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

will advans in omputing afft
How
To answr this qustion w nd
mathmatis?
hav a grasp on what mathmatis is, and
to
prisly, what mathmatiians do, and
mor
omputational powr fits with that. W
how
that formal proofs an partiularly
mntiond
vn impossil, for a mathmatiian to
long,
so omputr proof assistants hav om
vrify,
hav usd omputrs not just
Mathmatiians
vrify ut to gnrat proofs for som tim.
to
four-olour thorm ompltd in 1976
Th
11.6) was th first suh proof. It was
(Figur
ompliatd that no human ing ould
so
it without having to trust th softwar.
vrify
xampl is th 1998 proof of th Kplr
Anothr
whih ontaind ovr 3 gigayts of
onjtur,
“Proof y xhaustion” is a trm givn to
data.
whr a omputr has hkd all possil
proofs
Proofs of this natur ar onsidrd
ass.
aus thy an’t vrifid y a
prolmati
prson—som popl disrgard thm on
singl
asis, ut othrs apt thm.
this
thorm provrs (ATPs) ar
Automatd
that an prov mathmatial rsults
programs
using a st of axioms. ATPs an also
logially
millions of simulations in mission-ritial
run
suh as in nular powr plants,
appliations,
nsur a systm is working proprly—a task
to
would prov xtrmly tim-onsuming
that
human ings. ATPs ar hanging how w
for
Yt omputrs solv diffrntly
smathmatis.
humans, who look for graful onntions
from
possiilitis. Computr proofs an
twn
and awkward—as long as thy find a
lumsy
ths ar just th prolms assoiatd
Prhaps
a nw thnology; prhaps mathmatiians
with
omputr sintists may vntually al
and
od ATPs to dlivr autiful proofs that
to
nw insights to thir human dvloprs.
rval
prhaps not: whthr softwar is apal
Or
th fats of intuition and imagination that
of
th st xampls of human
haratriz
whthr ATPs will al to math
of
xd thir human ountrparts, th
or
of dvloping thm is dpning our
pursuit
of what it mans, and taks, to
undrstanding
is also possible to let computers loose to explore
It
on their own, and in some cases they
mathematics
come up with interesting conjectures that went
have
by mathematicians. We may be close to
unnoticed
how computers, rather than humans, would do
seeing
mathematics.
ommon far is, of ours, that on softwar is
A
nough at proving rsults, and in diding
good
rsults to prov, human mathmatiians
whih
om irrlvant. How would you did if
will
some supercomputer … reported a
Suppose
… which was so long and complex that no
proof
could understand it beyond the
mathematician
general terms. Could we have sucient faith
most
computers to accept this result, or would we say
in
the empirical evidence for their reliability is not
that
Tymozko oind th trm “nonsurvyal
Thomas
proof” to dsri proofs that ar
for a human mathmatiian to vrify,
infasil
as th 1979 omputr-assistd proof y
suh
and Hakn of th four-olour thorm.
Appl
argud that mathmatial proofs must
Tymozko
Conviningnss: th proof an prsuad a
•
vrifir of its onlusion.
rational
Survyaility: th proof is assil for
•
y humans.
vrifiation
Formalizaility: th proof uss only logial
•
twn onpts.
rlationships
III. Methods and tools
III. Methods and tools
III.1.1 Mathematical proof in the digital age
mak a mathmatial proof.
usful.
(American Mathematical Society 2008)
this far is misguidd or valid?
III.1.2 Non-surveyable proof
enough?
(Tymoczko 1979)
proof, thy hav don thir jo.
mt thr ritria.
mathmatis is an opn qustion. Rgardlss
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