scale (about 10-33 centimeters)portrayedit as
"space-timefoam": bubbles of space-timecurvature,sharinga complex
and ever-changingtopologyof interconnections(Wheeler 1964). But
perhapsbecause of
physicistswereunable to carrythisapproachfurther,
theinadequatedevelopmentat thattimeof topologyand manifoldtheory
(see below).
In the 1970s physiciststriedan even more conventionalapproach:
simplifythe Einsteinequationsby pretendingthattheyare almostlinear,
and then apply the standardmethods of quantum fieldtheoryto the
thusoversimplified
equations.But thismethod,too, failed:it turnedout
that Einstein's general relativityis, in technical language, "perturbatively
nonrenormalizable" (Isham 1991, sec. 3.1.4). This means that the strong
222
Alan D. Sokal
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are intrinsicto the theory;
nonlinearitiesof Einstein'sgeneralrelativity
are weak is simplyselfto
that
the
nonlinearities
any attempt pretend
contradictory.(This is not surprising: the almost-linearapproach
such as
destroysthe most characteristicfeaturesof general relativity,
black holes.)
In the 1980s a very differentapproach, known as stringtheory,
became popular: here the fundamentalconstituentsof matterare not
pointlikeparticlesbut rathertiny(Planck-scale)closed and open strings
(Green et al. 1987). In thistheory,thespace-timemanifolddoes not exist
as an objectivephysicalreality;rather,space-timeis a derivedconcept,an
approximationvalid only on large lengthscales (where "large" means
"much largerthan 10-33 centimeters"!).For a whilemanyenthusiastsof
stringtheorythoughttheywere closingin on a Theory of Everythingmodestyis not one oftheirvirtues-and some stillthinkso. But themathin stringtheoryare formidable,and it is far from
ematical difficulties
clearthattheywillbe resolvedanytimesoon.
More recently,a small group of physicistshas returnedto the full
nonlinearitiesof Einstein'sgeneralrelativity,
and-using a new mathematicalsymbolisminventedby AbhayAshtekar-theyhave attemptedto
visualizethe structureof the correspondingquantumtheory(Ashtekaret
al. 1992; Smolin 1992). The picturetheyobtainis intriguing:
as in string
the
manifold
is
an
valid
at large
only approximation
theory, space-time
distances,not an objectivereality;at small (Planck-scale)distances,the
ofthreads.
geometryof space-timeis a weave-a complexinterconnection
Finally,an excitingproposalhas been takingshape overthe past few
collaborationofmathematicians,
yearsin thehandsof an interdisciplinary
and
is
the
astrophysicists, biologists:this
theoryof the morphogenetic
field.22Since the mid-1980s evidence has been accumulatingthat this
field,firstconceptualizedby developmentalbiologists(Waddington1965;
Corner 1966; Giereret al. 1978), is in factcloselylinkedto the quantum
field:23(a) it pervadesall space; (b) it interactswithall matgravitational
of whetheror not thatmatter/energy
terand energy,irrespective
is magneticallycharged;and, most significantly,
(c) it is whatis knownmathematicallyas a "symmetricsecond-ranktensor."All threepropertiesare
of gravity;and it was provedsome yearsago thatthe only
characteristic
self-consistent
nonlinear
second-ranktensorfieldis,
theoryof a symmetric
at least at low energies,preciselyEinstein'sgeneralrelativity(Boulware
and Deser 1975). Thus, ifthe evidencefor(a), (b), and (c) holds up, we
can inferthat the morphogeneticfieldis the quantum counterpartof
Einstein'sgravitational
field.Untilrecently
thistheoryhas been ignoredor
even scorned by the high-energy-physicsestablishment,which has traditionally resented the encroachment of biologists (not to mention humanists) on its "turf."24However, some theoretical physicists have recently
Transgressingthe Boundaries
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223
begun to give this theory a second look, and there are good prospects for
progress in the near future.25
It is stilltoo soon to say whether stringtheory,the space-time weave,
or morphogenetic fields will be confirmed in the laboratory: the experiments are not easy to perform. But it is intriguingthat all three theories
have similar conceptual characteristics: strong nonlinearity, subjective
space-time, inexorable flux, and a stress on the topology of interconnectedness.
Differential
Topology and Homology
Unbeknownst to most outsiders, theoretical physics underwent a significant transformation-albeit not yet a true Kuhnian paradigm shift-in
the 1970s and 1980s: the traditional tools of mathematical physics (real
and complex analysis), which deal with the space-time manifold only
locally, were supplemented by topological approaches (more precisely,
methods fromdifferentialtopology26) that account forthe global (holistic)
structureof the universe. This trend was seen in the analysis of anomalies
in gauge theories (Alvarez-Gaum& 1985);27 in the theory of vortex-mediated phase transitions(Kosterlitz and Thouless 1973);28 and in stringand
superstring theories (Green et al. 1987). Numerous books and review
articles on "topology for physicists" were published during these years
(e.g., Nash and Sen 1983).
At about the same time, in the social and psychological sciences
Jacques Lacan pointed out the key role played by differentialtopology:
This diagram [the M6bius strip]can be consideredthe basis of a sort of
essentialinscriptionat the origin,in the knotwhichconstitutesthe subject.
This goes muchfurther
thanyou maythinkat first,because you can search
forthe sortof surfaceable to receivesuch inscriptions.
You can perhapssee
is unsuitable.A torus,a Klein
thatthe sphere,thatold symbolfortotality,
is
bottle,a cross-cutsurface,are able to receivesuch a cut.And thisdiversity
as
it
about
the
structure
of
mental
disexplainsmanythings
veryimportant
ease. If one can symbolizethe subjectby thisfundamentalcut,in the same
way one can showthata cut on a toruscorrespondsto the neuroticsubject,
and on a cross-cutsurfaceto anothersort of mentaldisease. (Lacan 1970,
192-93; lecturegivenin 1966)29
As Althusser (1993, 50) rightlycommented, "Lacan finallygives Freud's
thinkingthe scientificconcepts that it requires."30More recently,Lacan's
topologiedu sujet has been applied fruitfullyto cinema criticism (Miller
1977/78, esp. 24-25)31 and to the psychoanalysis of AIDS (Dean 1993,
esp. 107-8). In mathematical terms, Lacan is here pointing out that the
firsthomology group32of the sphere is trivial,while those of the other sur224
Alan D. Sokal
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facesare profound;and thishomologyis linkedwiththeconnectednessor
as
of the surfaceafterone or morecuts.33Furthermore,
disconnectedness
connectionbetweentheexternalstrucLacan suspected,thereis an intimate
tureof the physicalworldand its innerpsychologicalrepresentation
qua
been confirmed
Witten's
derivaknottheory:thishypothesis
has recently
by
tion of knotinvariants(in particularthe Jonespolynomial[Jones1985])
Chern-Simons
fromthree-dimensional
quantumfieldtheory(Witten1989).
Analogous topological structuresarise in quantum gravity,but
ratherthantwoinasmuchas themanifoldsinvolvedare multidimensional
well.
a
role
as
These multidimensional,higherhomologygroups play
dimensionalmanifoldsare no longeramenable to visualizationin conCartesianspace: forexample,the projective
ventionalthree-dimensional
which
arises
from
the ordinary3-sphere by identification
RP3,
space
of antipodes,would requirea Euclidean embeddingspace of dimension
at least 5 (James 1991, 271-72).34 Nevertheless,the higherhomology
via a suitablemultidigroups can be perceived,at least approximately,
mensional(nonlinear)logic (Kosko 1993).35
ManifoldTheory: (W)holes and Boundaries
Luce Irigaray(1987, 76-77), in her famous article "Is the Subject of
Science Sexed?" pointedout that
themathematical
desensembles],
sciences,in thetheoryof wholes[theorie
concernthemselves
withclosedand openspaces... Theyconcernthemselvesverylittlewiththequestionofthepartially
open,withwholesthatare
notclearlydelineated
[ensembles
flous],withanyanalysisoftheproblemof
borders[bords]..
