The Step to Rationality

33 Pages·2008·5.3 MB·English

CognitiveScience32(2008)3–35
(cid:1)
Copyright C 2008CognitiveScienceSociety,Inc.Allrightsreserved.
ISSN:0364-0213print/1551-6709online
DOI:10.1080/03640210701801917
The Step to Rationality: The Efficacy of Thought
Experiments in Science, Ethics, and Free Will
Roger N. Shepard
DepartmentofPsychology,StanfordUniversity,andTheArizonaSeniorAcademy
Abstract
ExamplesfromArchimedes,Galileo,Newton,Einstein,andotherssuggestthatfundamentallawsof
physicswere—or,atleast,couldhavebeen—discoveredbyexperimentsperformednotinthephysical
worldbutonlyinthemind.Althoughproblematicforastrictempiricist,theevolutionaryemergence
in humans of deeply internalized implicit knowledge of abstract principles of transformation and
symmetry may have been crucial for humankind’s step to rationality—including the discovery of
universalprinciplesofmathematics,physics,ethics,andanaccountoffreewillthatiscompatiblewith
determinism.
Keywords:Thoughtexperiments;Physicallaws;Imaginedtransformations;Mentalrotation;Symmetry;
Rationality;Morallaws;TheGoldenRule;Determinism;Agency;Freewill
1. Theproblem
Thoughtexperimentsarewidelyreportedtohaveplayedaprominentroleinthediscoveries
of physical laws. I begin by describing some specific thought experiments, similar to those
that were—or, at least, may have been—carried out by Archimedes, Galileo, Newton, and
Einstein and that appear to be sufficient to establish fundamental laws of physics, without
carrying out any of these experiments physically. How is this possible? Where does such
knowledge originate if not, as supposed by strict empiricists, from each individual’s own
directinteractionswiththephysicalworld?
TheanswerIproposegrewoutofmyevolutionaryperspectivetogetherwithmycognitive
psychological researches on mental transformations (Shepard & Cooper, 1982; Shepard &
Metzler, 1971) and on generalization (Shepard, 1987; also see the related far-reaching de-
velopmentssubsequentlyachievedbyTenenbaumandothers—e.g.,Chater&Vitanyi,2003;
Feldman,2000;Tenenbaum&Griffiths,2001a,2001b).Here,however,Ifocusprimarilyon
CorrespondenceshouldbeaddressedtoRogerN.Shepard,13805E.LangtryLane,Tucson,AZ85747.E-mail:
[email protected]
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theroleofmentaltransformationsandanassociatedsymmetryprincipleofinvarianceunder
transformation.
Ithenventuretosuggesthowthesameapproachmaybeextendable,beyondthediscovery
ofscientificlaws,toshedlightonthediscoveryofuniversalmoralprinciplesandtoaffordan
account of free will that is compatible with determinism. I shall argue that the emergence in
humans of the cognitive capabilities of discovering universal laws of science and principles
ofethics,andofexercisingfreewillaremanifestationofaterrestriallyunprecedented(ifonly
partially achieved) “step to rationality.” A crucial component of all three of these cognitive
capabilities is the ability to disengage from immediate self-interest and to imagine and to
evaluate alternative events or actions with respect to explicitly represented criteria. I invite
thoseengagedincognitivemodelingtothinkabouthowthekindsofabstractrepresentational
processes and symmetry principles I invoke here might be explicitly implemented in more
concrete,detailedandevenneuro-physiologicallyplausibleways.
2. Empiricalscienceandmathematics
Traditionally, a sharp distinction is maintained between the empirical sciences, on one
hand,andmathematicsandlogic,ontheother.Observations,measurements,andexperiments
onphysicalobjectsandphenomenaaregenerallyconsideredtobeessentialfortheadvance-
ment of the empirical sciences. In contrast, mathematics is supposed to be concerned with
what propositions are logically entailed by other propositions, regardless of whether these
propositionscorrespondtoanythinginthephysicalworld.
Also,inphysicswhataretakentobetheelementaryorprimitiveobjectsarealwaysprovi-
sionalandsubjecttolaterreconceptualization—usuallyintermsofstillmoreelementalenti-
ties.Thus,water,directlyexperiencedasacontinuousfluid,maybesuccessivelyreconceived
ascomposedofmorefundamentalentities:discretemolecules,thenatoms,thenelectronsand
protons,thenquarks,andthenperhapsmodesofvibrationinaconvolutedhigh-dimensional
manifold—withnodefiniteendinsight.
