Theory of Knowledge - Course Companion for Students Marija Uzunova Dang Arvin Singh Uzunov Dang
is a prime, not because we think so, or because317minds are shaped in one way rather than another,ourbecause it is, because mathematical reality isbutthat way.builtHardy’s prsptiv shapd y his ulturalIsor is it univrsal? Blow w xplorontxt,aout th origins of mathmatis. Dos itlaimsin th world outsid of human xprin, orxistit a produt of th human mind? W will larnisrogniz th assumptions that aompanytolaims, and xplor th impliations of thsthsprsptivs towards mathmatis.diffrntmany physical systems have mathematical…the segmented arrangements inrepresentations:pine cones and pineapples (Fibonaccisunowers,the curve of nautilus shells, elephantnumbers);and rams horns (logarithmic spiral); … atoms,tusksand galaxies, which all now have powerfulstarsdescriptors; even the cosmos as a whole,mathematicalrepresented by the equations of general relativity …?nowdoes the real world actualise maths at all? …Whyphysicists still explain this by some form ofMostPlatonism, which in its oldest formphilosophicalthat the universe is moulded by mathematicalsaysralism assrts that mathmatisMathmatialindpndntly of th human mind, thatxistsis disovrd, and not ratd, y humans.ita form of ralism, holds that thPlatonism,is omposd of astrat and trnalunivrsntitis”. This has n among“mathmatialmost datd topis in th philosophy ofthIf you ar a mathmatial ralist,mathmatis.unrasonal fftivnss of mathmatisthnot sm that unrasonal aftr all,mightth othr hand, anti-ralist prsptivsOnthat mathmatial ntitis do not xist,arguthat mathmatis is a languag humansandinvntd to talk aout quantity, strutur,havand hang. Th philosophr Hughspafamously dsrid mathmatis asLhman“thortial jui xtrator" that is usful foramaning out of things, ut has nosquzingargu that ths anti-ralist prsptivs,Othrsas nominalism, might appaling utsuhnot quit mak sns aus th truthsdoar rvald to us through mathmatiswhihfat xistd long for humans startd tointhm or had a languag with whihundrstandof th omplling anti-ralist argumnts isOnon th qustion of why humans ar al toasdastrat mathmatis in th first pla.“know”would w know th diffrn twnHowratd y th human mind andmathmatisindpndnt of it? Within TOKmathmatisnd not solv ths riddls. But anyonwalulating and modlling ral-worldmasuring,mathmatially should awar ofphnomnalaims and assumptions aout th natur ofthwhy dos any of this mattr?Ultimatly,xampls w look at nxt suggst thatThprsptivs influn attituds in ththsommunity and, thrfor, thmathmatisII. PerspectivesII. PerspectivesI I . P E R S P E C T I V E SII.1 Was mathematics discoveredor invented?(Hardy quoted in Tarlach 2014)aus th univrs is “mathmatial”.ontnt in itslf.to dsri thm.rality thy rst on.relationships which precede the material world.(Wertheim 2017)325
II. Perspectivesof mathmatial knowldg. Ourprodutionar shapd y, and in turn shap,prsptivsmathmatial ommunity around us, andthmathmatis intrats with th widrhowThrfor, it is worth xamining thworld.of having a divrsity of prsptivsvaluand mathmatiian JamsPhilosophrinvits us to onsidr whthr thrFranklinntitis of mathmatial natur at a timwrhumans thought aout mathmatisforhad a languag with whih to dsriand(Franklin 2014). H quikly suggsts thatit.thr wr many suh proprtis, andinddis among th ttr xampls. Thsymmtryilatral symmtry of animalsapproximatth irular symmtry of trs hav nandas xampls for th way mathmatisusdin th world indpndntmanifststhis vido, mathmatiian Marus du SautoyInsymmtry not in th imagind worldxplorshumans, ut in mid-13th ntury Islamiforart.gomtritrms: SautoySarhrality’s riddlSymmtry,taks us to th Alhamra in GranadaSautoyth south of Spain. Wath th vido andinmathmatis is an invntion, who gtsIffor it, and why do w tah it as if itrditunivrsal? If it is disovrd, has it nwrindpndntly, through diffrntdisovrdy various ulturs throughoutmthods,How hav diffrnt popls diffrntlyhistory?w know that th Moorish artists paintdToday17 mathmatially possil symmtris on thallilings and floors of th Alhamra. Thywalls,this six nturis for humans had provddid17 is th maximum numr of symmtristhata two-dimnsional plan.onIf two ulturs arriv at a mathmatial1.suh as symmtry sparatly, usingonptmthods, how dos this afft yourdiffrntof whthr mathmatis is disovrd,viwWhat might othr xplanations for th2.dsrid in qustion 1?oinidnConsidr th pattrns in th symmtris3.in th Alhamra.sn(a) In what sns ar th pattrns “ral”?