Plato Was Not a Mathematical Platonist

58 Pages·2023·1.101 MB·English

by  Elaine Landry

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This Element argues that Plato was not a mathematical n
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Platonist. It shows that Plato keeps a clear distinction between r
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mathematical and metaphysical realism, and the knife he uses
to slice the difference is method. The philosopher’s dialectical
method requires that we tether the truth of hypotheses
to existing metaphysical objects. The mathematician’s The Philosophy of
hypothetical method, by contrast, takes hypotheses as if they Mathematics
were first principles, so no metaphysical account of their truth is
needed. Thus, we come to Plato’s methodological as-if realism:
In mathematics, we treat our hypotheses as if they were first
principles, and, consequently, our objects as if they existed,
and we do this for the purpose of solving problems. Taking the
P Plato Was Not
road suggested by Plato’s Republic, the author shows that some la
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methodological commitments to mathematical objects are W
made in light of mathematical practice; some are made in light as
N a Mathematical
of foundational considerations; and some are made in light of o
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mathematical applicability. M
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about the Series Series Editors l P
This Cambridge Elements series provides Penelope Rush lat
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an extensive overview of the philosophy University of is
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of mathematics in its many and varied Tasmania Elaine Landry
forms. Distinguished authors will provide Stewart Shapiro
an up-to-date summary of the results of The Ohio State
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ElementsinthePhilosophyofMathematics
editedby
PenelopeRush
UniversityofTasmania
PLATO WAS NOT
A MATHEMATICAL
PLATONIST
Elaine Landry
’
University of California, Davis
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Not
Plato Was a Mathematical Platonist
PhilosophyofMathematics
DOI:10.1017/9781009313797
Firstpublishedonline:January2023
ElaineLandry
UniversityofCalifornia,Davis’
Authorforcorrespondence:ElaineLandry,[email protected]
Abstract:ThisElementarguesthatPlatowasnotamathematicalPlatonist.
ItshowsthatPlatokeepsacleardistinctionbetweenmathematicaland
metaphysicalrealism,andtheknifeheusestoslicethedifferenceis
method.Thephilosopher’sdialecticalmethodrequiresthatwetether
thetruthofhypothesestoexistingmetaphysicalobjects.The
mathematician’shypotheticalmethod,bycontrast,takeshypothesesas
iftheywerefirstprinciples,sonometaphysicalaccountoftheirtruthis
needed.Thus,wecometoPlato’smethodologicalas-ifrealism:In
mathematics,wetreatourhypothesesasiftheywerefirstprinciples,
and,consequently,ourobjectsasiftheyexisted,andwedothisforthe
purposeofsolvingproblems.TakingtheroadsuggestedbyPlato’s
Republic,theauthorshowsthatsomemethodologicalcommitmentsto
mathematicalobjectsaremadeinlightofmathematicalpractice;some
aremadeinlightoffoundationalconsiderations;andsomearemadein
lightofmathematicalapplicability.
Keywords:Plato,ancientphilosophyofmathematics,mathematicalrealism,
mathematicalplatonism,as-ifism
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ytisre ©ElaineLandry2023
vin ISBNs:9781009313780(PB),9781009313797(OC)
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e ISSNs:2514-3808(online),2399-2883(print)
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Contents
1 Introduction 1
2 TheInterpretiveLayoftheLand 3
3 TheDividedLine 9
4 Book7 16
5 TheGoodinMathematics 30
6 MathematicsVersusMetaphysics 41
References 47
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PlatoWasNotaMathematicalPlatonist 1
1Introduction
InthisElement,IwillarguethatPlatowasnotamathematicalPlatonist.1My
arguments will be based primarily on the evidence found in the Republic’s
DividedLineanalogyandinBook7.2IwillpresentPlato’sviewasitdevelops
in the text, which, while perhaps not as reader-friendly as one might like,3
emphasizes the significant changes that Plato intends to be surprising, even
shocking, to his reader – changes that are often missed in the interpretative
literature.First,IwillofferwhatItaketobeanaccuratetranslationofthetext,4
beforecriticallyconsideringwhichclaimsremainthesameandwhichchange,
especiallyaswetransitionfromtheDividedLinetoBook7.Finally,Iwillbring
theseclaimstogetherintoaconsistentpictureofPlato’sviewofmathematics,
demonstratingthathewasnotamathematicalPlatonist.
