Theory of Knowledge - Course Companion for Students Marija Uzunova Dang Arvin Singh Uzunov Dang
vidn for this unrasonal, vnAsfftivnss of mathmatis,miraulous,hav givn xampls of astratphilosophrsthat somhow om usful inonjtursappliations. For xampl, omplxral-worldwr long onsidrd to irrlvantnumrsrmovd from matrial rality, ut now smandto quantum mhanis.nssarylaw of gravitation is anothr xampl;Thonivd to xplain falling odisoriginallyEarth, it was xtndd on th asis of “vryonosrvations” to dsri th motionsantyplants, whr it “provd aurat yondofrasonal xptations” (Wignr 1960).allth 1970s dvlopmnts in ryptographyInyou might know as th mathmatis(whihWhatsApp nryption thnology andhinddrw on numr thory, andryptourrnis)fundamntal thorm of arithmti, whihthn onsidrd for 200 yars to amonghad“purst” of pur mathmatis for havingthral world valu. Suh xampls onrnnojust mathmatis’ fftivnss, ut also thnottwn dfining what is applid andtnsionis astrat mathmatis. On th on hand,whatHardy has said that pur mathmatis isG.H.usful than applid”, aus it trains“morthniqu so wll (Hardy 1940).mathmatialth othr hand, w hav Loahvsky’sOnthat thr is no suh thing asargumntmathmatis, aus all mathmatis“astrat”somday applid to th ral worldwillwll-known ritiqu of Wignr’s argumnt,Amathmatiian Rihard Hamming (1980),yfour “partial xplanations” to arguprovidsth fftivnss of mathmatis in ththatsins is nithr miraulous nornaturalunrasonal.We see what we look for: “muh of what w•oms from th glasss w put on” wrotsWe create the kind of mathematics•look for: for xampl, in fforts towephysial fors, physiistsundrstandtrid salars, thn vtors, and finallyfirsttnsors. And Nwton's thory ofinvntdwas rplad y th nwr, ttrgravityof gnral rlativity. Instad of amodlfftivnss, w s hr trialmiraulousMathematics and natural science address•a part of human existence, answeringonlyfew problems: this,comparativelyand politial philosophy, forasthtisar rtainly not fftivlyxampl,y mathmatis. Chaptr 8xplaindhow mathmatis may hav nxplorstoo far in th human sins.taknEvolution has primed humans to think•our mathmatial ailitymathematically:not a oinidn, w hav survivdisof it. This argumnt is on ofausI. ScopeI. Scope(Loahvsky quotd in dPillis 2002).(1980). Th xampls w find arisHammingth mathmatial tools usd, in a slf-fromrfrntial logi.Statu of Galilo Galili in Florn, ItalyFigure 11.1and rror.survival ias.323
I. Scopeof th arly xampls of this “unrasonalMostwr appliations of mathmatisfftivnss”physis. As th mathmatiian Isral Glfandinaov, it would misguidd to laim thatquipshas n miraulously fftiv inmathmatissins. That said, Glfand am on ofothrpionrs of iomathmatis. Towards th ndthth 20th ntury iologists wr systmatiallyofxtnsivly applying omplx mathmatisanda varity of filds. If you hav th intrst, doinrsarh on stohasti modlling of nzymsomswarm intllign and spatialdynamis,If mathmatis is th languag in whih1.xprsss itslf, is it ttrnaturas a mthod in th naturaldsridrathr than a ody of knowldgsins2. Whn onsidring knowing mathmatis:What vidn suggsts thr is mor(a)mathmatis than mthod?toIs knowing mathmatis knowing(b)to prform mathmatialhowor is thr mor to thisalulations,of knowldg?odyfar, our disussion has ntrtaind thThusthat mathmatis is th languagpossiilityth univrs; and that through it w gainofinto th strutur of th univrs. Thinsightsto this laim, suggstd y Hammingountrothrs, is that mathmatis is only our wayanddsriing and undrstanding th univrs,ofthr is nothing inhrntly mathmatialthatth univrs, or univrsal aoutaoutmathmatis.physiist Drk Aott (2013) hasThthat th supposd "unrasonalsuggstdof mathmatis is an illusionfftivnss"y human timsals; that w livinflunddi so quikly that th univrs appars toandgovrnd y mathmatial laws ut may notdisours aout mathmatis’ “unrasonalThisraiss two rlatd qustions:fftivnss”mathmatis is disovrd or invntd,whthrwhthr a numrial sns is iologiallyandthis xris to not your intuitionsUsmathmatis. Writ an nding toaoutfollowing sntn: “Mathmatialthis diffrnt from othr typs ofknowldgyou do this xris as a lass, olltIfompltd sntns andvryon’sth rang of viws and lifs. Thndisussth sntns into sts of laims andorganiz11There is only one thing which is more unreasonablethan the unreasonable eectiveness of mathematicsin physics, and this is the unreasonableineectiveness of mathematics in biology.(Gelfand quoted in Borovik 2018)Mathematics is biology’s next microscope, onlybetter; biology is mathematics’ next physics,only better.(Cohen 2004)atually so.vrsus ulturally ndowd.Practising skills: Constructingknowledge claimsmodlling of nural ntworks.For discussionWhat is mathematics about?knowldg in that ….”in itslf—and why?ountrlaims.324
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vidn for this unrasonal, vn
As
fftivnss of mathmatis,
miraulous,
hav givn xampls of astrat
philosophrs
that somhow om usful in
onjturs
appliations. For xampl, omplx
ral-world
wr long onsidrd to irrlvant
numrs
rmovd from matrial rality, ut now sm
and
to quantum mhanis.
nssary
law of gravitation is anothr xampl;
Th
onivd to xplain falling odis
originally
Earth, it was xtndd on th asis of “vry
on
osrvations” to dsri th motion
santy
plants, whr it “provd aurat yond
of
rasonal xptations” (Wignr 1960).
all
th 1970s dvlopmnts in ryptography
In
you might know as th mathmatis
(whih
WhatsApp nryption thnology and
hind
drw on numr thory, and
ryptourrnis)
fundamntal thorm of arithmti, whih
th
n onsidrd for 200 yars to among
had
“purst” of pur mathmatis for having
th
ral world valu. Suh xampls onrn
no
just mathmatis’ fftivnss, ut also th
not
twn dfining what is applid and
tnsion
is astrat mathmatis. On th on hand,
what
Hardy has said that pur mathmatis is
G.H.
usful than applid”, aus it trains
“mor
thniqu so wll (Hardy 1940).
mathmatial
th othr hand, w hav Loahvsky’s
On
that thr is no suh thing as
argumnt
mathmatis, aus all mathmatis
“astrat”
somday applid to th ral world
will
wll-known ritiqu of Wignr’s argumnt,
A
mathmatiian Rihard Hamming (1980),
y
four “partial xplanations” to argu
provids
th fftivnss of mathmatis in th
that
sins is nithr miraulous nor
natural
unrasonal.
We see what we look for: “muh of what w
•
oms from th glasss w put on” wrot
s
We create the kind of mathematics
•
look for: for xampl, in fforts to
we
physial fors, physiists
undrstand
trid salars, thn vtors, and finally
first
tnsors. And Nwton's thory of
invntd
was rplad y th nwr, ttr
gravity
of gnral rlativity. Instad of a
modl
fftivnss, w s hr trial
miraulous
Mathematics and natural science address
•
a part of human existence, answering
only
few problems: this,
comparatively
and politial philosophy, for
asthtis
ar rtainly not fftivly
xampl,
y mathmatis. Chaptr 8
xplaind
how mathmatis may hav n
xplors
too far in th human sins.
takn
Evolution has primed humans to think
•
our mathmatial aility
mathematically:
not a oinidn, w hav survivd
is
of it. This argumnt is on of
aus
I. Scope
I. Scope
(Loahvsky quotd in dPillis 2002).
(1980). Th xampls w find aris
Hamming
th mathmatial tools usd, in a slf-
from
rfrntial logi.
Statu of Galilo Galili in Florn, Italy
Figure 11.1
and rror.
survival ias.
323