However, in mathematics the elementary or primitive objects are specified by the mathe-
matician rather than by nature. Such objects as the integers of arithmetic or number theory
or the points and lines of geometry are themselves completely transparent from the outset.
Ordinarily,wedonotsupposethatapointoralineorthatanintegersuchas1,2,or3(setting
asideFrege’sandRussell’sabstract,set-theoryofnaturalnumbers)willbefoundtobecom-
posedofsomepreviouslyunsuspectedmoreelementalcomponents.Instead,theadvancesyet
to be made in number theory or in geometry are expected to concern what relations among
suchelementarynumbersorpointsandlineswillbefoundtobeentailedbywhateveraxioms
wehaveformulatedfornumbertheoryorgeometry.
Nevertheless, mathematics and physics are alike in that we aspire, in both cases, to a
consistent theoretical system of basic assumptions and derivable implications. But whereas
in physics such a system is valued to the extent that it provides the simplest explanation or
prediction of the widest range of what we observe or measure in the physical world, in pure
mathematicsthesystemmaybevaluedforitsownsimplicity,elegance,symmetry,orbeauty—
independently of the extent to which it corresponds to what we observe or measure in the
R.N.Shepard/CognitiveScience32(2008) 5
physicalworld.Nevertheless,resultsofpuremathematicsthatwereoriginallyvaluedfortheir
intrinsic beauty are often later found to be most useful or even crucial for the development
of physical theory. Notable examples include the non-Euclidean geometries of Minkowski
andofRiemannforspecialandgeneralrelativity;thecomplexnumbers,matrixalgebra,and
infinite-dimensional Hilbert spaces for quantum theory; and the theory of groups for both
relativityandquantummechanics.
Moreover,theexplicitformalizationsofthelawsofphysicsandofmathematicsare,alike,
creationsofthehumanmind.Absentacomprehendingmind,suchequationswouldnotexist
and, even if by chance they did, they would exist only as meaningless configurations of
physical matter. How do we humans come to formulate, to comprehend, and to evaluate the
formalizations ofmathematical andphysicallaws? Whenwe try todothis forourselves,we
mayfindthatthecognitiveprocessesinthecasesofphysicsandmathematicsareoftenmore
similarthanthetraditionaldistinctionbetweenmathematicalandtheempiricalscienceswould
suggest. To illustrate, I now present simple examples, first, from mathematics (specifically,
geometry)and,then,fromphysics(primarily,mechanics).
3. Pythagoras’stheoremforrighttriangles
ThePythagoreantheoremrelatesthelengthofthehypotenuse(c)ofarighttriangletothe
lengthsofitstwoothersides(a&b)bytheequation(a2 +b2 =c2).Itissurelyatheoremof
mathematics. It may not hold exactly in the physical world. Long before Einstein, it clearly
didnotholdforlargetrianglesonthesphericalsurfaceoftheearth.(Theequilateralspherical
triangle formed by traveling due south from the North Pole to the equator then due east one
quarter of the way around the equator and then due north back to the North Pole is in clear
violationthePythagoreantheorem.Eachpairofthethreeequallegsofthis“triangular”journey
formsarightangle.)Now,accordingtoEinstein’sgeneraltheoryofrelativity,thePythagorean
theoremdoesnotexactlyholdmoregenerallyfortrianglesinthree-dimensionalphysicalspace,
moreorlesscurvedasthatspaceisinthevicinityofmassivebodies.Nevertheless,according
to pure geometry, it does precisely hold for triangles in a flat plane. How can we convince
ourselvesofthislatter,mathematicalfact?
There have been numerous intuitive demonstrations of the validity of the Pythagorean
theorem. Fig. 1 illustrates one attributed to Pythagoras himself. A right triangle (with its 2
sidesofarbitrarylengthsaandbandhypotenuseofcorrespondinglengthc,asshowninA)is
imaginedtobereplicatedthreetimes,yieldingtheidenticalcopies(labeled“1,”“2,”and“3”)
eachrigidlyrotatedand/orflippedtofitwithinasquarewithsidesoflengtha+b(asshownin
B).Onecanimmediatelyseethatthetwoemptyportionsofthatsquare(displayedasshaded
inC)arebothsquareandthatonenecessarilyhasareaa2andtheothernecessarilyhasareab2.