.36
In 1982, when Irigaray'sessay firstappeared, thiswas an incisivecriticism: differential
topologyhas traditionally
privilegedthe studyof what
are knowntechnicallyas "manifoldswithoutboundary."However,in the
past decade, undertheimpetusof thefeminist
critique,some mathematicians have given renewed attentionto the theoryof "manifoldswith
boundary"[Fr. varitis a' bord](see, forexample,Hamza 1990; McAvity
and Osborn 1991; Alexanderet al. 1993). Perhapsnotcoincidentally,
it is
preciselythesemanifoldsthatarise in the new physicsof conformalfield
theory,superstring
theory,and quantumgravity.
In stringtheory,the quantum-mechanical
amplitudeforthe interaction of n closed or open strings is represented by a functional integral
(basically, a sum) over fields living on a two-dimensional manifold with
boundary (Green et al. 1987). In quantum gravity,we may expect that a
similar representation will hold, except that the two-dimensional man-
Transgressingthe Boundaries
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225
one. Unfortuifoldwithboundarywillbe replacedby a multidimensional
goes againstthe grainof conventionallinear
nately,multidimensionality
mathematicalthought,and despite a recent broadening of attitudes
nonlinearphe(notablyassociated withthe studyof multidimensional
manifoldswith
nomenain chaos theory),the theoryof multidimensional
boundaryremainssomewhatunderdeveloped.Nevertheless,physicists'
workon the functional-integral
approach to quantumgravitycontinues
apace (Hamber 1992; Nabutoskyand Ben-Av 1993; Kontsevich1994),
ofmathematicians.37
and thisworkis likelyto stimulatetheattention
As Irigarayanticipated,an importantquestionin all of thesetheories
is: can the boundarybe transgressed(crossed), and if so, whathappens
then?Technically,thisis knownas the problemof boundaryconditions
level,themostsalientaspectofboundary
(b.c.). At a purelymathematical
conditionsis the greatdiversityof possibilities:forexample,"freeb.c."
b.c." (specularreflection
as in a mir(no obstacleto crossing),"reflecting
in
and
another
of
the
b.c."
manifold),
ror), "periodic
(re-entrance
part
with
The
b.c."
"antiperiodic
(re-entrance
question
180-degreetwist).
is: of all theseconceivableboundaryconditions,which
posed byphysicists
of quantum gravity?Or perones actuallyoccur in the representation
of
and
on an equal footing,as sugdo
them
occur
all
simultaneously
haps,
the
complementarity
principle?38
gestedby
At thispointmysummaryof developmentsin physicsmuststop,for
thesimplereasonthattheanswersto thesequestions-ifindeedtheyhave
univocal answers-are not yet known.In the remainderof this essay,I
proposeto takeas mystartingpointthosefeaturesof thetheoryof quanwellestablished(at leastby thestandards
tumgravitywhichare relatively
of conventionalscience), and attemptto drawout theirphilosophicaland
politicalimplications.
Transgressingthe Boundaries: Toward a LiberatoryScience
Over the past two decades therehas been extensivediscussionamong
of modernistversus
criticaltheoristswithregardto the characteristics
culture;and in recentyearsthesedialogueshave begunto
postmodernist
devote detailed attentionto the specificproblemsposed by the natural
sciences (see especially Merchant 1980; Keller 1985; Harding 1986;
Aronowitz1988b; Haraway1991; and Ross 1991). In particular,Madsen
and Madsen have recentlygivena veryclear summaryofthecharacterisscience.They posittwocriteriafor
ticsofmodernistversuspostmodernist
a postmodern science: "A simple criterion for science to qualify as postmodern is that it be free from any dependence on the concept of objective truth. By this criterion, for example, the complementarity interpre226
Alan D. Sokal
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tationof quantumphysicsdue to Niels Bohr and theCopenhagenschool
is seen as postmodernist"(1990, 471).39 Clearly,quantumgravityis in
this respect an archetypalpostmodernistscience. Second, "The other
conceptwhichcan be takenas beingfundamentalto postmodernscience
Postmodernscientific
theoriesare constructedfrom
is thatof essentiality.
theoretical
elements
which
are
essential
for the consistencyand
those
utilityof the theory"(1990, 471-72). Thus quantitiesor objectswhich
are in principleunobservable-such as space-timepoints,exact particle
positions,or quarksand gluons-ought notto be introducedintothetheory.40Whilemuch of modernphysicsis excludedby thiscriterion,
quantumgravityagain qualifies:in thepassage fromclassicalgeneralrelativity
to the quantized theory,space-timepoints (and indeed the space-time
manifolditself)have disappearedfromthetheory.
fora
However,these criteria,admirableas theyare, are insufficient
liberatorypostmodern science: they liberate human beings from the
tyrannyof "absolute truth"and "objectivereality,"but not necessarily
fromthe tyrannyof otherhuman beings. In AndrewRoss's words,we
need a science "thatwill be publiclyanswerableand of some serviceto
progressiveinterests"(1991, 29).41 From a feministstandpoint,Kelly
Oliver(1989, 146) makesa similarargument:
In orderto be revolutionary,
feminist
theorycannotclaimto describewhat
"natural
facts."
feminist
theories
shouldbe political
exists,or,
Rather,
tools,
in
for
concrete
situations.
The
strategies overcoming
oppression specific
of
feminist
should
be
to
theories-not
goal,then,
theory,
developstrategic
truetheories,
notfalsetheories,
butstrategic
theories.
How, then,is thisto be done?
In what follows,I would like to discuss the outlinesof a liberatory
postmodernscience on two levels: first,withregardto generalthemes
and attitudes;and second, withregardto politicalgoals and strategies.
One characteristic
of theemergingpostmodernscienceis itsstresson
and discontinuity:
thisis evident,forexample,in chaos thenonlinearity
as wellas in quantumgravity.42At
oryand thetheoryof phase transitions
the same time,feministthinkershave pointedout the need foran adein particularturbulentfluidity(Irigaray1985;
quate analysisof fluidity,
as it mightat
Hayles 1992).43These two themesare not as contradictory
firstappear: turbulenceconnectswithstrongnonlinearity,
and smoothis sometimesassociatedwithdiscontinuity
ness/fluidity
(e.g., in catastrophe theory[Thom 1975, 1990; Arnol'd 1992]); so a synthesisis by no
means out of the question.
Second, the postmodern sciences deconstruct and transcend the
Cartesian metaphysical distinctions between humankind and Nature,
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227
One characteristic observer and observed, Subject and Object. Already quantum mechanics,
oftheemerging
postmodern
scienceis itsstress
on nonlinearity
and discontinuity:
thisis evident,for
example,inchaos
theoryand the
theoryofphase
transitions
as well
as inquantum
gravity.
