• "ConcerningtheUltimateFoundationoftheDifferentiationofRegionsofSpace"(1768).
12.KantandHandedness
ImmanuelKant(1724-1804)
[Anincongruentcounterpartis]...anobjectwhichiscompletelylikeandsimilartoanother,althoughitcannotbeincludedexactlywithinthesamelimits."
Claim:Absolutespaceisnecessarytoexplaintheexistenceofincongruentcounterparts.
• Anincongruentcounterpartisacertaintype ofmirrorimage.
1
1. IncongruentCounterparts2. Kant'sArgumentforAbsoluteSpace
1.IncongruentCounterparts
• Maps(1)and(2)reproducethesamerelationsbetweenobjects.• Arelationistmustsaytheyarethesame.• Anabsolutistcansaytheyaredifferent;namely,theydifferintheirlocationswithrespecttoabsolutespace.
Example1
(1) (2)
2
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
3
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
3
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
3
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
3
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
3
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
3
(2) O' isanincongruentcounterpart ofO ifitcannot bemadetocoincidewithObyrigidmotions.
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
4
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
(2) O' isanincongruentcounterpart ofO ifitcannot bemadetocoincidewithObyrigidmotions.
4
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
(2) O' isanincongruentcounterpart ofO ifitcannot bemadetocoincidewithObyrigidmotions.
4
ImportantFact:Whetherornotamirrorimageisanincongruentcounterpartdependsonthepropertiesofthespaceitislocatedin.
TwotypesofmirrorimageLetO beanobjectandletO' beitsmirrorimage.(1) O' isacongruentcounterpart ofO ifitcanbemadetocoincidewithO byrigid
motions.
(2) O' isanincongruentcounterpart ofO ifitcannot bemadetocoincidewithObyrigidmotions.
5
• Ifthespaceis3-dimensional,thenFanditsmirrorimagearecongruentcounterparts.
6
• Ifthespaceis3-dimensional,thenFanditsmirrorimagearecongruentcounterparts.
6
• Ifthespaceis3-dimensional,thenFanditsmirrorimagearecongruentcounterparts.
6
• Ifthespaceis3-dimensional,thenFanditsmirrorimagearecongruentcounterparts.
6
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.
7
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
y••
x
x'••
y'
• Ifthespaceis2-dimensionalandnon-orientable,thenFanditsmirrorimagearecongruentcounterparts.
Example:AMöbiusstrip.• Obtainedbyidentifyingedgepointsxwithx',andywithy' ona2-dimstrip.• Result:Aglobal"twist"thatallowsthemirrorimageofFtoberigidlytransportedaroundtheentirespacebackontoF.
8
Modifieddefinition:
Anobjectisanincongruentcounterpart ofanotheriftheycannotbemadetooccupythesameplacebyrigidmotionsinalocal(closelysurrounding)regionofspace.
• Anobjectissaidtopossesshandedness (chirality)justwhenitanditsmirrorimageareincongruentcounterparts.
• Anobjectissaidtolackhandedness (chirality)justwhenitanditsmirrorimagearecongruentcounterparts.
sphericalcow 9
2.Kant'sArgumentforAbsoluteSpace
"Letitbeimaginedthatthefirstcreatedthingwereahumanhand,thenitmustnecessarilybeeitherarighthandoralefthand."
• But:Arelationistcannotdeterminethehandednessofanobjectintheabsenceofotherobjects.
• So:Relationalismisnotadequate.
• Leftandrighthandsagreeonallrelationalproperties.• Absolutist:Theydisagreeontheirlocationswithrespecttoabsolutespace.
10
LetterExampleAgain
• DoFanditsmirrorimagehavethesamerelationalproperties?• Dependsonhowmanypropertiesoneiswillingtoconsiderasrelational.
THEQUICKBROWNOXJUMPEDOVERTHELAZYDOG.
THEQUICKBROWNFOXJUMPEDOVERTHELAZYDOG.
• Fanditsmirrorimagedifferintheirrelationalpropertiestotheotherlettersinthesentence.- ThereisnowaytomakethemirrorimageofFfitintothesentenceinthesamewaythatFdoes.
- Similarly,thereisnowaytofitarighthandintoaleft-handed(FreddyKrueger)glove(andviceversa).