Now the four identical triangles are imagined to be rigidly translated and rotated within the
samesquaretofitintothefourcornersofthesquareasillustratedinD.Withthisrearrangement
of the four identical triangles within the same square, the remaining, empty portion of the
square(againshaded,asshowninD)isimmediatelyseentobeatippedquadrilateralwhose
sides are each of length c (the length of the hypotenuse of the original triangle). Because
◦
the whole configuration (in D) is obviously invariant under 90 rotations, all angles of the
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Fig.1. AproofofthePythagoreantheorem.
tipped quadrilateral must be identical and, hence, this quadrilateral is a square. Because its
sides are of length c, the area of this square is c2. Whatever area within the original larger
squarecontainingthefourtrianglesisnotoccupiedbythosetrianglesmustbeinvariantunder
non-overlapping rearrangements of those triangles within that larger square. Hence the sum
oftheareasofthetwosquareshadedareasinC(i.e.,a2+b2)mustequaltheareaofthesingle
shadedlargerrotatedsquareinD(i.e.,c2),Q.E.D.
Fig. 2 illustrates another, quite different and in some ways simpler demonstration. (First
broughttomyattentionbyDouglasHofstadter,personalcommunication.)Asindicatedinthe
figure,thePythagoreantheorem(a2 +b2 =c2)isequivalenttothestatementthatthesumof
theareasofthesquaresconstructedonthetwoshortersidesofthetriangleequalstheareaof
thelarger,tiltedsquareconstructedonthehypotenuse.
Again, the truth of the theorem can be confirmed by imagining a few simple operations.
First,imagineastraightlineconstructedorthogonaltothehypotenuseandpassingthroughthe
oppositevertex(asindicatedinthefigurebythedottedsegmentthatthusdividestheoriginal
triangle into the two areas labeled “A” and “B”). Second, imagine each constructed square
(of area a2, b2, or c2) and its adjacent triangle (of area A, B, or A + B) rigidly rotated (and
also flipped in the case of the largest square-plus-triangle) as a rigid unit to yield the three
house-shapedobjectslinedupacrossthebottomofthefigure.Third,fromaconsiderationof
thecomplementaryanglesoftheoriginaltriangle,confirmthatthethreetrianglesatthetops
ofthethreesquares(linedupbelowthat),ofareasA,B,andA+B,arenecessarilyidentical
in shape. Finally, from this identity of shape, reach the conclusion for the proportional areas
ofthesquaresthata2 +b2 =c2 (QED).
I suggest that the operations imagined in these two proofs of the Pythagorean theorem
are of essentially the kind that my students and I investigated in our studies of imagined
R.N.Shepard/CognitiveScience32(2008) 7
Fig.2. AnalternativeproofofthePythagoreantheorem.
transformations,including“mentalrotation”andapparentmotion(Shepard&Cooper,1982;
Shepard&Metzler,1971).Alsocrucial,here,istherealizationthatshapeandsizeareinvari-
ant under these rigid transformations of translation, rotation, and horizontal flipping (which
is just a rotation in depth). I have suggested, too, that the visual salience of symmetries
arisesfromtheimplicitrecognitionthatanobjectisinvariantunderthesetransformationsof
reflection, translation, rotation, or, more generally, screw displacement in three-dimensional
space (as in the 4 illustrative examples displayed in Fig. 3). Such invariance under trans-
formation also applies to the case of the mere exchange or permutation of identical objects,
whichwillfigureprominentlyinsomeoftheensuingexamples(bothfromphysicsandfrom
meta-ethics).
4. Archimedes’slawofthelever
UnlikethetheoremofPythagoras,whichbelongstopuremathematics,Archimedeslawof
the lever appears to be a law of physics. It can be stated as follows: Physical objects placed
alongabeamrestingonacentralfulcrumwillbalanceifandonlyifthesumoftheproducts
of the weights and their distances from the fulcrum is equal for the objects on the left and
fortheobjectsontherightofthefulcrum.Archimedesmayhaveverifiedthislawbyplacing
actual physical objects on actual physical balance beams. But he need not have done so. He
mayverywellhaveseenthatthislawmustholdbythoughtexperimentsandtheprincipleof
symmetry,asIillustrateinAthroughEofFig.4.
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Fig.3. Symmetryasinvarianceundertransformation.
