Logos and Alogon: Thinkable and the Unthinkable in Mathematics, from the Pythagoreans to the Moderns

307 Pages·2023·3.52 MB·English

by  Arkady Plotnitsky

Arkady Plotnitsky
Logos and
Alogon
Thinkable and the Unthinkable in
Mathematics, from the Pythagoreans
to the Moderns
Logos and Alogon
Arkady Plotnitsky
Logos and Alogon
Thinkable and the Unthinkable
in Mathematics, from the Pythagoreans
to the Moderns
ArkadyPlotnitsky
Literature,Theory,andCulturalStudies
Program,PhilosophyandLiteratureProgram
PurdueUniversity
WestLafayette,IN,USA
ISBN978-3-031-13677-1 ISBN978-3-031-13678-8 (eBook)
https://doi.org/10.1007/978-3-031-13678-8
MathematicsSubjectClassification:00A30;01A20;01A60;01A65;81-03;81P05
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Mathematik. – Wir wollen die Feinheit und
Strenge der Mathematik in alle
Wissenschaften hineintreiben, so weit dies
nur irgend möglich ist; nicht im Glauben,
daß wir auf diesem Wege die Dinge
erkennenwerden,sondernumdamitunsere
menschliche Relation zu den
Dingen festzustellen. Die Mathematik ist
nur das Mittel der allgemeinen und letzten
Menschenkenntnis.
—Friedrich Nietzsche, Die Fröhliche
Wissenschaft
Preface
This book is a philosophical study of mathematics, pursued by considering and
relating two aspects, the first conceptual and the second epistemological, of math-
ematical thinking and practice, especially in modern mathematics. Defined by its
abstract nature, modern mathematics began to emerge, roughly, around 1800, and
took the center stage of mathematics by 1900 and has occupied it since. The book
will also consider earlier historical occasions, where both these aspects and their
relationships played a shaping role, beginning with ancient Greek, specifically
Pythagorean, mathematics, thus making the history of mathematics part of the
philosophy of mathematics. Bringing the philosophy and history of mathematics
together is essential to my argument in this book, as discussed in the Introduction.
Here,myfocusisonthetwoaspectsinquestionandtheirrelationships.
The first aspect is that of grounding mathematics, or in the language of ancient
Greekphilosophyusedinmytitle,thelogosofmathematicsinconceptualthinking
rather than in logical propositions or making calculations. Both logic and calcula-
tions are of course essential to mathematics and its logos, including conceptual
thinking there, first, because the structure of mathematical concepts is logical, and
secondly, because calculations require concepts and can lead to new concepts.
While, however, fully respecting these features, this study will give priority to
concepts in mathematical thinking, especially creative mathematical thinking, to
beprimarilyconsideredinthisstudyastheinventionofnewconcepts,thusmaking
thelogosofmathematicsprimarilyconceptual.Theconceptualgroundingofmath-
ematics has been studied previously, in particular in conjunction with the work of
Bernhard Riemann, who is the central figure of this study in this context. His
thinking and work, however, and this feature itself, will be considered from a new
perspective, defined by a new understanding of mathematical concepts and by the
Pythagorean aspect of mathematics, which is the second aspect of mathematics
definingthisstudy’sargument.
This aspect is in turn twofold: the first part is epistemological and the second,
again,conceptual.Thefirstpartrepresentsthemostepistemologicalradicalformof
vii
viii Preface
modernmathematics,whichemergedinthetwentiethcenturyandisdesignatedby
thisstudy as“radicalPythagoreanmathematics.”Thismathematicsisgroundedin
thefundamentalroleoftheunthinkableinit,orinancientGreekterms,usedbythe
Pythagoreans and adopted by this study’s title, an alogon in the logos of this
mathematics. By the unthinkable I refer, in affinity with the ancient Greek alogon,
tothatwhichisstrictlyinconceivableorincomprehensible,unthinkable:itisbeyond
thereachofthought,atleastatagivenpointintimeandpossiblyever.Ishallalso
adoptanew,alsoetymologicallyancient-Greek-based,termforthisunthinkable:the
“alogonal.” (As theoretical terms, rather than merely ancient Greek words,
“alogonal”and“alogon”willhereafterbeusedwithoutitalicsorquotationmarks.)