earlier in this century, shattered the ingenuous Newtonian faith in an
objective, prelinguistic world of material objects "out there"; no longer
could we ask, as Heisenberg put it, whether "particles exist in space and
time objectively." But Heisenberg's formulation still presupposes the
objective existence of space and time as the neutral, unproblematic arena
in which quantized particle-waves interact (albeit indeterministically);and
it is preciselythis would-be arena that quantum gravityproblematizes.Just
as quantum mechanics informsus that the position and momentum of a
particle are brought into being only by the act of observation, so quantum
gravity informs us that space and time themselves are contextual, their
meaning defined only relative to the mode of observation.44
Third, the postmodern sciences overthrowthe static ontological categories and hierarchies characteristic of modernist science. In place of
atomism and reductionism, the new sciences stress the dynamic web of
relationships between the whole and the part; in place of fixed individual
essences (e.g., Newtonian particles), they conceptualize interactions and
flows (e.g., quantum fields). Intriguingly,these homologous featuresarise
in numerous seemingly disparate areas of science, from quantum gravity
to chaos theory to the biophysics of self-organizingsystems. In this way,
the postmodern sciences appear to be converging on a new epistemological paradigm, one that may be termed an ecologicalperspective, broadly
understood as "recogniz[ing] the fundamentalinterdependence of all phenomena and the embeddedness of individuals and societies in the cyclical
patterns of nature" (Capra 1988, 145).45
A fourthaspect of postmodern science is its self-conscious stress on
symbolism and representation. As Robert Markley (1992, 264) points
out, the postmodern sciences are increasingly transgressingdisciplinary
boundaries, taking on characteristics that had heretofore been the
province of the humanities:
Quantum physics,hadron bootstraptheory,complex numbertheory,and
chaos theorysharethe basic assumptionthatrealitycannotbe describedin
linearterms,thatnonlinear-and unsolvable-equations are the onlymeans
possibleto describea complex,chaotic,and non-deterministic
reality.These
in
theories
metacritical
the
sense that
postmodern
are-significantly-all
themselves
as
rather
than
as
"accurate"
theyforeground
metaphors
descriptions of reality.In termsthatare more familiarto literarytheoriststhanto
theoreticalphysicists,we mightsay that these attemptsby scientiststo
develop new strategiesof descriptionrepresentnotes towardsa theoryof
and verbal-is
theories,of how representation-mathematical,
experimental,
not a solutionbut partof the semiinherently
complexand problematizing,
otics of investigating
the universe.46
228
Alan D. Sokal
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From a differentstartingpoint, Aronowitz (1988b, 344) likewise suggests
that a liberatoryscience may arise from interdisciplinarysharing of epistemologies:
Natural objects are also sociallyconstructed.It is not a questionof whether
thesenaturalobjects,or, to be moreprecise,the objectsof naturalscientific
knowledge,exist independentlyof the act of knowing.This question is
answeredbytheassumptionof "real" timeas opposed to thepresupposition,
commonamong neo-Kantians,thattimealwayshas a referent,
thattempoa relative,notan unconditioned,category.Surely,theearth
ralityis therefore
evolvedlong beforelifeon earth.The questionis whetherobjectsof natural
scientific
knowledgeare constitutedoutsidethesocial field.Ifthisis possible,
we can assume thatscience or art may develop proceduresthateffectively
neutralize the effectsemanatingfromthe means by which we produce
Performanceartmaybe such an attempt.
knowledge/art.
Finally,postmodernscience provides a powerfulrefutationof the authoritarianismand elitisminherentin traditionalscience, as well as an empirical
basis for a democratic approach to scientificwork. For, as Bohr noted, "a
complete elucidation of one and the same object may require diverse points
of view which defya unique description";thisis quite simplya factabout the
world, much as the self-proclaimedempiricistsof modernistscience might
preferto deny it. In such a situation,how can a self-perpetuatingsecular
priesthood of credentialed "scientists" purport to maintain a monopoly on
the production of scientificknowledge? (Let me emphasize that I am in no
way opposed to specialized scientifictraining;I object only when an elite
caste seeks to impose its canon of "high science," withthe aim of excluding
a priorialternativeformsof scientificproduction by nonmembers.)47
The content and methodology of postmodern science thus provide
powerful intellectual support for the progressive political project, understood in its broadest sense: the transgressingof boundaries, the breaking
down of barriers,the radical democratization of all aspects of social, economic, political, and culturallife (see, for example, Aronowitz 1994). Conversely,one part of this project must involve the constructionof a new and
trulyprogressive science that can serve the needs of such a democratized
society-to-be. As Markley observes, there seem to be two more-or-less
mutually exclusive choices available to the progressive community:
On theone hand,politicallyprogressivescientistscan tryto recuperateexisteneing practicesformoralvaluestheyuphold,arguingthattheirright-wing
mies are defacingnatureand thatthey,the counter-movement,
have access
to the truth.[But] the stateof the biosphere-air pollution,waterpollution,
disappearingrain forests,thousandsof species on the verge of extinction,
large areas of land burdened far beyond theircarryingcapacity,nuclear
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229
power plants,nuclear weapons, clearcutswhere thereused to be forests,
malnutrition,
starvation,
disappearingwetlands,nonexistent
grasslands,and
caused diseases-suggests thattherealistdreamof
a rashof environmentally
scientificprogress, of recapturingrather than revolutionizingexisting
to a politicalstruggle
methodologiesand technologies,is, at worst,irrelevant
thatseeks somethingmore than a reenactmentof statesocialism.(Markley
1992, 271)
The alternativeis a profound reconception of science as well as politics:
The dialogical move towardsredefiningsystems,of seeing the world not
onlyas an ecologicalwholebut as a set of competingsystems-a worldheld
togetherby thetensionsamongvariousnaturaland humaninterests-offers
thepossibilityof redefining
whatscienceis and whatit does, of restructuring
deterministic
schemesof scientificeducationin favorof ongoingdialogues
about how we intervenein our environment.
(Markley1992, 271)48
It goes without saying that postmodernist science unequivocally favors
the latter,deeper approach.
In addition to redefiningthe content of science, it is imperative to
restructure and redefine the institutional loci in which scientific labor
takes place-universities, government labs, and corporations-and
reframethe reward system that pushes scientiststo become, often against
theirown betterinstincts,the hired guns of capitalists and the military.As
Aronowitz (1988b, 351) has noted, "One-third of the 11,000 physics
graduate students in the United States are in the single subfield of solid
state physics, and all of them will be able to get jobs in that subfield."
(Although this observation appeared in 1988, it is all the more true today.)
By contrast,there are few jobs available in eitherquantum gravityor environmental physics.
But all this is only a firststep: the fundamental goal of any emancipatory movement must be to demystifyand democratize the production of
scientific knowledge, to break down the artificialbarriers that separate
"scientists" from "the public." Realistically, this task must start with the
younger generation,througha profound reformof the educational system
(Freire 1970; Aronowitz and Giroux 1991, 1993). The teaching of science
and mathematics must be purged of its authoritarianand elitistcharacteristics,49 and the content of these subjects enriched by incorporating the
insights of the feminist,queer, multiculturalist,and ecological critiques.50
Finally, the content of any science is profoundly constrained by the
language within which its discourses are formulated; and mainstream
Westernphysical science has, since Galileo, been formulatedin the language
of mathematics.51But whose mathematics? The question is fundamental,
for, as Aronowitz has observed, "neither logic nor mathematics escapes
230
Alan D. Sokal
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the 'contamination' of the social" (Aronowitz 1988b, 346).52 And as feminist thinkershave repeatedly pointed out, in the present culture this contamination is overwhelmingly capitalist, patriarchal, and militaristic:
"Mathematics is portrayed as a woman whose nature desires to be the
conquered Other" (Campbell and Campbell-Wright 1995, 135).53 Thus,
a liberatoryscience cannot be complete without a profound revision of the
canon of mathematics.54As yet no such emancipatory mathematics exists,
and we can only speculate upon its eventual content. We can see hints of
it in the multidimensional and nonlinear logic of fuzzy systems theory
(Kosko 1993); but this approach is still heavily marked by its origins in
the crisis of late-capitalist production relations.55 Catastrophe theory
(Thom 1975, 1990; Arnol'd 1992), with its dialectical emphases on
smoothness/discontinuityand metamorphosis/unfolding,will indubitably
play a major role in the future mathematics; but much theoretical work
remains to be done before this approach can become a concrete tool of
progressive political praxis (see Schubert 1989 for an interestingstart).
Finally, chaos theory-which provides our deepest insightsinto the ubiquitous yet mysterious phenomenon of nonlinearity-will be central to all
futuremathematics. And yet,these images of the futuremathematicsmust
remain but the haziest glimmer: for,alongside these three young branches
in the tree of science, there will arise new trunks and branches-entire
new theoretical frameworks-of which we, with our present ideological
blinders, cannot yet even conceive.
