Theory of Knowledge - Course Companion for Students Marija Uzunova Dang Arvin Singh Uzunov Dang
apparnt timlssnss of knowldg inThmay lad on to th assumptionmathmatismathmatial knowldg is produd in athatway. In rality, th prati of mathmatistimlssmddd in ultural ontxts and has hangdisovr tim.markdly18th-ntury Europ th ida of mathmatisInth hiddn languag of th univrs wasasaptd, and mathmatiians fashiondwidlyas “natural mn” (AlxandrthmslvsPrsptivs on mathmatis shift, along2010).th norms and valus of mathmatialwithRvaling this “hiddn ordr“ inommunitis.mathmatial strutur of rality may attratthpartiular kind of mathmatiian: somon alagrappl with astrat thoris whil grounddtoth ral world. Ths Enlightnmnt-rainwr dsrid as unorruptdmathmatiiansth formalitis, sptals, rivalris andyof soity; and thir ontmporarisjalousisiographrs praisd th “simpliity” andandof thir pursuit. Ths dpitions,“purity”ontrast, th prsptiv that mathmatisInonrnd with th rlationships twnisonpts may hav attratd a vryastratprsonality. Historian of mathmatisdiffrntAlxandr xamins how, at th ginningAmirth 19th ntury, th Enlightnmnt’sofmn” gav way to a nw gnration“naturalofmathmatiians:prsptiv of mathmatis, AlxandrThishas haratrizd th disiplin sin thargus,of th 19th ntury: it has “lgitimizdginningallowd for a nw typ of mathmatialandimpratial, slf-rfrntial,knowldg:to worldly lif, and judgd only y itsirrlvantTOK w oftn sarh for disagrmnts toIna light on diffrnt prsptivs. Howvr,shinaout mathmatis appars to maksomthingasir to rsolv; w do not oftndisagrmntsof mathmatial ontrovrsis. Could it harmathmatis has lss sop for disagrmnt?thatthat th prati of mathmatis, or thOrof mathmatiians, is partiularlyommunityin avoiding, silning or rsolvingfftivprsptivs? Kp ths qustions inonflitingthorms and onjturs inManyarry th nams of individuals.mathmatisDo individuals hav an aility to mak1.disproportionat impat in th fildamathmatis as ompard to othrofdisiplins?How has th rol and influn of2.in mathmatis hangd ovrindividualsIs th fild of mathmatis lss or mor3.ompard to othr disiplins?ollaorativHas mathmatis n disproportionatly4.y partiular ulturs? If so, dosshapdII. PerspectivesII. PerspectivesII.2 Individuals and the mathematicalcommunitypurity, its truth, and its auty” (2010).mind as you rad through Box 11.1.For discussionThe humans of mathematicsturn, shapd soial idals and idas aoutinand mathmatiians, with long-mathmatislasting ffts.tim?it ontinu to ?mathematics . . . was a wondrous alternativereality governed solely by the eternal laws of puremathematics, unsullied by the crass realities ofthe world around us. Unlike their elders, the newmathematicians were not so much interested inacquiring new and useful results as they werefocused on the internal architecture of mathematicsitself, its interconnections, and the precise meaningof its statements. Mathematics, for them, was itsown self-contained world and could be judged bymathematical standards alone.(Alexander 2010)327
II. Perspectivesonpts an appar so stal,Mathmatialand timlss that thy sldom attratastratattntion of non-spialists or prompt soialthAn important xampl to thontrovrsis.involvs mathmatial indivisils,ontrarywr on at th hart of intlltual,whihand rligious lif in 17th-nturypolitialAording to Alxandr, th rsult ofEurop.ontrovrsy around thm “hlpd opn ththto a nw and dynami sin, to rligiouswayand to politial frdoms unknowntolration,human history”(2014).inwr mathmatial indivisils a mattrWhyonrn to politial and rligious authoritisofwll as to mathmatiians and sintists?asfirst xamin what w man y anLt’sImagin a straight lin that isindivisil.of tiny littl lins, so small that thyomposd dividd. Prhaps you might hav aannotlittl pis on this lin, in whih asillionsiz of ah is 1 illionth of th ig lin.thdividing it into illions is aritrary, youButdivid it into two and thn divid it intoouldin whih as you would hav twoillions,pis. You ould oviously divid thillionin many diffrnt ways, and hav diffrntlinof indivisils. But ould you arrivnumrsan infinit numr of indivisils? It is aatqustion, aus an infinitonfoundingof tiny pis ould an infinitlynumrlin. Could ah littl pi of th linlongzro siz? If so, how ould thy add uphava positiv magnitud? This prolm is mortoposd y Zno’s paradox.suintlytortois hallngs Ahills to a ra, startingAa 10-mtr had start. Aording to Zno,withditats that Ahills an nvr athlogiBy th tim h has ovrd th 10 mtrs,up.tortois will hav movd a tiny it mor,th4 ntimtrs. By th tim Ahillsprhapsthos 4 ntimtrs, th tortois willovrsadvand a littl furthr, and so on, adhavan infinit numr of finithallng:thus an infinit numr of tims—distans,argus Zno, adds up to an infinitwhih,of tim.amounta short vido xplanation of Zno’sForvisit this link.paradox,trms: KllhrSarhis Zno’s paradox?”