11
• But:Howcanarelationistdeterminethehandednessofanobjectwhentherearenootherreferenceobjectstodefinedistinguishingrelationalproperties?
• Moreover:Whatifsuchreferenceobjectsthemselveshavebeenreflected?
• SinceFanditsmirrorimageshareallrelationalpropertiesinthesesentences,arelationistwillnotbeabletodistinguishthem.
• Absolutistintuition:Aren'tFanditsmirrorimagedistinct,independentoftheirrelationstootherobjects?
12
Relationist'sReflectionArgument
• Suppose:Absolutespaceexists.
• Anabsolutistmustclaimthereflectionproducesdistinctworlds.- Theydifferontheirvaluesofabsoluteposition.
• Arelationistwillclaimthatthereflectiondoesnotproducedistinctworlds.- Sincetherelationsbetweenmaterialobjectsareunaffected(andthere'snosuchthingasabsolutespace),theworldsarenotdistinct.
AB
Universe1
absolutespace
world1(unreflected)
Universe2
absolutespace
world2(reflected)
A
• Then:Thefollowingtwouniversesmustbepossible:
13
PossibleAbsolutistRetort:
• Wouldareflectedworldbeindiscerniblefromanunreflectedworld?
Spock MirrorSpock
• ReplaceSpockwithadecayingCobalt-60atom:
Co60®Ni60 +e− +νe
Co60
e−
νe
Co60
e−
νe
Co60 decay MirrorCo60 decay
• Co60 decay(electronemittedindirectionofnuclearspin)isobservedmoreoftenthanMirrorCo60 decay(electronemittedinoppositedirectionofnuclearspin)(Wuetal.1957).
Chien-ShiungWu(1912-1997)
Designed, run by C-S Wu. 1957 Nobel prize to Lee & Yang (men). 1978 Wolf prize finally to Wu.• Evidencethattheweakforce(thatgovernsdecay)violates
mirrorsymmetry(i.e.,"parity").14
• AbsolutistClaim:Thereflectedandunreflectedworldsarenot observationallyindiscernible.- Inworld1,theCo60 atomdecayoccursmorefrequentlythaninworld2.
• Onusisnowontherelationisttoexplainthephysicaldifferencebetweenworlds1and2.- RecallClarke'sDynamicShift,withparity-violatingexperimentsnowreplacinginertialeffects.
B
Universe1
absolutespace
world1(unreflected)
Universe2
absolutespace
world2(reflected)
A A
15
LetitbeimaginedthatthefirstcreatedthingwereaCo60 decayprocess,thenitmustnecessarilybeeitheraright-handedCo60decayprocess,oraleft-handedCo60 decayprocess...and there'salaw-likephysicaldifferencebetweenthetwo!
• Canarelationistbothground thedistinctionbetweenright- andleft-handedprocessesandexplainwhyoneismoreprobablethantheother?
(a) Claimthatthedifferenceisintrinsic:Co60 decayprocessespossessanintrinsicmonadic(non-relational)propertythatboth determinestheirhandednessand theirweak-force-governedbehavior.
(b) Claimthedifferenceisextrinsic:- WhatdetermineswhetherthefirstcreatedCo60 decayprocessisright- orleft-handedisitsrelationtoallsubsequentCo60 decayprocesses.
- And:Itisabrutelawlikefact(inneedofnofurtherexplanation)thatoneofthesedecayprocessesismoreprobablethantheother.
16
ALingeringConcernaboutOption(b)
• WhatexplainsNewton'sFirstLaw?Howdoesaforce-freeobjectknowtomoveinertially?
• Similarly:Whatexplainstheparity-violatingweakforce?WhydoCo60 atomspreferdecaymodesofonechiralityratherthananother(givenchiralityisnotintrinsic)?
- Absolutist:Alocal interactionbetweenspacetimeandtheobject(localspacetime"feelers"?).
- Relationist:Anonlocal correlationbetweentheobjectandotherobjects(nonlocalinertial antennae?).
- Absolutist:Alocal interactionbetweenspacetimeandtheobject.
- Relationist:Anonlocal correlationbetweentheobjectandotherobjects(nonlocalweak-forceantennae).
• Isonesetofmysteriousantennae(absolutist)betterthantwo(relationist)?17