The immediate issues raised by

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Oct 10, 2013 (4 years and 1 month ago)

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The immediate issues raised by
Gödel’s Theorems


Gödel’s

Incompleteness Theorems dealt a devastating blow to the generally held
belief that every branch of mathematics could be fully axiomatised, that is provided
with a

base


of axioms
, accepted without p
roof,

from which all
other
propositions of
the system can be derived.
Given

that the
complete
axiomatisation of geometry has
been established,
it was fairly natural to assume the same could be achieved in other
fields,
and
this is certainly

a

desirable

ai
m
as it renders all propositions in the system
true s
imply
if the

relatively few

axioms are
.

Gödel
, through an inventive method of
proof using the mapping of meta
-
mathematical statements to statements within
mathematics itself, showed that the Euclidean
paradigm of axiomatisation could never
be realised for important branches of mathematics, notably set theory and arithmetic.
He further showed tha
t it is impossible to establish

t
he internal consistency of
arithmetic (or similarly complex deductive syst
ems
) without appeal to principle
s
whose internal consistency is just as open to doubt as those being investigated.

As
well as ramifications for the
foundations

of mathematics, these theorems give rise to
other philosophical issues such as Lucas’ argument that

the human brain is superior to
any computer, and provide some difficulty for Dummett’s theory of meaning.


Given an axiomatic base for a deductive system, there is a further need to establish the
consistency of the axioms laid down. This
may seem

simple
in

the case of Euclidean
geometry

whose axioms are ostensibly true of space, hence consistent, as no mutually
inconsistent statements could be simultaneously true of any set of objects.
The
discovery of meaningful geometries which adopt axioms different to

and incompatible
with Euclid’s parallel postulate, leads to the question of whether these
further
systems
are consistent. They could be shown to be so by finding a model, that is, some
interpretation of the objects and relations of the formal system by re
al world objects
and relations, on which all the axioms hold true. For example, Riemannian geometry
can be modelled by a Euclidean sphere, so that ‘plane’ in Riemannian geometry is
interpreted as the surface of a Euclidean sphere, ‘points’ as points on the

sphere, etc
Then
a demonstration of
Riemannian geometry
’s consistency

reduces to the
demonstration of Euclidean geometry’s consistency. However a problem remains in
this latter case, for the model we have accepted, ordinary space, is an infinite one, of
w
hich we can only observe a finite portion and then reason inductively to the
universal case.
In fact, the foundations of most significant areas of mathematics can
only be mirrored in infinite models, thus seeking a proof of consistency in this way
does not

provide
an indubitable

answer.


Hilbert

sought an ‘absolute’
proof
of the consistency of arithmetic, by first com
pletely
formalising the system,
reducing it to meaningless marks and patterns devoid of
content, enabling the analysis of a finite number of s
tructural features of expressions
in order to sho
w no
contradictory formulas
could be derived
.

Using this ‘finitistic’
method of proof it can be shown that the sentential calculus is consistent and
complete, as is a system of arithmetic of cardinals permit
ting addition but not
multiplication.
However
,

Gödel’s

theorem shows that Hilbert’s aim of producing a
similar proof for a
ll of arithmetic is impossible, by removing the possibility of casting
it in axiomatic form
,

and casting doubt on the possibility of a
ny finitary
characterisation of the system.


The proof of Gödel’s theorem
s depend
s

on

mapping meta
-
mathematical statements
about the system,
for example “arithmetic is consistent”
, to actual mathematical
statements in the system itself.
Then we can specif
y an arithmetical statement, s,
corresponding to the meta
-
arithmetical statement that “s cannot be proved”. We can
co
nvince ourselves that s is true;

for suppose otherwise, then s could be proved, which
contradicts the assumption of its falsehood. So s is
true, hence the meta
-
mathematical
statement “s cannot be proved” is true.
A statement of the first theorem can be given
as follows:

F
or any consistent formal theory that proves basic arithmetical truths, it is possible to
construct an arithmetical stateme
nt that is true but not provable in the theory. That is,
any consistent theory of a certain expressive strength is incomplete.