(b) What vidn is thr for this?11disovrd mathmatis?inmathmatis.For discussionWas symmetry there before wefound it?ofhumans.as opposd to ing invntd?onsidr th qustions low.Making connectionsthe theme “Knowledge and religion”. How do artistic,religious and mathematical knowledge practicesKnowledge in mathematics, art and religioninteract with one another in this example? WhatThe example of Islamic geometrical art placeschallenges or opportunities arise when they intersect?mathematics at an intersection with AOK “The arts” and326
- Page 281 and 282: IV. Ethicsin Chaptr 5. Grgory Young
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is a prime, not because we think so, or because
317
minds are shaped in one way rather than another,
our
because it is, because mathematical reality is
but
that way.
built
Hardy’s prsptiv shapd y his ultural
Is
or is it univrsal? Blow w xplor
ontxt,
aout th origins of mathmatis. Dos it
laims
in th world outsid of human xprin, or
xist
it a produt of th human mind? W will larn
is
rogniz th assumptions that aompany
to
laims, and xplor th impliations of ths
ths
prsptivs towards mathmatis.
diffrnt
many physical systems have mathematical
…
the segmented arrangements in
representations:
pine cones and pineapples (Fibonacci
sunowers,
the curve of nautilus shells, elephant
numbers);
and rams horns (logarithmic spiral); … atoms,
tusks
and galaxies, which all now have powerful
stars
descriptors; even the cosmos as a whole,
mathematical
represented by the equations of general relativity …?
now
does the real world actualise maths at all? …
Why
physicists still explain this by some form of
Most
Platonism, which in its oldest form
philosophical
that the universe is moulded by mathematical
says
ralism assrts that mathmatis
Mathmatial
indpndntly of th human mind, that
xists
is disovrd, and not ratd, y humans.
it
a form of ralism, holds that th
Platonism,
is omposd of astrat and trnal
univrs
ntitis”. This has n among
“mathmatial
most datd topis in th philosophy of
th
If you ar a mathmatial ralist,
mathmatis.
unrasonal fftivnss of mathmatis
th
not sm that unrasonal aftr all,
might
th othr hand, anti-ralist prsptivs
On
that mathmatial ntitis do not xist,
argu
that mathmatis is a languag humans
and
invntd to talk aout quantity, strutur,
hav
and hang. Th philosophr Hugh
spa
famously dsrid mathmatis as
Lhman
“thortial jui xtrator" that is usful for
a
maning out of things, ut has no
squzing
argu that ths anti-ralist prsptivs,
Othrs
as nominalism, might appaling ut
suh
not quit mak sns aus th truths
do
ar rvald to us through mathmatis
whih
fat xistd long for humans startd to
in
thm or had a languag with whih
undrstand
of th omplling anti-ralist argumnts is
On
on th qustion of why humans ar al to
asd
astrat mathmatis in th first pla.
“know”
would w know th diffrn twn
How
ratd y th human mind and
mathmatis
indpndnt of it? Within TOK
mathmatis
nd not solv ths riddls. But anyon
w
alulating and modlling ral-world
masuring,
mathmatially should awar of
phnomna
laims and assumptions aout th natur of
th
why dos any of this mattr?
Ultimatly,
xampls w look at nxt suggst that
Th
prsptivs influn attituds in th
ths
ommunity and, thrfor, th
mathmatis
II. Perspectives
II. Perspectives
I I . P E R S P E C T I V E S
II.1 Was mathematics discovered
or invented?
(Hardy quoted in Tarlach 2014)
aus th univrs is “mathmatial”.
ontnt in itslf.
to dsri thm.
rality thy rst on.
relationships which precede the material world.
(Wertheim 2017)
325