Notes
I thankGiacomo Caracciolo,Lucia Fernindez-Santoro,Lia Gutierrez,and Elizabeth Meiklejohnforenjoyablediscussionswhichhave contributedgreatlyto this
essay.Needlessto say,thesepeople shouldnotbe assumedto be in totalagreement
withthe scientificand politicalviewsexpressedhere,nor are theyresponsiblefor
remain.
anyerrorsor obscuritieswhichmayinadvertently
1. For a samplingof views,see Jammer1974; Bell 1987; Albert1992; Diirr
et al. 1992; Weinberg1992 (chap. 4); Coleman 1993; Maudlin 1994; Bricmont
1994.
2. See also Overstreet1980; Craige 1982; Hayles 1984; Booker 1990;
of ideas
Greenberg 1990; and Porter 1990 for examples of cross-fertilization
betweenrelativistic
quantumtheoryand literarycriticism.
been misUnfortunately,
Heisenberg'suncertainty
principlehas frequently
interpretedby amateur philosophers. As Gilles Deleuze and Felix Guattari
(1994, 129-30) lucidlypointout,
The contentand
of
methodology
postmodern
sciencethus
providepowerful
intellectual
supportforthe
progressive
politicalproject,
understoodinits
broadestsense:
thetransgressing
of boundaries,
thebreaking
down ofbarriers,
theradical
democratization
ofallaspectsof
social,economic,
and
political,
life.
cultural
in quantumphysics,Heisenberg'sdemon does not expressthe impossibility
of measuringboththe speed and thepositionof a particleon thegroundsof
the Boundaries
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231
a subjectiveinterference
of the measurewiththe measured,but it measures
exactlyan objectivestateof affairsthatleaves the respectivepositionof two
of its particlesoutside of the fieldof its actualization,the numberof independentvariablesbeingreducedand thevaluesofthecoordinateshavingthe
same probability Perspectivisim,
or scientific
is neverrelative
relativism,
....
to a subject: it constitutesnot a relativity
of truthbut, on the contrary,a
truthof the relative,thatis to say,of variableswhose cases it ordersaccording to thevalues it extractsfromthemin its systemof coordinates.
3. See also Porush (1989) fora fascinatingaccount of how a second group
of scientistsand engineers-cyberneticists-contrived,
withconsiderablesuccess,
to subvertthe most revolutionary
of
implications quantum physics.The main
limitationof Porush's critiqueis thatit remainssolelyon a culturaland philoby an analystrengthened
sophicalplane; his conclusionswould be immeasurably
sis of economicand politicalfactors.(For example,Porush failsto mentionthat
Claude Shannonworkedforthethentelephonemonopoly
engineer-cyberneticist
AT&T.) A carefulanalysiswould show,I think,thatthe victoryof cybernetics
overquantumphysicsin the 1940s and 1950s can be explainedin largepart by
the centralityof cyberneticsto the ongoing capitalistdrive for automationof
industrialproduction,comparedwiththe marginalindustrialrelevanceof quantummechanics.
4. Aronowitz(1981, 28) has noted that wave-particledualityrendersthe
"willto totalityin modernscience" severelyproblematic:
The differences
withinphysicsbetweenwave and particletheoriesof matter,
the indeterminacy
principlediscoveredby Heisenberg,Einstein'srelativity
of arrivingat a unified
theory,all are accommodationsto the impossibility
fieldtheory,one in whichthe "anomaly" of differencefora theorywhich
posits identitymay be resolvedwithoutchallengingthe presuppositionsof
science itself.
For furtherdevelopmentof theseideas, see Aronowitz1988a, 524-25, 533.
5. Bohr's analysisof the complementarity
principlealso led him to a social
outlookthatwas, forits time and place, notablyprogressive.Consider the followingexcerptfroma 1938 lecture(Bohr 1958, 30):
I mayperhapshereremindyou of theextentto whichin certainsocietiesthe
roles of men and women are reversed,not only regardingdomestic and
social dutiesbut also regardingbehaviourand mentality.
Even ifmanyof us,
in such a situation,mightperhapsat firstshrinkfromadmitting
thepossibilitythatit is entirelya caprice of fatethatthe people concernedhave their
specificcultureand not ours,and we nottheirsinsteadof our own,it is clear
that even the slightestsuspicion in this respect implies a betrayalof the
nationalcomplacencyinherentin any humanculturerestingin itself.
6. This impressiveworkalso explainsthe intimateconnectionswithG6idel's
proofof theincompletenessof formalsystemsand withSkolem'sconstructionof
nonstandard models of arithmetic, as well as with Bataille's general economy.
For furtherdiscussion of Bataille's physics see Hochroth 1995.
Numerous other examples could be adduced. For instance, Barbara Johnson
(1989, 12) makes no specific reference to quantum physics; but her description
of deconstruction is an eerily exact summary of the complementarity principle:
232
Alan D. Sokal
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"Instead of a simple either/or
structure,deconstructionattemptsto elaboratea
discoursethatsays neither
nor'both/and'nor even 'neither/nor'
whileat
'either/or,
the same time not totallyabandoningthese logics either."See also McCarthy
1992 fora thought-provoking
analysisthatraises disturbingquestionsabout the
"complicity"between(nonrelativistic)
quantumphysicsand deconstruction.
7. Permitme in thisregarda personalrecollection:Fifteenyearsago, whenI
was a graduatestudent,myresearchin relativistic
quantumfieldtheoryled me to
an approachthatI called "de[con]structivequantumfieldtheory"(Sokal 1982).
Of course, at thattimeI was completelyignorantof JacquesDerrida's workon
in philosophyand literarytheory.In retrospect,
deconstruction
however,thereis
a strikingaffinity:
my workcan be read as an explorationof how the orthodox
discourse (e.g., Itzyksonand Zuber 1980) on scalar quantum fieldtheoryin
four-dimensional
thespace-time(in technicalterms,"renormalizedperturbation
and therebyto
ory" forthe(p44 theory)can be seen to assertits own unreliability
undermineits own affirmations.
Since then,myworkhas shiftedto otherquestions,mostlyconnectedwithphase transitions;but subtlehomologiesbetween
the two fieldscan be discerned,notablythe themeof discontinuity
(see n. 42).
For furtherexamples of deconstructionin quantumfieldtheory,see Merz and
KnorrCetina 1994.
8. Bell 1987, especiallychaps. 10 and 16. See also Maudlin 1994 (chap. 1)
fora clear account presupposingno specialized knowledgebeyond high-school
algebra.
9. Greenbergeret al. 1989, 1990; Mermin 1990, 1993.
10. Aronowitz(1988b, 331) has made a provocativeobservationconcerning
nonlinearcausalityin quantummechanicsand itsrelationto the social constructionof time:
Linear causality assumes that the relation of cause and effectcan be
expressedas a functionof temporalsuccession. Owing to recentdevelopmentsin quantummechanics,we can postulatethatit is possibleto knowthe
effectsof absentcauses; thatis, speakingmetaphorically,
effectsmay anticipate causes so thatour perceptionof themmay precedethe physicaloccurrenceof a "cause." The hypothesisthatchallengesour conventionalconception of linear time and causalityand that asserts the possibilityof time's
reversalalso raisesthequestionofthedegreeto whichtheconceptof "time's
arrow"is inherentin all scientific
are successful,
theory.If theseexperiments
the conclusions about the way time as "clock-time"has been constituted
historicallywill be open to question. We will have "proved" by means of
experimentwhathas longbeen suspectedby philosophers,literaryand social
critics:thattime is, in part, a conventionalconstruction,its segmentation
into hours and minutesa productof the need forindustrialdiscipline,for
rationalorganizationof social labor in the earlybourgeoisepoch.
The theoreticalanalysesof Greenbergeret al. (1989, 1990) and Mermin (1990,
1993) providea strikingexample of thisphenomenon;see Maudlin 1994 fora
detailedanalysisof the implicationsforconceptsof causalityand temporality.
An
experimentaltest,extendingtheworkof Aspect et al. (1982), willlikelybe forthcoming within the next few years.
11. The intimate relations between quantum mechanics and the mind-body
problem are discussed in Goldstein 1983, chaps. 7 and 8.