“Whatdo w ronil this logi with ourHowAhills oviously dos ath upxprin?th tortois, ut to solv th paradox wwithknow what is wrong with th argumnt,mustas is widly livd, th prolm lis inIf,laim that th sum of an infinit numrZno’sthings is an infinit thing, thn th solutionoffairly straightforward: th alulus ofissris shows us that th sum ndonvrgnt infinit. Think aout utting a pi ofnotinto infinitly small pis of string: thstringof ths is still finit, quivalnt to thsum11Box 11.1: A bitter dispute regarding human liberty and the infinitely smallnot just its onlusion.lngth of th original string.Zeno’s paradox and infinityZno’s paradox assrtd that Ahills ould nvrFigure 11.2ath up with th tortois, aus vry tim h got los, thtortois movd a littl furthr, in an innit numr of nitstps. An anint Chins paradox dsris th sam prolm:“a on-foot stik, vry day tak away half of it, in a myriad agsit will not xhaustd” (Frasr 2017). Th paradox is attriutdto philosophrs of th Mohism shool twn 500–200 bce,around th sam priod that Zno was ativ.infinitum. Ahills is fad with an impossil328
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apparnt timlssnss of knowldg in
Th
may lad on to th assumption
mathmatis
mathmatial knowldg is produd in a
that
way. In rality, th prati of mathmatis
timlss
mddd in ultural ontxts and has hangd
is
ovr tim.
markdly
18th-ntury Europ th ida of mathmatis
In
th hiddn languag of th univrs was
as
aptd, and mathmatiians fashiond
widly
as “natural mn” (Alxandr
thmslvs
Prsptivs on mathmatis shift, along
2010).
th norms and valus of mathmatial
with
Rvaling this “hiddn ordr“ in
ommunitis.
mathmatial strutur of rality may attrat
th
partiular kind of mathmatiian: somon al
a
grappl with astrat thoris whil groundd
to
th ral world. Ths Enlightnmnt-ra
in
wr dsrid as unorruptd
mathmatiians
th formalitis, sptals, rivalris and
y
of soity; and thir ontmporaris
jalousis
iographrs praisd th “simpliity” and
and
of thir pursuit. Ths dpitions,
“purity”
ontrast, th prsptiv that mathmatis
In
onrnd with th rlationships twn
is
onpts may hav attratd a vry
astrat
prsonality. Historian of mathmatis
diffrnt
Alxandr xamins how, at th ginning
Amir
th 19th ntury, th Enlightnmnt’s
of
mn” gav way to a nw gnration
“natural
ofmathmatiians:
prsptiv of mathmatis, Alxandr
This
has haratrizd th disiplin sin th
argus,
of th 19th ntury: it has “lgitimizd
ginning
allowd for a nw typ of mathmatial
and
impratial, slf-rfrntial,
knowldg:
to worldly lif, and judgd only y its
irrlvant
TOK w oftn sarh for disagrmnts to
In
a light on diffrnt prsptivs. Howvr,
shin
aout mathmatis appars to mak
somthing
asir to rsolv; w do not oftn
disagrmnts
of mathmatial ontrovrsis. Could it
har
mathmatis has lss sop for disagrmnt?
that
that th prati of mathmatis, or th
Or
of mathmatiians, is partiularly
ommunity
in avoiding, silning or rsolving
fftiv
prsptivs? Kp ths qustions in
onfliting
thorms and onjturs in
Many
arry th nams of individuals.
mathmatis
Do individuals hav an aility to mak
1.
disproportionat impat in th fild
a
mathmatis as ompard to othr
of
disiplins?
How has th rol and influn of
2.
in mathmatis hangd ovr
individuals
Is th fild of mathmatis lss or mor
3.
ompard to othr disiplins?
ollaorativ
Has mathmatis n disproportionatly
4.
y partiular ulturs? If so, dos
shapd
II. Perspectives
II. Perspectives
II.2 Individuals and the mathematical
community
purity, its truth, and its auty” (2010).
mind as you rad through Box 11.1.
For discussion
The humans of mathematics
turn, shapd soial idals and idas aout
in
and mathmatiians, with long-
mathmatis
lasting ffts.
tim?
it ontinu to ?
mathematics . . . was a wondrous alternative
reality governed solely by the eternal laws of pure
mathematics, unsullied by the crass realities of
the world around us. Unlike their elders, the new
mathematicians were not so much interested in
acquiring new and useful results as they were
focused on the internal architecture of mathematics
itself, its interconnections, and the precise meaning
of its statements. Mathematics, for them, was its
own self-contained world and could be judged by
mathematical standards alone.
(Alexander 2010)
327