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

TOK w rgularly ngag with th onpts
In
rtainty, truth and ojtivity. Mathmatis, it
of
laimd, has a spial rlationship with ths
is
ing al to ahiv thm through th
onpts,
of mathmatial proof. A proof is tru
pursuit
if it is always tru. This standard of rigour
only
rtainty, arrivd at through rasoning
and
from mpirial argumnts, is oftn said
dtahd
proofs ar uilt using logial
Mathmatial
from thorms, whih ar prviously
infrns
onjturs, and axioms, whih ar ithr
provn
or assumd statmnts usd as th
slf-vidnt
point. Th ovious “wak spot”, if thr is
starting
in mathmatial proof is in th axioms usd.
on,
1. Consid mathmatial axioms.
(a) On what asis an axioms hosn?
(b) Who hooss axioms, using what
Is all of mathmatis prdiatd on our
2.
hoi of axioms?
prdssors’
If our prdssors had hosn a diffrnt
3.
of axioms, would w hav a diffrnt
st
Do mathmatial ralists and anti-ralists
4.
diffrnt answrs to qustion 3?
hav
5. (a) Do mathmatiians hav faith in
To what xtnt dos faith play a rol
(b)
mathmatis?
in
all th stps in th proof ar logially sound,
If
oms a thorm. Mathmatial proof is
it
from th vidn-asd proof in law,
diffrnt
xampl, in that on provn a thorm is
for
and final. Th mathmatial ommunity
onlusiv
hks th stps and judgs th ompltnss,
thn
and originality of th proof. That last
auray
is important: for all th rigour and rtainty
point
to mathmatial proofs, thorms ar
attriutd
in a soial pross that is not infallil.
vrifid
mathematicians prove theorems . . . . The
When
of the arguments is determined by
correctness
scrutiny of other mathematicians, in informal
the
in lectures, or in journals … the means
discussions,
which mathematical results are veried is
by
a social process and is thus fallible.
essentially
[T]he history of mathematics has many stories
…
false results that went undetected for a long
about
In addition, … important theorems have
time.
such long and complicated proofs that very
required
people have the time, energy and necessary
few
to check through them. And some proofs
background
extensive computer code to, for example,
contain
a lot of cases that would be infeasible to check
check
hand. How can mathematicians be sure that such
by
are reliable? To get around these problems,
proofs
scientists and mathematicians began
computer
develop the eld of formal proof. A formal proof
to
one in which every logical inference has been
is
all the way back to the fundamental axioms
checked
mathematics. … [S]uch proofs are so long and
of
that it would be impossible to have
cumbersome
checked by human mathematicians. But now
them
can get “computer proof assistants” to do the
one
this nxt stion w xamin how th us of
In
is affting mathmatis.
omputrs
III. Methods and tools
III. Methods and tools
I I I . M E T H O D S A N D T O O LS
III.1 Once proven: The eternal truth
of mathematics
to a hallmark of mathmatis.
For discussion
A closer look at axioms
ritria?
mathmatis today?
checking.
(American Mathematical Society 2008)
axioms?
337