Whileanegativeofthelogical(or“logonal”),thealogonalisdifferentfromandeven
opposite to what is designated as the illogical: the alogonal is based on logical
thinking,arisingfromthecharacterofcertainphenomena,eithermental,specifically
mathematical,ormaterial,specificallyasconsideredinphysics,andoftherelation-
shipsamongthem.Thepossibilityofthealogonalisdeducedfromthischaracter.At
thesametime,thealogonalaspectofagiventheorymayberesponsibleforthekey
conceptsdefiningthistheoryand,hence,forwhatonecanthinkandknowbymeans
ofthistheory.Inotherwords,partofwhatisthinkable,insomesituationsevenits
most important part, emerges from the unthinkable, as a set of its effects, even
thoughitisnotpossibletoestablish,think,howtheseeffectscomeabout,whilethe
nature of this emergence may be unthinkable in turn. Accordingly, at stake in this
study is not merely considering the unthinkable, the alogonal, as a general idea,
intriguingasitmaybephilosophically,butalsoandprimarilyitspossiblyunavoid-
able role in the advancement of thinking and knowledge in mathematics and
mathematicalsciences,suchasphysics.Thisadvancementmayrequire,first,assum-
ingthattheunthinkableisanessentialandpossiblypermanentpartofthought,and
second,working,thinking,withtheunthinkable,ratherthanseeingtheunthinkable
or,inthiscase,thatwhichismerelyasyetunthought,assomethingtobeeventually
reachable by thought. The latter type of thinking is possible, too, and often neces-
sary, but it pertains to a form of thinking different from thinking that involves the
alogonal as its irreducible component, in which case one must think with the
unthinkableandadjustone’sknowledgeaccordingly.
Radical Pythagorean mathematics, however, contains a second definingfeature:
the interaction between geometry and algebra in its conceptual architecture. The
terms geometry and algebra are broadly conceived in this study (beyond their
standard meanings, while preserving and emphasizing their essential origins and
features)toincludeundertheheadingofgeometryallmathematics ofspatiality,in
particular topology, and under the heading of algebra such fields as arithmetic,
number theory, or analysis, as mathematics defined by working with mathematical
symbols.Thischoiceofterminologyisstrategicbutisjustifiedgiventhehistoryof
mathematics from ancient Greek mathematics on. The interaction between both
forms of mathematics was central to the mathematics of the Pythagoreans (in their
case, between geometry and arithmetic), and has continued to remain central to
mathematics.Itisthecombinationofboththeroleoftheunthinkableinthinking,of
thealogonalinthelogosofmathematics,andtheinteractionofgeometryandalgebra
Preface ix
thatdefinedradicalPythagoreanmathematicsand,withittheprojectofthisstudy,
along with grounding mathematics in conceptual thinking, in the invention of and
workingwithconcepts.
Two related concepts defined by the same alogonal epistemology, but more
general in nature (insofar as they need not contain Pythagorean aspects) will be
adoptedinthisstudy—thatof“realitywithoutrealism”(RWR)andthatof“ideality
without idealism” (IWI), a form of reality without realism specifically referring to
mentalreality,suchasthatofmathematics.Theseconceptspresupposemoregeneral
concepts of reality and existence, material or mental (or ideal), as discussed in
Chap. 1. Briefly, by reality, I understand that which is assumed to exist without
making any claims concerning the character of this existence. Such claims define
realism, which, in the case of mental or ideal reality, such as that considered in
mathematics, becomes a form of idealism, such as that of mathematical Platonism
(a twentieth-century conception which should not be confused with Plato’s own
philosophy). The term mathematical realism is sometimes used as well. In most
understandingsoftheterm,realismoridealismassumesthepossibilityofforminga
representationoratleastaconceptionofthereality responsibleforthephenomena
considered. By contrast, the absence of such a claim allows one to place a given
realityorpartofitbeyondrepresentationorknowledge,orevenconception.Inthe
firsteventuality,IshallspeakoftheweakRWR/IWIview,andinthesecondofthe
strong or alogonal RWR/IWI view, generally assumed in this study. Either view
corresponds to RWR or IWI-type theories or interpretations of theories. In such
theoriesorinterpretations,theunthinkableinthinkingbecomesaproductivecondi-
tionofthinkingandknowledge,ratherthaninhibitingorevenprecludingthem,asin
realist theories or philosophical positions defined by realism. The latter positions
have been adopted by many, even most, mathematicians, physicists, and philoso-
phers,beginningwithPlato,andhaveshapedtheirskepticalattitudetowardRWR/
IWI-typetheoriesorinterpretationsoftheories.Forthem,suchtheoriesorinterpre-
tationsareundesirableorevenunacceptable,asstandinginthewayofadvancement
ofthoughtandknowledge,oracceptableonlyasstopgaptemporaryexpedients,with
ahopethattheywilleventuallybereplacedbyoneoranotherrealistalternative.The
philosophical position of this study is the opposite: it affirms the unthinkable in
thinking,thealogonwithinalogos,asafriendofthought,whichhelpstoadvanceit,
under the assumption that this advancement may not be possible otherwise. One
cannot, of course, be certain, because a realist alternative cannot be excluded,
especially in the future. We cannot know in advance what nature or thought will
makeusconfront,forexample,inscienceormathematics,orhowourthoughtwill
handlethisconfrontation,ifitcan,becausethiscannotbeguaranteedeither.