IfhehadimaginedthesituationshowninA,Archimedeswouldimmediatelyhaveseenby
symmetrythatthefouridenticalweightsmustbalance.Becausetheweightsareidentical,any
permutation of them must leave the situation unchanged, with no reason for one side or the
othertotiltdown.Nowimaginethatwithoutalteringtheweightofanypartofthebeamitself,
itismodifiedtohaveasecondaryfulcrumasshowninB.Ifthefouridenticalweightsarenow
placed as illustrated in C, everything will still balance as it did in A. Moreover, this remains
trueforanyplacementofthetwoweightsonthesecondaryBeam2thatissymmetricalaround
that secondary fulcrum. For any such symmetrical placement, the combined weight of those
twoobjectsiscommunicatedtotheprimarybeam,asbefore,atthelocationofthatsecondary
fulcrum. Each of the cases exhibited (C, D, & E) thus satisfies Archimedes’s condition. As
shown by the equation over each case, the negative sum of the products on the left of the
fulcrumcancelsthepositivesumoftheproductsontheright.
More generally, beginning with any distribution of any number of weights along a beam,
we can use the same symmetry principle to confirm the following: First, we note (from the
preceding)thatwhetherArchimedes’sconditionholdsornot(i.e.,whetherthebeambalances
ornot),thefactofitsbalanceorimbalanceispreservedunderanytransformationofmoving
anytwoweightstogetheratthemidpointbetweenthem.Second,asillustratedforaparticular,
arbitrarydistributionofidenticalweightsinFig.5,iterationofsuchtransformationsconverges
R.N.Shepard/CognitiveScience32(2008) 9
Fig.4. ThoughtexperimentleadingtoArchimedes’slawofthelever.
towardthesituationinwhichalltheweightsarelocatedatArchimedes’scentroidoftheoriginal
distribution. At every stage of this process, the centroid will be at the fulcrum if and only if
Archimedes’s condition holds and, hence, the beam balances—as it surely will if at the end
alltheweightsarestackeddirectlyoverthefulcrum(asatthebottomofthefigure).
Thus, are we able to verify Archimedes law, without ever placing any actual weights on
any actual beam. What appeared to be an empirical fact about the physical world turns out
tobeentailedbyanabstract,mathematicalprincipleofinvarianceundertransformation—or,
equivalently, of symmetry. We are able to verify the truth of this law of physics by thought
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Fig.5. ConfirmationofArchimedes’slawforanarbitrarydistributionofweights.
alone,muchaswedidinthecaseofPythagoras’smathematicallawconcerningrighttriangles.
Inbothcasesthelawsareuniversalundertheconditionsspecified.ThelawofPythagorasis
universal,aswesaw,onlyforrighttrianglesinaflatplane.Likewise,thelawofArchimedes
is universal only for weights distributed on a beam that is rigid and balanced on the fulcrum
priortotheweightsbeingplaceduponit.
R.N.Shepard/CognitiveScience32(2008) 11
After choosing the mathematical example of Pythagoras and the physical example of
Archimedes, I noticed a striking connection between them: Imagine that you wish to verify
that three long physical rods of the same uniform diameter and rigid material would form a
righttriangle.But(a)thelongnarrowroominwhichyoumustdothisisnotwideenoughto
accommodate the triangle itself and (b) no ruler or measuring tape is available. I leave it as
anexerciseforthereadertouseArchimedes’slawtoverifythefollowingstatement:Ifalong
(unmarked)beamandfulcrumareavailable,andthetwoshorterbarsarelaidside-by-sideon
onesideofthebeamextendingfromthefulcrumandthelongestbarislaidontheotherside
extending from the fulcrum (as illustrated in Fig. 6), then if a triangle were formed by the
threebars,itwouldbearighttriangleifandonlyifthebarsonthebeamachieveabalance.
5. Galileo’slawoffallingbodies
PerhapsthemostthoroughlyanalyzedthoughtexperimentofalltimeistheoneGalileoused
torefutetheclaimattributedtoAristotlethatfallingbodiesdropwithspeedsproportionalto
theirweights(seeGendler,1998).Presumably,Aristotledidnotreachanerroneousconclusion
byperforminganactualexperiment.Ifhereacheditfromathoughtexperiment,itevidently
was one to which he did not devote enough thought. Possibly by imagining the hefting of a
lightobjectinonehandandaheavyobjectintheotherhandhewasledtothehastyconclusion
that if the objects were released, the greater downward force he felt from the heavier object
would manifest itself as a faster descent on the release of that object. But such a conclusion
ignores inertia—something that Artistotle might have realized before Galileo if he had also
thoughtaboutthegreaterforceneededtoaccelerateamoremassiveobjecttothesamespeed
asalighterobject(e.g.,ifbothobjectswererestingontheslipperysurfaceofafrozenpond).