A lot of the proof that s is true
can

be expressed within the system being examined,
but finally we use the fact of the axioms’

consistency to arrive at
out conclusion
. As
Moore says, of any sound axiomatic base
, “our very recognition of its soundness
propels us beyond it”.


So any axiomatic base we can provide for arithmetic, even if
augmented

with
indefinitely many consistent a
xioms, will never constitute a foundation for all the
truths of the subject. This does not refute, as some have supposed
, the antirealist
doctrine that a statement’s truth consists in its provability
-

it does not entail

the
existence of a generally non
-
pro
vable

true

statement beyond the reach of all axiomatic
bases,
simply the existence
for each

base

of a true statement than cannot be derived
from that base.


Lucas uses this result to argue for the superiority of the human mind over any
machine. The reasoni
ng is that computers operate by decision procedures, and
Gödel’s

theorem shows
that
no such decision procedure exists for arithmetical truth.
At most a computer could only determine the truth values of statements within a
certain limited range. We humans,
however, could always recognise the truth of a
statement outside the range, as outlined above.

What is it that enables us to
go beyond
the computer in this respect, and recognise a further truth? Moore describes it as a
kind of self
-
conscious reflection, f
or we reflect on the truth of the given axioms and
hence accept the consistency of the system.



Lucas’ argument is not entirely compelling. For one thing, we may conceive of a
machine that could mimic human self
-
consciousness in recognising the consistenc
y of
any principles it simultaneously accepts, and thus be capable of deriving the same
truths as we do.

Furthermore, it may be that the processes by which we determine
truth can be characterised by some more complex decision procedure which we
ourselves c
annot grasp!


A further motivation for axiomatisation is the need to assimilate all truths of a subject
in order to grasp the concepts employed there.
Our inability to completely axiomatise
arithmetic can be seen to pose a challenge to Dummett’s

view
, orig
inally
Wittgenstein’s, that “meaning is use”. In the case of mathematics, this
is stating

tha
t
the meaning of an expression

is equivalent to the ways in which it

figures in the truths
of
a

formal system.
Then we cannot grasp this meaning without some finit
e
characterisation of these truths, which
Gödel’s

theorem seems to show cannot be
provided.

The problem is not irresolvable,

for w
e could still do this by a means other
than axiom
atisation;

Moore suggests the supplementation of an incomplete base by
a
gene
ral instruction
to the effect that
whatever principles we accept are
jointly
consistent.
But even without being able to assimilate
all

truths of arithmetic, must this
theory of meaning be abandoned? Any meaning can be described as possessing a kind
of infi
nitude, since an expression can be used in infinitely many contexts, and then
surely to require a finite description of that use is too demanding. We may indeed
grasp the meaning of

a concept through its use, but we never see the infinite range of
its uses
.


Axiomatisation remains the most desirable
approach to
mathematics;

however
,

Gödel’s second theorem

that the consistency of a formal system cannot be proved
within the system itself means we
often
have no recourse but appeal to
intuition in
order to sati
sfy ourselves a base is consistent
.

If we examine our reasons for
supposing the Zermelo
-
Frankel axioms of set theory are consistent, it is because we
consider the axioms to be accurate statements about sets, but this is simply an appeal
to our fallible in
tuition of sets.

In geometry it
is by extrapolating from the finite to the
infinite that we verify the axioms as true of space and hence consistent.
Gödel’s

results are some of the most impressive and significant proofs of the 20
th

century,
dealing a
heavy

blow to the pursuit of foundation
s

for mathematics.

Whether or not
they allow us to conclude that the mathematical ability of the human mind necessarily
outstrips that of a machine, they certainly
seem to illustrate the creativity of human
intellect
-

whil
e
any bit of reasoning can be codified into a set of rules, there will
always be further exercises of reason which go beyond the rules and yet are evidently
reasonable and right.