12. Among the voluminous literature,Capra 1975 can be recommended for
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233
its scientificaccuracy and its accessibilityto nonspecialists.In addition,Sheldrake 1981, while occasionallyspeculative,is in generalsound. For a sympatheticbut criticalanalysisof New Age theories,see Ross 1991, chap. 1. For a critique of Capra's workfroma Third Worldperspective,see Alvares1992, chap. 6.
13. Newtonianatomismtreatsparticlesas hyperseparated
in space and time,
backgroundingtheirinterconnectedness
(Plumwood 1993a, 125); indeed, "the
is thatof kineticenergyonly'force'allowedwithinthemechanisticframework
theenergyof motionby contact-all otherpurportedforces,includingactionat a
distance,beingregardedas occult" (Mathews 1991, 17). For criticalanalysesof
the Newtonian mechanisticworldview,see Weil 1968, esp. chap. 1; Merchant
1980; Berman 1981; Keller 1985, chaps. 2 and 3; Mathews 1991, chap. 1; and
Plumwood 1993a, chap. 5.
14. Accordingto the traditionaltextbookaccount,special relativity
is concerned withthe coordinatetransformations
relatingtwoframesof referencein
uniformrelativemotion. But this is a misleadingoversimplification,
as Bruno
Latour has pointedout:
How can one decide whetheran observationmade in a train about the
behaviourof a fallingstone can be made to coincide withthe observation
If thereare onlyone,
made of the same fallingstonefromtheembankment?
or even two,framesof reference,no solutioncan be foundsince the man in
thetrainclaimshe observesa straightline and the man on the embankment
a parabola. . . . Einstein'ssolutionis to consider threeactors: one in the
train,one on the embankmentand a thirdone, the author[enunciator]or
who triesto superimposethe coded observations
one of its representants,
sentback bythetwoothers. . . Withouttheenunciator'sposition(hiddenin
Einstein'saccount), and withoutthe notionof centresof calculation,Einstein's own technical argumentis ununderstandable.(1988, 10-11, 35;
emphasisin original)
In the end, as Latour wittily
but accuratelyobserves,special relativity
boils down
to the propositionthat "more framesof referencewith less privilegecan be
accessed, reduced,accumulatedand combined,observerscan be delegatedto a
fewmoreplaces in theinfinitely
small (eleclarge (the cosmos) and theinfinitely
trons),and the readingstheysend willbe understandable.His [Einstein's]book
could well be titled:'New InstructionsforBringingBack Long-Distance ScientificTravellers'"(22-23). Latour's criticalanalysisof Einstein'slogic providesan
to special relativity
fornon-scientists.
eminentlyaccessibleintroduction
15. It goes withoutsayingthatspecial relativity
proposes new conceptsnot
as Virilio
only of space and time but also of mechanics. In special relativity,
(1991, 136) has noted, "the dromosphericspace, space-speed, is physically
describedby whatis called the 'logisticequation,'theresultof theproductofthe
mass displacedby the speed of itsdisplacement,MxV." This radicalalterationof
in thequantum
the Newtonianformulahas profoundconsequences,particularly
discussion.
theory;see Lorentzet al. 1952 and Weinberg1992 forfurther
16. StevenBest (1991, 225) has put his fingeron the crux of the difficulty,
whichis that"unlikethe linearequationsused in Newtonianand even quantum
mechanics, nonlinear equations do [not] have the simple additive property
wherebychains of solutions can be constructedout of simple, independent
parts." For this reason, the strategies of atomization, reductionism, and contextstripping that underlie the Newtonian scientificmethodology simply do not work
in general relativity.
234
Alan D. Sokal
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17. G6del 1949. For a summaryof recentworkin this area, see 't Hooft
1993.
18. These new notionsof space, time,and causalityare inpartforeshadowed
Thus, AlexanderArgyros(1991, 137) has notedthat
alreadyin special relativity.
"in a universedominatedby photons,gravitons,and neutrinos,that is, in the
veryearlyuniverse,the theoryof special relativity
suggeststhatany distinction
betweenbeforeand afteris impossible.For a particletravelingat the speed of
light,or one traversinga distance thatis in the orderof the Planck length,all
events are simultaneous."However, I cannot agree withArgyros'sconclusion
that Derridean deconstructionis thereforeinapplicableto the hermeneuticsof
early-universe
cosmology:Argyros'sargumentto thiseffectis based on an impermissiblytotalizinguse of special relativity
(in technicalterms,"light-conecooris inescapable. (For a similarbut
dinates") in a contextwheregeneralrelativity
less innocenterror,see n. 20.)
Jean-FranoisLyotard(1989, 5-6) has also pointedout thatnot onlygeneral
but also modernelementary-particle
relativity,
physics,imposes new notionsof
time:
In contemporaryphysicsand astrophysics. . . a particlehas a sort of eleThis is whycontempomentarymemoryand consequentlya temporalfilter.
raryphysiciststendto thinkthattimeemanatesfrommatteritself,and thatit
is not an entityoutsideor insidethe universewhose functionit would be to
timesinto universalhistory.It is onlyin certainregions
gatherall different
thatsuch-only partial-synthesescould be detected.There would on this
wherecomplexityis increasing.
view be areas of determinism
Michel Serres (1992, 89-91) has noted thatchaos theory(Gleick
Furthermore,
1987) and percolationtheory(Stauffer1985) have contestedthe traditionallinear conceptof time:
Time does not alwaysflowalong a line. . . or a plane, but along an extraordinarilycomplex manifold,as if it showed stoppingpoints,ruptures,sinks
[puits], funnels of overwhelmingacceleration [chemindesd'accelration
rips,lacunae, all sown randomly.... Time flowsin a turbulent
foudroyante],
and chaoticmanner;it percolates.(Translationmine.Note thatin thetheory
of dynamicalsystems,"puits" is a technicaltermmeaning"sink," i.e. the
oppositeof "source.")
These multipleinsightsintothenatureof time,providedby different
branchesof
of the complementarity
illustration
physics,are a further
principle.
General relativitycan arguablybe read as corroboratingthe Nietzschean
deconstructionof causality(see, e.g., Culler 1982, 86-88), althoughsome relativistsfindthis interpretation
problematic.In quantummechanics,by contrast,
this phenomenonis ratherfirmlyestablished(see n. 10). General relativityis
also, of course, the startingpoint for contemporaryastrophysicsand physical
cosmology.See Mathews 1991 (59-90, 109-16, 142-63) fora detailedanalysis
of the connections between general relativity(and its generalizationscalled
"geometrodynamics")and an ecologicalworldview.For an astrophysicist's
speculationsalong similarlines,see Primackand Abrams 1995.
19. Discussion of Derrida (1970, 265-66).
20. Right-wingcriticsGross and Levitt(1994, 79) have ridiculedthisstateit as an assertionabout specialrelativity,
in which
ment,willfully
misinterpreting
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235
the Einsteinianconstantc (the speed of lightin vacuum) is of course constant.
No readereven minimallyconversantwithmodernphysics-except an ideologically biased one-could fail to understandDerrida's unequivocal referenceto
generalrelativity.
21. Luce Irigaray(1987, 77-78) has pointed out thatthe contradictions
betweenquantumtheoryand fieldtheoryare in factthe culminationof a historical processthatbegan withNewtonianmechanics:
The Newtonianbreakhas usheredscientificenterpriseinto a worldwhere
sense perceptionis worthlittle,a worldwhichcan lead to theannihilationof
theverystakesof physics'object:thematter(whateverthepredicates)of the
universeand of the bodies thatconstituteit. In thisveryscience,moreover
[d'ailleurs],cleavages exist: quantum theory/field
theory,mechanics of
of fluids,forexample.But theimperceptibility
of thematter
solids/dynamics
under studyoftenbringswithit the paradoxicalprivilegeof solidityin discoveriesand a delay,even an abandoningof the analysisof theinfinity
[l'infini] of the fieldsof force.