“Thought...isthemeasureoftheUniverse,”thegreatestgiftofPrometheus,the
first Pythagorean, who gave humans numbers and geometry as well, P. B. Shelley
says in Prometheus Unbound (II, iv. 73; Shelley 1977, p. 174). Shelley’s phrase
echoes and arguably refers to Protagoras’ “man is the measure of all things,” but
givesitthePythagoreanmeaningofalogosthatmaycontainthealogonwithinit.As
the poem says a bit later, “the deep truth is imageless” and may even be beyond
x Preface
conception rather than only image, although Shelley himself might not have been
willing to go that far (II, iv. 116; Shelley 1977, p. 175). More precisely, Shelley’s
statementis:
He[Prometheus]gavemanspeech,andspeechcreatedthought,
WhichisthemeasureoftheUniverse
(PrometheusUnoundII,iv.73).
Evenifoneassumesthatsuchanontologicaloranthropologicalprimacyoflanguage
is possible, which is doubtful, my main point here will not be affected. In the
RWR/IWIview,wecannotspeakor,atthealogonallimit,eventhinkoftheultimate
constitution of nature or thought, including in terms of mathematical or scientific
concepts, assuming that they can be dissociated from ordinary concepts and lan-
guage. Language and thought are indissociable from and shape each other even in
mathematics,althoughthelatterhasthecapacity,atleastinprinciple,ofrepresenting
reality,mental(whichisitsown)orphysical,apartfromordinaryor,inthecaseof
physicalreality,physicalconcepts.ThiscapacityisdisabledbytheRWR/IWIview,
at least as concerns some strata of the reality considered, strata placed beyond all
representation or, alogonally, beyond conception. This view, however, still gives
mathematicsacapacitytorelatetothesestrataotherwise,througheffectsthesestrata
have on those part of reality that can be thought about, known, represented, or
spoken about. Indeed, in RWR/IWI-type theories (or interpretations of certain
theories)consideredinthisstudy,theexistenceofalogonalstrataofrealityis,rather
thanmerelyassumedorimagined,rigorouslyinferredonthebasisofsucheffects,in
thefirstplace,suchastheexistenceofundecidablepropositions inmathematicsor
thecharacterofthedataobservedinquantumexperimentsinquantumphysics,data
thatcanonlybepredictedprobabilisticallybymeansofthemathematicalformalism
ofquantumtheory.