Typically,5themathematicalPlatoniststoryistoldonthebasisofthreerealist
components:(a)that mathematical objects, asPlatonicforms,existindepend-
entlyofusinthemetaphysicalrealmofforms;(b)thatthewaythingsareinthis
metaphysicalrealmfixesthetruthofmathematicalstatements;and(c)thatwe
cometoknowsuchtruthsby,somehoworother,“recollecting”thewaythings
areinthemetaphysicalrealm.6ThisPlatoniststory,byconfusingthehypothet-
icalmethodofmathematicswiththedialecticalmethodofphilosophy,conflates
thetwotypesofrealismatplayinPlato:methodologicalrealismandmetaphys-
icalrealism.MyaimistoshowthatwhilePlatoisaphilosophicalPlatonist–
that is, he adopts metaphysical realism for philosophical inquiry – he is a
mathematicalas-ifist–thatis,headoptsmethodologicalrealismformathemat-
icalinquiry.Thus,itisbykeepingthesemethodsdistinctthatwewillseethat,as
sserP regards(c),wecometoknowmathematicalobjectsbytreatingourhypotheses
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gd 1 JustasWhitehead(1929,p.39),claimedthatthehistoryofphilosophyconsistsofaseriesof
irb footnotestoPlato,thisElement,forthemostpart,consistsofaseriesoffootnotes(literaland
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aC figurative)toBurnyeat(2000);but,aswewillsee,withimportantdifferences.Mostsignificant
yb among these is that Burnyeat holds that Plato leaves open the question of the existence of
e
n mathematicalobjects.Idisagree.Platoisclear:mathematicals,orasIwillcallthemmathematical
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o objects,arenotforms.
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eh 2 Aswewillsee,therearealsoassertionsfoundintheMenoandtheTheaetetusthatfurtherwitness
silb myarguments,butmyprimaryfocusistheRepublic.
uP 3 Toprovidethereaderwithaconsistentlyflowinginterpretationthatalsofollowstheorderof
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97 Plato’sarguments,Ihaveoptedtoplace someofthecriticaldiscussionsandanalyses ofthe
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13 interpretativeliteratureinfootnotes.
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00 4 AlltranslationsarefromReeve(2004),unlessotherwiseindicated.
187 5 Anotableexceptiontothestandardstoryisfoundinthehistoricallyrichandphilosophically
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/7 robustbookbyPanzaandSereni(2013).
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01 6 IhavedemonstratedthatrecollectionintheMenoisnotofferedasamethodformathematical
.01 knowledge(Landry2012).Aswewillsee,whatisofferedasthemathematician’smethodfor
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2 ThePhilosophyofMathematics
as if they were true first principles for the purpose of using these to solve
mathematicalproblems.
Asregards(a),Iwillshowthatmathematicalobjectsdependonthemathem-
atical problem that we are attempting to solve. It is the problem at hand that
gives rise to the needed hypotheses that themselves are taken as if they were
true,anditisthesehypothesesthatgiverisetotheneededobjectsofthoughtthat
we take asifthey exist for thepurpose ofsolving theproblem. Mathematical
objects,then,existinamethodologicalsensebutnotinametaphysicalsense.
Against(b),whatfixesthetruthofamathematicalstatementisitsmethod,not
its metaphysics – that is, mathematical truth is fixed by a demonstration that
showsthattheanswertoourproblemcanbededucedfromourhypothesis;itis
notfixedbythewaythingsareinthemetaphysicalrealmofforms.Aswewill
see, in mathematics, existence is a consequence of truth – that is, is a conse-
quenceoftakingourhypothesesasiftheyweretrueforthepurposeofsolvinga
problem.Inphilosophy,bycontrast,truthisaconsequenceofexistence,thatis,
is a consequence of our tethering our hypotheses to independently existing
forms.Itistheseconsiderations,whicharisebykeepingdistinctthemathemat-
ician’sandthephilosopher’smethods,thatallowustoseethatPlatowasnota
mathematical metaphysical realist; rather, he was a mathematical methodo-
logicalrealist.
Myaimistoarguethatsincethemethodusedbythemathematicianisdistinct
fromthatofthephilosopher,thensotoomustbetheirobjects.Fromamethodo-
logical standpoint, I will show that the mathematician uses the hypothetical
methodandtravelsdownwardfromahypothesis,takenasifitwereatruefirst
sse principle, toward a conclusion. The philosopher, on the other hand, uses the
rP y dialecticalmethodtofirsttravelupwardfromahypothesis,takenasahypoth-
tisre esis,towardafirstprinciple,thetruthofwhichisfixedbyaform,andtheythen
v
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e
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dirb will further show that, as a result of these methodological differences, the
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a mathematician, in their goal of solving mathematical problems, needs only
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b e take their objects as if they exist. This is why, now from an epistemological
n
iln standpoint,mathematicalobjectsaretobetakenasobjectsofthought,whereas
o
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eh philosophicalobjectsaretobetakenasobjectsofunderstanding(or,attheend
silbu ofBook7,asobjectsofknowledge).Bringingthesetwostandpointstogether,I
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79 willarguethatmathematicalobjects,asthingsthatarisefrom“images,”orfrom
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3 drawnorconstructeddiagrams,arenonethelesstobetakenasdistinctfromsuch
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01 “images” and so are to be taken as “things themselves.” However, even as
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01.0 selves”;theyaremethodologicallyreal–thatis,wetreatthemasiftheyexist
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PlatoWasNotaMathematicalPlatonist 3
Indeed,thisiswhy,attheendofBook7,Platolikensthefacultyofthoughtto
that of imagination and, as a consequence, comes to reserve the term “know-
ledge”forphilosophical knowledge only.Thus,taking myevidence primarily
from the Divided Line analogy and Book 7, I will argue that Plato was not a
mathematicalPlatonist;mathematicalobjectsarenotforms,theydonoteither
exist in some metaphysical realm or fix the truth of mathematical statements,
andwedonotcometoknowthemviarecollection.