I have here correctedthe translationof d'ailleurs,whichmeans "moreover"or
"besides" (not "however").
22. Sheldrake1981, 1991; Briggsand Peat 1984, chap. 4; Granero-Porati
and Porati 1984; Kazarinoff1985; Schiffmann1989; Psarev 1990; Brooks and
of
Castor 1990; Heinonen et al. 1992; Rensing 1993. For an in-depthtreatment
the mathematicalbackgroundto this theory,see Thom 1975, 1990; and fora
briefbut insightful
of thisand related
analysisof thephilosophicalunderpinnings
approaches,see Ross 1991 (40-42, 253 n. 20).
23. Some early workersthoughtthat the morphogeneticfield mightbe
relatedto the electromagnetic
field,but it is now understoodthatthisis merelya
suggestiveanalogy:see Sheldrake1981 (77, 90) fora clear exposition.Note also
point (b) below.
24. For anotherexampleof the "turf"effect,see Chomsky1979 (6-7).
I should mention
25. To be fairto the high-energy-physics
establishment,
thatthereis also an honestintellectualreason fortheiroppositionto thistheory:
inasmuchas it posits a subquantuminteractionlinkingpatternsthroughoutthe
a "nonlocalfieldtheory."Now, thehisuniverse,it is, in physicists'terminology,
toryof classicaltheoreticalphysicssincetheearly1800s, fromMaxwell'selectrocan be read in a verydeep sense as a
dynamicsto Einstein'sgeneralrelativity,
in
trendaway fromaction-at-a-distance
theoriesand towardlocalfieldtheories:
technicalterms,theoriesexpressiblebypartialdifferential
equations(Einsteinand
Infeld 1961; Hayles 1984). So a nonlocal fieldtheorydefinitely
goes againstthe
argued,the
grain.On theotherhand,as Bell (1987) and othershave convincingly
as expressedin
keypropertyof quantummechanicsis preciselyits non-locality,
Bell's theoremand itsgeneralizations
(see nn. 8 and 9). Therefore,a nonlocalfield
is not onlynaturalbut in
theory,althoughjarringto physicists'classicalintuition,
in thequantumcontext.This is why
factpreferred
(and possiblyevenmandatory?)
is a local fieldtheory,whilequantumgravity(whether
classical generalrelativity
nonlocal.
field)is inherently
string,weave,or morphogenetic
26. Differential topology is the branch of mathematics concerned with those
properties of surfaces (and higher-dimensional manifolds) that are unaffected
by smooth deformations. The properties it studies are thereforeprimarily qualitative rather than quantitative, and its methods are holistic rather than Cartesian.
236
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27. The alertreaderwill noticethatanomaliesin "normalscience" are the
usual harbingerof a futureparadigmshift(Kuhn 1970).
28. The floweringof the theoryof phase transitionsin the 1970s probably
reflectsan increasedemphasison discontinuity
and rupturein the widerculture
(see n. 42).
29. For an in-depthanalysis of Lacan's use of ideas frommathematical
topology,see Juranville1984 (chap. 7); Granon-Lafont1985, 1990; Vappereau
1985; and Nasio 1987, 1992; a briefsummaryis given by Leupin 1991. See
Hayles 1990 (80) foran intriguingconnectionbetweenLacanian topologyand
she does not pursue it. See also Zizek 1991 (38-39,
chaos theory;unfortunately
45-47) forsome further
homologiesbetweenLacanian theoryand contemporary
number
physics.Lacan also made extensiveuse of concepts fromset-theoretic
theory:see, forexample,Miller 1977/78and Ragland-Sullivan1990.
In bourgeois social psychology,topological ideas had been employed by
Kurt Lewin as earlyas the 1930s, but thisworkfounderedfortwo reasons:first,
because of its individualistideological preconceptions;and second, because it
reliedon old-fashionedpoint-settopologyratherthanmoderndifferential
topology and catastrophetheory.Regardingthe second point,see Back 1992.
30. "Il suffit,
a cettefin,reconnaltreque Lacan conf&reenfina la pensee de
Freud,les conceptsscientifiques
qu'elle exige."This famousessayon "Freud and
Lacan" was firstpublishedin 1964, beforeLacan's workhad reachedits highest
level of mathematicalrigor.It was reprintedin Englishtranslationin Althusser
1969.
31. This articlehas become quite influential
in filmtheory:see, forexample,
Jameson1982 (27-28) and thereferencescitedthere.As Strathausen(1994, 69)
indicates,Miller's articleis tough going for the reader not well versed in the
mathematicsof set theory.But it is well worththe effort.For a gentleintroductionto set theory,see Bourbaki1970.
32. Homologytheoryis one of the two main branchesof the mathematical
fieldcalled algebraictopology.
For an excellentintroductionto homologytheory,
see Munkres 1984; or fora more popular account,see Eilenbergand Steenrod
1952. A fullyrelativistic
homologytheoryis discussed,forexample,in Eilenberg
and Moore 1965. For a dialecticalapproach to homologytheoryand its dual,
cohomologytheory,see Massey 1978. For a cyberneticapproach to homology,
see Saludes i Closa 1984.
33. For the relationof homologyto cuts,see Hirsch 1976 (205-8); and for
an applicationto collectivemovementsin quantumfieldtheory,see Caracciolo et
al. 1993 (especiallyAppendixA. 1).
34. It is, however,worthnotingthatthe space RP3 is homeomorphicto the
of conventionalthree-dimensional
Euclidgroup SO(3) of rotationalsymmetries
ean space. Thus, some aspects of three-dimensional
Euclidicityare preserved
(albeit in modifiedform) in the postmodernphysics,just as some aspects of
Newtonianmechanicswerepreservedin modifiedformin Einsteinianphysics.
35. See also Johnson1977 (481-82) foran analysisof Derrida's and Lacan's
effortstowardtranscendingthe Euclidean spatiallogic.
Alongrelatedlines,Eve Seguin (1994, 61) has notedthat"logic saysnothing
about the world and attributes to the world properties that are but constructs of
theoretical thought. This explains why physics since Einstein has relied on alternative logics, such as trivalent logic which rejects the principle of the excluded
middle." A pioneering (and unjustly forgotten) work in this direction, likewise
the Boundaries
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237
inspiredby quantum mechanics,is Lupasco 1951. See also Plumwood 1993b
feministperspectiveon nonclassicallogics.For a crit(453-59) fora specifically
ical analysisof one nonclassicallogic ("boundarylogic") and its relationto the
ideologyof cyberspace,see Markley1994.
36. This essay originallyappeared in French in Irigaray1982. Irigaray's
can also be renderedas "theoryof sets,"and bordsis
phrase theoriedes ensembles
usually translatedin the mathematicalcontext as "boundaries." Her phrase
ensembles
flousmay referto the new mathematicalfieldof "fuzzy sets" (Kaufmann 1973; Kosko 1993).
37. In the historyof mathematicstherehas been a long-standingdialectic
betweenthedevelopmentof its "pure" and "applied" branches(Struik1987). Of
course,the "applications"traditionally
privilegedin thiscontexthave been those
profitableto capitalistsor usefulto theirmilitaryforces:for example,number
(Loxton
theoryhas been developed largelyforits applicationsin cryptography
1990). See also Hardy 1967 (120-21, 131-'32).
of all boundaryconditionsis also suggestedby
38. The equal representation
Chew's bootstraptheoryof "subatomicdemocracy":see Chew 1977 foran introduction,and see Morris 1988 and Markley1992 forphilosophicalanalysis.
39. The main limitationof the Madsen-Madsen analysisis thatit is essentiallyapolitical;and it hardlyneeds to be pointedout thatdisputesoverwhatis
truecan have a profoundeffecton, and are in turnprofoundlyaffectedby,disputes overpoliticalprojects.Thus Markley(1992, 270) makes a pointsimilarto
situatesit in its politicalcontext:
thatof Madsen-Madsen,but rightly
Radical critiquesof science thatseek to escape the constraintsof deterministicdialecticsmustalso giveovernarrowlyconceiveddebatesabout realism
and truthto investigatewhat kind of realities-politicalrealities-mightbe
engenderedby a dialogical bootstrapping.Within a dialogicallyagitated
debates about realitybecome, in practicalterms,irrelevant.
environment,
is a historicalconstruct.