MathematicswasanequallyimportantgiftofPrometheus.Aeschylus,inProme-
theus Bound, whom Shelley cites (displacing Aeschylus’s version toward the pri-
macy of language), says so. He first affirms thought and consciousness as
Prometheus’ greatest gift: “But harken to the miseries that beset mortals—who
were formerly senseless and I made them to have sense and to be endowed with
thinking.”Consciousnesscomesnext,enablinghumanstodiscern“therisingofthe
starsandtheirsetting,thecosmos.”Thenhespeaksofhisgreatest,Pythagorean,gift,
through which the harmony of the cosmos can be discovered, too: “And I also
discovered for them number, that most excellent of intellectual inventions,” which
thus precedes language and arts, which come next, “the combining of letters, with
which one has memory of everything, that worker who is mother of the Muses”
(PrometheusBound,ll.444–462).Itisthisstatementthatmayhavebeentheorigin
ofShelley’s“Hegavemanspeech,andspeechcreatedthought,whichisthemeasure
oftheUniverse,”becauseforShelley,alltruethoughtisart,poetry.Thismaywellbe
true,insofarasonekeepsinmind,asShelleydoes,themeaningoftheancientGreek
“poiein” as making, creating something new—a new poem, a new philosophical
concept, a new poetic idea (an idea of a reform), a new mathematical or scientific
concept,inshort,anythingnewthatadvancedthoughtandhelpedtomakelifebetter.
Preface xi
Shelley was a political revolutionary as well. Numbers, “that most excellent of
intellectual inventions,” represent our measurements in geometry and physics, but
they,orgeometry,aninventionhardlyanylessexcellent,areonlypossiblebecause
ofthought,whichisourprimordialmeasure ofeverything.Bothconcepts, number
and measurement, were given their alogonal complexities by the Pythagoreans’
discoveryoftheincommensurablenumbers,andaswillbediscussedinthisstudy,
became radically questioned and rethought by modern mathematics, especially in
view of Gödel’s incompleteness theorems, and quantum theory, which radically
redefinedtheconceptofmeasurement,tothepointofmakingaquantum“measure-
ment” something to which no previous concept of measurement could any longer
apply.
ThoughtremainsourmeasureoftheUniverse,eveniftheUniverse,theultimate
natureoftheUniverse,itselfmay,asthePythagoreansmighthavesensed,bebeyond
thought,andthusbe,perhapsforeveralogonal,immeasurable.Thismaybeequally
trueabouttheultimatenatureoflifeorofourthought,whichisassumedinthisstudy
tobetheproductofourbrainsandthusstillmaterial,eventhoughwehavenotheory
ofthoughtasafunctionoftheneurologicalandthusultimatelyphysical(evenifnot
reducibletophysics)processesofourbrains.Thethoughtofthis“beyondthought”
isstillonlyathought,thisalogonisstillpartofourlogos.Theanalysisofquantum
physics given by this book confirms and amplifies one of Heraclitus’s famous
maxims: “nature loves to hide.” But the discussion of mathematics given in this
book suggests that thought, the ultimate nature of thought, likes to hide, too. This,
however,neednotmeanthattheultimatelevelofrealityisdefinedbypermanenceor
oneness, the One, as Parmenides, another key pre-Socratic thinker, and Plato
thought: the concept of permanence is no more applicable to this “it” than that of
changeoranyotherconcept,including“somethinghappened,”or“it,”“hidden,”or
“many.” Thinking in terms of the Heraclitean vision of the hidden, becoming, and
themanyappearstobemuchmoreinaccordwiththeultimateconstitutionofreality,
material or mental. Parmenides and, following him, Plato would have seen this
multiplicityandbecomingasillusionsofoursensesratherthanthecharacteristicsof
the ultimate nature of reality, hidden from our senses, if for them, not, at least not
entirely, from thought. In the present, strong or alogonal RWR, view one would
accepttheirfirstclaim,whenconsideringtheultimateconstitutionofreality,butnot
their second claim. For it does not follow that this reality is defined by absolute
permanence, any more than by change, or as anything hidden because this word
impliessomething,a“secret,”howeverdeep,thatcanberevealedoneday.Normay
itbeaccessibletothought.Hence,ratherthan referring tosomething,“reality”isa
namewithoutaconceptassociatedtoit,orevenanamethatisnotaname,becauseit
isdifficultforanametoavoidaconceptassociatedwithit.Itdoesnotrefertosome
single “secret” determinate entity to be discovered some day or even never to be
discovered but assumed to be hidden somewhere as such an entity. The “hidden”
(if one can still use the term) of the alogonal only refers to the beyond of thought.
There is always part of the real, possibly each time different, that is, as something
alogonal, beyond thought, but that has effects on what can be thought and that, as
such, must be part of a theory, a logos, working with these effects to advance
thinkingandknowledge,andtomakethisadvancementpossible.