2TheInterpretiveLayof theLand
The number of interpretations of Plato’s views of mathematics is vast. Some
consider the whole of Plato’s works, others focus on specific dialogues. My
interpretationwillfocusprimarilyonwhatPlatosaysintheRepublic’sDivided
LineandBook7.Thereasonforthisistwofold;exceptfortheMeno,theseare
the onlyplaces where Plato presents a sustained account ofmathematics, and
there seems little debate that this dialogue was written by Plato.7 In a broad
stroke,myinterpretationisintendedtocutamidpointbetweenthetwoprevail-
ing and competing views. The first is the view of Cornford (1932), White
(1976), Tait (2002), and Benson (2006; 2010; 2012) that the hypothetical
methodispartofthedialecticalmethodsothatmathematicalobjectsmust,in
somesense,bepartoftherealmofforms.ThesecondistheviewofBurnyeat
(2000) and Broadie (2020) that the mathematician’s hypothetical method is
distinct from the philosopher’s dialectical method, but that Plato adopts a
quietist stance on the ontological status of mathematical objects – that is, on
the question of whether mathematical objects are to be taken as distinct from
sse forms.
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7 betweentwodifferentapplicationsofthesamemethod.Boththedianoetician
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31 and the dialectician applyor use the method of hypothesis, but the former
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90 does so inadequately and incorrectly. The dianoetician [as exemplified by
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01 7 IhaveanalyzedwhatPlatotellsusofthebenefitsandlimitsofthemathematician’smethodinthe
.01 Meno(Landry2012).WhilethereisadiscussionofmathematicsintheSeventhLetter,itisfar
/gro fromclearwhetherthisworkisPlato’s.
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4 ThePhilosophyofMathematics
“currentpractitioners”ofmathematics],unlikethedialectician,…mistakes
herhypothesisforarchai[forunhypotheticalfirstprinciples].
(pp.1–2;italicsadded)
Most part of interpreters hold that these unhypothetical first principles are
unhypotheticalbecausetheyaretetheredto,orfixedby,astablemetaphysical
domain(i.e.,byarealmofmathematicalobjectstakenasphilosophicalforms,
or,likeTait,byarealmoffoundationalmathematicalobjectstakenasgeometric
forms).
Burnyeat(2000),incontrast,useshisdistinctfrominterpretationtopointto
twostancesthatonemayadoptasregardstheontologicalstatusofmathematical
objects: the “internal” stance taken by practicing mathematicians and the
“external” metaphysical stance taken by the philosopher of mathematics. He
remains oddly silent on what the practicing mathematician’s internal stance
comestobut,asregardsthelatter,holdsthatPlato“leavestheexternalquestion
tantalisinglyopen”(p.22).Likewise,Broadie(2020)holdsthat“Platoshowsno
interestinthismetaphysicalquestion”(p.15).
Benson(2000)similarlyholdsthat“Platoislessconcernedtoofferafourfold
ontologyassociatedwiththefoursectionsoftheLine,thanheistodescribethe
correct method of the greatest mathēma – the knowledge of the Form of the
Good”(p. 1).But, aswe have noted, Benson, as manyother interpreters who
believe that the philosopher’s method must be adopted by the mathematician,
holdsthatthisexternalquestionmustbeanswered.Thevarious,whatIwillcall,
metaphysical interpretations that seek to answer this external question agree
that one must adopt a mathematical Platonist position but bifurcate over
sse whether this should be answered at a philosophical or at a metamathematical
rP y level–thatis,whetheronemustadopttheviewthatmathematicalobjectsare
tisrev philosophical forms themselves or are to be founded on an ontologically
in
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silbu mathematicalobjectsasintermediates:8
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13 8 AstoodoesBroadie:“Platoalsopostulatestwocorrespondinglydifferentlevelsofintelligible
900 reality,theformsproperandthedistinct‘intermediate’ormathematicalswhichweknowfrom
187 Aristotle came to be posited in Plato’s school” (p. 15). McLarty (2005) also argues for an
9/7 “intermediates”position:“GlauconinPlato’sRepublicfailstograspintermediates.Heconfused
1
01 pursuingagoal(ofsearchingforfirstprinciples)withachievingit,andsohe(mistakenly)adopts
.01 ‘mathematicalplatonism’”(p.115).SeealsoFoley’s(2008)article,foranilluminatingdiscussion
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ro of how the ratios and the proportions of the line can be used to partition debates about the
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