"Reality,"finally,
See Markley1992 (266-72) and Hobsbawm 1993 (63-64) forfurther
discussion
of thepoliticalimplications.
40. Aronowitz (1988b, 292-93) makes a slightlydifferent,but equally
cogent,criticismof quantumchromodynamics(the currently
hegemonictheory
bound statesof quarksand gluons): drawnucleonsas permanently
representing
ing on theworkof Pickering(1984), he notesthat
in his [Pickering's]account,quarksare the name assignedto (absent) phenomena thatcohere withparticleratherthan fieldtheories,which,in each
case, offerdifferent,
althoughequally plausible,explanationsforthe same
(inferred)observation.That the majorityof the scientificcommunitychose
forthetraditionrather
one overanotheris a functionof scientists'preference
thanthevalidityof explanation.
However,Pickeringdoes not reach back farenoughintothe historyof
physicsto findthe basis of the researchtraditionfromwhich the quark
explanationemanates.It maynotbe foundinsidethetraditionbutin theideology of science, in the differencesbehind field versus particle theories, simple versus complex explanations, the bias toward certaintyrather than indeterminateness.
238
Alan D. Sokal
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Alongverysimilarlines,Markley(1992, 269) observesthatphysicists'preference
for quantum chromodynamicsover Chew's bootstrap theoryof "subatomic
democracy"(Chew 1977) is a resultof ideologyratherthandata:
It is not surprising,in thisregard,thatbootstraptheoryhas fallenintorelative disfavoramong physicistsseekinga GUT (Grand UnifiedTheory) or
TOE (Theory of Everything)to explainthe structureof theuniverse.Comare productsof theprivileging
prehensivetheoriesthatexplain"everything"
of coherenceand orderin westernscience. The choice betweenbootstrap
thatconfrontsphysicistsdoes nothave to
theoryand theoriesof everything
offeredby theseaccountsof availabledata
do primarilywiththetruth-value
but with the narrativestructures-indeterminateor deterministic-into
whichthesedata are placed and by whichtheyare interpreted.
the vast majorityof physicistsare not yetaware of these incisive
Unfortunately,
held dogmas.
critiquesof one of theirmostfervently
For another critique of the hidden ideology of contemporaryparticle
physics,see Kroker et al. 1989 (158-62, 204-7). The styleof this critiqueis
rathertoo Baudrillardianformy staid taste,but the contentis (except fora few
minorinaccuracies)righton target.
41. For an amusingexample of how thismodestdemand has drivenrightStalinist"is the chosen epiwing scientistsinto fitsof apoplexy ("frighteningly
thet),see Gross and Levitt 1994 (91).
42. Whilechaos theoryhas been deeplystudiedby culturalanalysts-see, for
example,Hayles 1990, 1991; Argyros1991; Best 1991; Young 1991, 1992; Assad
1993 among many others-the theoryof phase transitionshas passed largely
unremarked.(One exceptionis the discussionof the renormalization
group in
and the emergence
Hayles 1990 [154-58].) This is a pity,because discontinuity
to
of multiplescales are centralfeaturesin thistheory;and itwould be interesting
is conknowhow the developmentof thesethemesin the 1970s and afterwards
nectedto trendsin the widerculture.I therefore
suggestthistheoryas a fruitful
fieldfor futureresearchby culturalanalysts.Some theoremson discontinuity
whichmaybe relevantto thisanalysiscan be foundin Van Enteret al. 1993.
43. See, however,Schor 1989 for a critiqueof Irigaray'sundue deference
towardconventional(male) science,particularly
physics.
RobertMarkley(1991,
44. ConcerningtheCartesian/Baconian
metaphysics,
6) has observedthat
Narrativesof scientific
progressdependupon imposingbinaryoppositionstrue/false,
right/wrong-ontheoreticaland experimentalknowledge,priviovermetaphor,monologicalauthority
legingmeaningovernoise,metonymy
overdialogicalcontention. . . These attemptsto fixnatureare ideologically
coercive as well as descriptively
limited.They focus attentiononly on the
smallrangeof phenomena-say, lineardynamics-whichseem to offereasy,
oftenidealized ways of modelingand interpreting
humankind'srelationship
to theuniverse.
While this observation is informed primarily by chaos theory-and secondarily
by nonrelativistic quantum mechanics-it in fact summarizes beautifully the radical challenge to modernist metaphysics posed by quantum gravity.
45. One caveat: I have strong reservations about Capra's use here of the
the Boundaries
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239
wordcyclical,whichifinterpreted
too literallycould promotea politicallyregressive quietism.For furtheranalyses of these issues, see Bohm 1980; Merchant
1980, 1992; Berman 1981; Prigogineand Stengers1984; Bowen 1985; Griffin
1988; Kitchener1988; Callicott1989 (chaps. 6 and 9); Shiva 1990; Best 1991;
Haraway 1991, 1994; Mathews 1991; Morin 1992; Santos 1992; and Wright
1992.
46. A minor quibble: it is not clear to me that complex numbertheory,
whichis a new and stillquite speculativebranchof mathematicalphysics,ought
to be accordedthe same epistemologicalstatusas thethreefirmly
establishedsciences citedby Markley.
See Wallerstein1993 (17-20) foran incisiveand closelyanalogous account
of how the postmodernphysicsis beginningto borrowideas fromthe historical
social sciences; and see Santos 1989 and 1992 fora moredetaileddevelopment.
47. At this point,the traditionalscientist'sresponseis thatworknot conformingto the evidentiarystandardsof conventionalscience is fundamentally
not worthyof credence.But this
thatis, logicallyflawedand therefore
irrational,
refutationis insufficient:
for,as Porush (1993) has lucidlyobserved,modern
admitteda powerful"intrusionof the
mathematicsand physicshave themselves
irrational"in quantummechanicsand G6del's theorem-although,understandcenturiesago, modernistscientistshave
ably,like the Pythagoreanstwenty-four
attemptedto exorcisethisunwantedirrationalelementas besttheycould. Porush
makes a powerfulplea fora "post-rationalepistemology"thatwould retainthe
bestof conventionalWesternsciencewhilevalidatingalternative
waysofknowing.
Note also thatJacques Lacan, froma quite different
startingpoint, came
in modern
long ago to a similarappreciationof theinevitablerole of irrationality
mathematics:
If you'll permitme to use one of those formulaswhich come to me as I
writemynotes,humanlifecould be definedas a calculus in whichzero was
irrational.This formulais just an image,a mathematicalmetaphor.When I
not to some unfathomable
emotionalstatebut
say "irrational,"I'm referring
preciselyto whatis called an imaginarynumber.The square root of minus
one doesn'tcorrespondto anythingthatis subjectto our intuition,anything
real-in themathematicalsense of theterm-and yet,it mustbe conserved,
along withits fullfunction.(Lacan 1977, 28-29; seminaroriginallygivenin
1959)
in modernmathematics,
For further
reflections
on irrationality
see Solomon 1988
(76) and Bloor 1991 (122-25).
48. Along parallel lines, Donna Haraway (1991, 191-92) has argued eloquentlyfora democraticscience comprising"partial,locatable,criticalknowledges sustainingthepossibilityofwebs of connectionscalled solidarityin politics
in epistemology"and foundedon "a doctrineand pracand sharedconversations
tice of objectivitythat privilegescontestation,deconstruction,passionate conof systemsof knowlwebbedconnections,and hope fortransformation
struction,
edge and ways of seeing." These ideas are furtherdeveloped in Haraway 1994
and Doyle 1994.
49. For an example in the contextof the Sandinistarevolution,see Sokal
1987.
