Theory of Knowledge - Course Companion for Students Marija Uzunova Dang Arvin Singh Uzunov Dang

II. Perspectives
onpts an appar so stal,
Mathmatial
and timlss that thy sldom attrat
astrat
attntion of non-spialists or prompt soial
th
An important xampl to th
ontrovrsis.
involvs mathmatial indivisils,
ontrary
wr on at th hart of intlltual,
whih
and rligious lif in 17th-ntury
politial
Aording to Alxandr, th rsult of
Europ.
ontrovrsy around thm “hlpd opn th
th
to a nw and dynami sin, to rligious
way
and to politial frdoms unknown
tolration,
human history”(2014).
in
wr mathmatial indivisils a mattr
Why
onrn to politial and rligious authoritis
of
wll as to mathmatiians and sintists?
as
first xamin what w man y an
Lt’s
Imagin a straight lin that is
indivisil.
of tiny littl lins, so small that thy
omposd
dividd. Prhaps you might hav a
annot
littl pis on this lin, in whih as
illion
siz of ah is 1 illionth of th ig lin.
th
dividing it into illions is aritrary, you
But
divid it into two and thn divid it into
ould
in whih as you would hav two
illions,
pis. You ould oviously divid th
illion
in many diffrnt ways, and hav diffrnt
lin
of indivisils. But ould you arriv
numrs
an infinit numr of indivisils? It is a
at
qustion, aus an infinit
onfounding
of tiny pis ould an infinitly
numr
lin. Could ah littl pi of th lin
long
zro siz? If so, how ould thy add up
hav
a positiv magnitud? This prolm is mor
to
posd y Zno’s paradox.
suintly
tortois hallngs Ahills to a ra, starting
A
a 10-mtr had start. Aording to Zno,
with
ditats that Ahills an nvr ath
logi
By th tim h has ovrd th 10 mtrs,
up.
tortois will hav movd a tiny it mor,
th
4 ntimtrs. By th tim Ahills
prhaps
thos 4 ntimtrs, th tortois will
ovrs
advand a littl furthr, and so on, ad
hav
an infinit numr of finit
hallng:
thus an infinit numr of tims—
distans,
argus Zno, adds up to an infinit
whih,
of tim.
amount
a short vido xplanation of Zno’s
For
visit this link.
paradox,
trms: Kllhr
Sarh
is Zno’s paradox?”
“What
do w ronil this logi with our
How
Ahills oviously dos ath up
xprin?
th tortois, ut to solv th paradox w
with
know what is wrong with th argumnt,
must
as is widly livd, th prolm lis in
If,
laim that th sum of an infinit numr
Zno’s
things is an infinit thing, thn th solution
of
fairly straightforward: th alulus of
is
sris shows us that th sum nd
onvrgnt
infinit. Think aout utting a pi of
not
into infinitly small pis of string: th
string
of ths is still finit, quivalnt to th
sum
11
Box 11.1: A bitter dispute regarding human liberty and the infinitely small
not just its onlusion.
lngth of th original string.
Zeno’s paradox and infinity
Zno’s paradox assrtd that Ahills ould nvr
Figure 11.2
ath up with th tortois, aus vry tim h got los, th
tortois movd a littl furthr, in an innit numr of nit
stps. An anint Chins paradox dsris th sam prolm:
“a on-foot stik, vry day tak away half of it, in a myriad ags
it will not xhaustd” (Frasr 2017). Th paradox is attriutd
to philosophrs of th Mohism shool twn 500–200 bce,
around th sam priod that Zno was ativ.
infinitum. Ahills is fad with an impossil
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