 Gödel: ‘How did you
receive your revolutionary
insights?’
 Einstein: ‘By raising
questions that children
are told not to ask’.
 Gödel raised questions
that mathematical
logicians and
philosophers weren’t
supposed to ask.
 Peirceanly speaking (in order to
introduce a variation of Gödel):
1. Generality is that which is accepted as
true from the local perspective but
possibly true and possibly false from
the global perspective. A generality is,
then, neither true nor false when
considering both perspectives (i.e. it
may be ‘falsified’ within some future
timespace context). Hence classical
logical principles don’t necessarily
apply.
2. Particularities that are taken as true from
a local perspective are not demonstrably
true from the global perspective. Yet
within that local perspective they are
customarily construed as either true or
false (i.e. classical logical principles tend
to apply).
3. Possibilities stand a chance of
emerging within some future timespace
juncture; yet, as possibilities, some will
contradict others; nevertheless, both
one possibility and another mutually
exclusive possibility can coexist as
comfortable bedfellow. Hence classical
logical principles don’t necessarily
apply here either.
How can these three
apparently unruly
misfits be brought
together?
 Don Quixote sees a threatening ‘giant’.
Sancho Panza sees a ‘windmill’. Who
is right? Both, and neither. They are
both right within their local
perspective; neither is demonstrably
right or wrong within the global
perspective.
 Cervantes’s fictional world is right
within its particular perspective. But
we can’t simply say it’s wrong when
placed within the global, ‘real world’
perspective.
In this light, what,
then, is the nature of
the fiction/‘real world’ distinction
with respect to
Gödel’s proof?
 Jaako Hintikka (2000): Gödel
numbering is like staging a play. The
actors have their normal life outside the
play, but they also have a role in the
fiction in question. Consequently what
they say in the play can be taken in at
least two different ways: either as it
would be understood in her everyday
life, or as a line within the fictive world.
 Likewise, in Gödel numbering one and
the same number string can be taken in
two different ways: either as a
proposition about numbers in their
everyday life as numbers, or as a
statement about the formulas that
those same numbers represent when
they play different characters in their
Gödelian play.
 In both cases we are asked to ‘suspend
our belief in the everyday domain’ and
‘suspend our disbelief in the imaginary
domain’. We must take numbers or
characters to be acting simultaneously
in their natural and their artificial roles.
 In an alternative way of putting this,
Zeno’s paradox holds within the local
imaginary domain, and as long as we
consider Achilles and the Tortoise to
be actors within that domain, the
Tortoise will always win (we suspended
disbelief in the imaginary domain).
However, within our own ‘real world’ we
know that all we have to do is put the
chimerical Achilles in hot pursuit of the
Tortoise and it is a no contest (we
suspended belief in our everyday
domain by including an imaginary
situation).
 Like Zeno’s paradox, Gödel’s sentence
says of itself, ‘I am undecidable’, and
within that domain there is no
decidability; however, we can add another
theorem to that domain and thus render it
decidable; hence the ‘paradox’, testifying
to its ‘inconsistency’, was merely
‘incomplete’, and it was we who
imperiously ‘completed’ it; but now, the
domain finds itself caught up in the same
dilemma anew.
 Quixote says ‘Giant!’; Sancho says
‘Windmill!’, and within the domain Cervantes
offers us there is no solution. However we
‘suspend disbelief’ in the fictive world and
give Quixote the benefit of the doubt, or, we
place stock in Sancho’s view. Yet, within the
fictive domain undecidability ruled. Outside
that domain, we would ordinarily opt for
Sancho’s reality and Quixote’s dementia; yet,
interpreted from another perspective, by
‘suspending our ordinary belief in our levelheaded reason and logic’, we become aware
of a little bit of Sancho and a little bit of
Quixote in all of us, so the paradox remains.
How do we resolve
this apparent
dilemma?
 The Gödel sentence is like a statement
made by a character in a play about
himself and at the same time about real
people. Suppose Clint Eastwood says,
playing a role in a movie, “In this
situation even Clint Eastwood couldn’t
keep a straight face”. This statement is
part of the movie, not of real life. Yet
the meaning and the truth of that
statement have to be judged by
reference to real persons, in this case
Clint Eastwood, the actor.
 The truth or falsity of the fictivecharacter/real-life-person can only be
decided by examining when the actual
real-life person would smile in a certain
situation, not when the fictive-character
in a movie might break out in a grin.
Likewise, the Gödelian sentence is a
part of a self-referential play, and its
truth or provability must be judged as if
it were an ordinary arithmetical
statement.
 Peirceanly speaking, once again:
1. A sentence may be considered true
within a local perspective (i.e. ‘Atoms are
indivisible spheres’), but its truth can’t
be absolutely provable. From the global
perspective it is neither true nor false, for
at some future timespace juncture, there
is no knowing which of the range of
possible alternate sentences may be
considered a more viable candidate (i.e.
‘Atoms are solid indivisible spheres, they
are vortices, they are like a solar system,
they are a probability amplitude, etc.).
2. A particular sentence, strictly within the
norms of a local context, can be
considered either true or false, with the
stipulation that the sentence may turn
out to be either inconsistent or
incomplete or both within that local
context.
3. The range of pure, unactualized
possibilities holds the promise that,
however inconsistent and/or incomplete
the sentences within a particular context
that have been given generality status
may be, some alternate sentence will be
available for selection at some future
timespace context.