Galileo’sthoughtexperimentwasconclusive.Ityieldedthecorrectconclusion:Thespeed
of descent is independent of the weight of the object (to the extent that air resistance is
negligible). This implies that the greater downward force that Aristotle imagined he would
feelfromtheheavierobjectisinfactexactlytheforceneededtoacceleratethemoremassive
object to the very same speed. I shall describe a thought experiment that is slightly different
from the one actually offered by Galileo but that is, I believe, equally conclusive and more
revealingoftherelevanceoftheprincipleofsymmetry.
I imagine Galileo imagining himself at the top of the leaning tower at Pisa with three
identical bricks. By the symmetry principle of invariance under permutation of identical
objects, if the three bricks were dropped together, they should reach the ground at the same
time. As in the case of Archimedes’s identical weights equally distant on each side of the
Fig.6. AcorrespondencebetweenthelawsofArchimedesandPythagoras.
12 R.N.Shepard/CognitiveScience32(2008)
fulcrum,thereisnoreasonforonetodescendmorethananother.Nowsupposetwoofthese
three bricks are glued together to form a single, twice-as-heavy brick. Surely, the virtually
weightlessfilmofglueusedtomakethetwobricksintoonewouldnotcausethatnowheavier
bricktofalltwiceasrapidlyastheseparate,thirdbrick.Onceagain,thecorrectconclusionis
reachedwithouthavingtoperformanactualexperiment.
6. Newton’slawofactionandreaction
Newton’sinspirationforhisuniversallawofgravitationissometimesattributedtoathought
experiment in which he imagined throwing an apple (according to legend, one that dropped
on him from a tree under which he was sitting in an orchard). Newton would have realized
that if he threw the apple with greater and greater force, the apple would fall to ground at
a greater and greater distance from him. Knowing that the earth is a sphere, he would have
concludedthatifhewereabletothrowtheapplewithsufficientforce,itwouldfallnottoearth
butaroundtheearth.Hemightthenconjecturethatthemoonwashurlingwithsuchspeedas
tobesimilarlyeverfallingaroundtheearth.
I now consider, instead, a different thought experiment from which Newton might have
arrived at his Third Law of Motion—the law that every action has an equal and opposite
reaction.Ifocusonthislawfortworeasons:First,ofhisthreelawsofmotion,thisistheone
thatwasmostoriginalwithNewton.Second,thisthoughtexperimentderivesinaparticularly
simpleandtransparentwayfromtheprincipleofsymmetry.
IimagineNewtonimagininghimselfarchedoverthewaterwithhisfeetonthegunwaleof
hisboatandhishandsonthegunwaleofanotherboatofthesamesizethatheisendeavoringto
pushawayfromhisboat.Fromtheobvioussymmetryofthesituation,Newtonwouldrealize
thatthereisnowaythathecanpushtheotherboatawayfromhiswithoutequallypushinghis
boatawayfromtheother.Hemightalsogoontorealizethatifhewerestrandedinthemiddle
ofalakewithanoar-lessboatloaded,say,withapples,hecouldpropelhimselfbacktoward
shorebyhurlingtheapples,onebyone,totherearwithgreatforce(justasspacevehiclesnow
acceleratethroughemptyspacebyejectingmoleculesofthegaseousproductsofcombustion
rearwardatextremelyhighvelocity).
7. Einstein’stheoryofrelativity
Einstein,likeGalileo,wasamasterofthethoughtexperiment.Hearrivedatspecialrelativity
throughthoughtexperimentsabouthowthe sameeventswouldbeexperiencedbyobservers
indifferentstatesofmotion.Galileohadassertedthateventsonashipwouldappearthesame
toshipboardobserverswhethertheshipwereatrestorunderfullsail.Ineithercase,hesaid,
an iron ball dropped from the top of the mast would appearto drop straight down relative to
the ship, and not (as some had supposed) further back on the deck if the ship were moving
swiftlyforward.
Einstein proposed that the same principle should continue to apply even if the velocity of
thevesselweretoapproachthespeedoflight.Fig.7illustratesarelevantthoughtexperiment