50. For feminist critiques, see Merchant 1980; Easlea 1981; Keller 1985,
1992; Harding 1986, 1991; Haraway 1989, 1991; Plumwood 1993a. See Wylie et
al. 1990 for an extensive bibliography. The feminist critique of science has, not
240
Alan D. Sokal
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been the objectof a bitterright-wing
For a sampling,
counterattack.
surprisingly,
see Levin 1988; Haack 1992, 1993; Sommers 1994; Gross and Levitt 1994
(chap. 5); and Patai and Koertge 1994.
For queer critiques,see Trebilcot1988 and Hamill 1994.
For multiculturalist
critiques,see Ezeabasili 1977; Van Sertima 1983; Frye
1987; Sardar 1988; Adams 1990; Nandy 1990; Alvares 1992; Harding 1994. As
withthe feministcritique,the multiculturalist
perspectivehas been ridiculedby
critics,witha condescensionthatin some cases borderson racism.
right-wing
See, for example, Ortiz de Montellano 1991; Martel 1991/92; Hughes 1993
(chap. 2); and Gross and Levitt1994 (203-14).
For ecological critiques,see Merchant1980, 1992; Berman 1981; Callicott
1989 (chaps. 6 and 9); Mathews 1991; Wright1992; Plumwood 1993a; Ross
1994.
51. See Wojciehowski1991 for a deconstructionof Galileo's rhetoric,in
method can lead to direct
particularhis claim that the mathematico-scientific
and reliableknowledgeof "reality."
A veryrecentbut importantcontribution
to the philosophyof mathematics
can be found in the workof Deleuze and Guattari(1994, chap. 5). Here they
introducethephilosophically
fruitful
notionof a "functive"[Fr.fonctif],
whichis
neithera function[Fr. fonction]nor a functional[Fr. fonctionnelle]
but rathera
more basic conceptualentity:"The object of science is not conceptsbut rather
functionsthatare presentedas propositionsin discursivesystems.The elements
of functionsare calledfunctives"(117). This apparentlysimpleidea has surprisconsequences;its elucidationrequiresa detourinto
inglysubtleand far-reaching
chaos theory(see also Rosenberg1993 and Canning 1994):
The firstdifference
betweenscience and philosophyis theirrespectiveattitudes towardchaos. Chaos is definednot so much by its disorderas by the
infinitespeed withwhicheveryformtakingshape in it vanishes.It is a void
thatis not a nothingnessbut a virtual,containingall possible particlesand
drawingout all possible forms,whichspringup onlyto disappear immediately,withoutconsistencyor reference,withoutconsequence. Chaos is an
infinitespeed of birthand disappearance.(117-18)
But science,unlikephilosophy,cannotcope withinfinitespeeds:
It is by slowingdown thatmatter,as well as the scientificthoughtable to
penetrateit [sic] with propositions,is actualized. A functionis a Slowmotion. Of course, science constantlyadvances accelerations,not only in
catalysisbut in particleacceleratorsand expansionsthatmovegalaxiesapart.
However,the primordialslowingdown is not forthese phenomenaa zeroinstantwithwhichtheybreakbut rathera conditioncoextensivewiththeir
whole development.To slow down is to set a limitin chaos to which all
speeds are subject,so thattheyforma variabledeterminedas abscissa,at the
same time as the limit forms a universal constant that cannot be gone
beyond (forexample,a maximumdegreeof contraction).Thefirstfunctives
are therefore
thelimitand thevariable,and referenceis a relationshipbetween
values of the variable or, more profoundly,the relationship of the variable, as
abscissa of speeds, with the limit. (118-19; emphasis mine)
A rather intricate furtheranalysis (too lengthy to quote here) leads to a conclusion of profound methodological importance for those sciences based on mathe-
Transgressingthe Boundaries
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241
maticalmodeling:"The respectiveindependenceof variablesappears in mathematicswhen one of themis at a higherpowerthan the first.That is whyHegel
shows that variabilityin the functionis not confinedto values that can be
changed (2/3 and 4/6) or are leftundetermined(a = 2b) but requiresone of the
variablesto be at a higherpower(y2/x= P)" (122). (Note thattheEnglishtransF an amusingerrorthatthoroughly
lationinadvertently
writesy2/x P,
manglesthe
logic of the argument.)
Surprisinglyfora technicalphilosophicalwork,thisbook (Qu'est-ceque la
was a best-sellerin France in 1991. It has recentlyappearedin Engphilosophie?)
lish translation,but is, alas, unlikelyto compete successfullywith Rush Limbaugh and Howard Sternforthebest-sellerlistsin thiscountry.
52. For a viciousright-wing
attackon thisproposition,see Gross and Levitt
1994 (52-54). See Ginzberg 1989; Cope-Kasten 1989; Nye 1990; and Plumwood 1993b forlucid feministcritiquesof conventional(masculinist)mathematical logic,in particularthe modusponensand thesyllogism.Concerningthe modus
ponens,see also Woolgar 1988 (45-46) and Bloor 1991 (182); and concerning
the syllogism,see also Woolgar 1988 (47-48) and Bloor 1991 (131-35). For an
mathematicalconceptionsof infinity,
see
analysisof the social imagesunderlying
ofthe social contextuality
of mathematHarding 1986 (50). For a demonstration
ical statements,
see Woolgar1988 (43) and Bloor 1991 (107-30).
53. See Merchant1980 fora detailedanalysisof the themesof controland
dominationin Westernmathematicsand science.
Let me mentionin passingtwo otherexamplesof sexismand militarismin
mathematicsthatto my knowledgehave not been noticedpreviously.The first
concernsthe theoryof branchingprocesses,which arose in VictorianEngland
fromthe "problemof the extinctionof families"and whichnow playsa keyrole
interalia in the analysisof nuclearchain reactions(Harris 1963). In the seminal
(and thissexistword is apt) paper on the subject,Francis Galton and the ReverendH. W. Watson(1874) wrote:
The decay of the familiesof men who occupied conspicuous positionsin
past timeshas been a subjectof frequentresearch,and has givenriseto various conjectures . . . The instances are very numerous in which surnames
that were once common have since become scarce or have whollydisappeared. The tendencyis universal,and, in explanationof it,the conclusion
has hastilybeen drawnthata risein physicalcomfortand intellectualcapacityis necessarilyaccompaniedby a diminutionin 'fertility'..
Let p0 p1,P2, . . . be the respective probabilities that a man has 0, 1, 2,
.
let each son have the same probabilityof sons of his own, and so
.sons,
on. What is the probabilitythatthe male line is extinctafterr generations,
and moregenerallywhatis the probabilityforany givennumberof descendantsin the male line in any givengeneration?
One cannotfailto be charmedbythequaintimplicationthathumanmalesreprothe classism,social-Darwinism,and sexismin this
duce asexually;nevertheless,
passage are obvious.
The second example is Laurent Schwartz's 1973 book Radon Measures.
While technically quite interesting,this work is imbued, as its title makes plain,
with the pro-nuclear-energy worldview that has been characteristic of French
science since the early 1960s. Sadly, the French left-especially but by no means
solely the PCF-has
traditionally been as enthusiastic for nuclear energy as the
et
al. 1980).
Touraine
right (see
54. Just as liberal feministsare frequentlycontent with a minimal agenda of
242
Alan D. Sokal
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legal and social equalityforwomen and "pro-choice,"so liberal(and even some
socialist) mathematiciansare oftencontentto workwithinthe hegemonicZermelo-Fraenkelframework(which, reflectingits nineteenth-century
liberal origins,alreadyincorporatestheaxiom of equality)supplementedonlybytheaxiom
is grosslyinsufficient
fora liberatory
of choice. But thisframework
mathematics,
as was provenlong ago by Cohen 1966.
55. Fuzzy systemstheoryhas been heavilydeveloped by transnationalcorporations-firstin Japanand laterelsewhere-to solve practicalproblemsof efficiencyin labor-displacingautomation.
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