Gödel: ‘How did you Gödel: ‘How did you receive your revolutionary receive your revolutionary insights?’insights?’

Einstein: ‘By raising Einstein: ‘By raising questions that children questions that children are told not to ask’.are told not to ask’.

Gödel raised questions Gödel raised questions that mathematical that mathematical logicians and logicians and philosophers weren’t philosophers weren’t supposed to ask.supposed to ask.

Peirceanly speaking (in order to introduce a Peirceanly speaking (in order to introduce a variation of Gödel):variation of Gödel):

1. 1. GeneralityGenerality is that which is accepted as true is that which is accepted as true from the local perspective but possibly true from the local perspective but possibly true and possibly false from the global and possibly false from the global perspective. A generality is, then, perspective. A generality is, then, neitherneither true true nornor false when considering both perspectives false when considering both perspectives (i.e. it may be ‘falsified’ within some future (i.e. it may be ‘falsified’ within some future timespace context). Hence classical logical timespace context). Hence classical logical principles don’t necessarily apply.principles don’t necessarily apply.

2. 2. ParticularitiesParticularities that are taken as true from that are taken as true from a local perspective are not demonstrably a local perspective are not demonstrably true from the global perspective. Yet true from the global perspective. Yet within that local perspective they are within that local perspective they are customarily construed as customarily construed as eithereither true true oror false (i.e. classical logical principles tend false (i.e. classical logical principles tend to apply).to apply).

3. 3. PossibilitiesPossibilities stand a chance of stand a chance of emerging within some future timespace emerging within some future timespace juncture; yet, as possibilities, some will juncture; yet, as possibilities, some will contradict others; nevertheless, contradict others; nevertheless, bothboth one possibility one possibility andand another mutually another mutually exclusive possibility can coexist as exclusive possibility can coexist as comfortable bedfellow. Hence classical comfortable bedfellow. Hence classical logical principles don’t necessarily logical principles don’t necessarily apply here either.apply here either.

How can these three How can these three apparently unruly apparently unruly misfits be brought misfits be brought together?together?

Don Quixote sees a threatening ‘giant’. Don Quixote sees a threatening ‘giant’. Sancho Panza sees a ‘windmill’. Who is Sancho Panza sees a ‘windmill’. Who is right? right? BothBoth, and , and neitherneither. They are . They are bothboth right within their local perspective; right within their local perspective; neitherneither is is demonstrably right or wrong within the demonstrably right or wrong within the global perspective.global perspective.

Cervantes’s fictional world is right within its Cervantes’s fictional world is right within its particular perspective. But we can’t simply particular perspective. But we can’t simply say it’s wrong when placed within the global, say it’s wrong when placed within the global, ‘real world’ perspective.‘real world’ perspective.

In this light, what, then, In this light, what, then, is the nature of the is the nature of the fiction/‘real -world’ fiction/‘real -world’ distinction with respect distinction with respect to Gödel’s proof?to Gödel’s proof?

Jaako Hintikka (2000): Gödel Jaako Hintikka (2000): Gödel numbering is like staging a play. The numbering is like staging a play. The actors have their normal life outside the actors have their normal life outside the play, but they also have a role in the play, but they also have a role in the fiction in question. Consequently what fiction in question. Consequently what they say in the play can be taken in at they say in the play can be taken in at least two different ways: either as it least two different ways: either as it would be understood in her everyday would be understood in her everyday life, or as a line within the fictive world.life, or as a line within the fictive world.

Likewise, in Gödel numbering one and Likewise, in Gödel numbering one and the same number string can be taken in the same number string can be taken in two different ways: either as a two different ways: either as a proposition about numbers in their proposition about numbers in their everyday life as numbers, or as a everyday life as numbers, or as a statement about the formulas that statement about the formulas that those same numbers represent when those same numbers represent when they play different characters in their they play different characters in their Gödelian play.Gödelian play.

In both cases we are asked to ‘suspend In both cases we are asked to ‘suspend our belief in the everyday domain’ and our belief in the everyday domain’ and ‘suspend our disbelief in the imaginary ‘suspend our disbelief in the imaginary domain’. We must take numbers or domain’. We must take numbers or characters to be acting simultaneously characters to be acting simultaneously in their natural and their artificial roles.in their natural and their artificial roles.

In an alternative way of putting this, Zeno’s In an alternative way of putting this, Zeno’s paradox holds within the local imaginary paradox holds within the local imaginary domain, and as long as we consider domain, and as long as we consider Achilles and the Tortoise to be actors within Achilles and the Tortoise to be actors within that domain, the Tortoise will always win that domain, the Tortoise will always win (we suspended disbelief in the imaginary (we suspended disbelief in the imaginary domain). However, within our own ‘real domain). However, within our own ‘real world’ we know that all we have to do is put world’ we know that all we have to do is put the chimerical Achilles in hot pursuit of the the chimerical Achilles in hot pursuit of the Tortoise and it is a no contest (we Tortoise and it is a no contest (we suspended belief in our everyday domain by suspended belief in our everyday domain by including an imaginary situation).including an imaginary situation).

Like Zeno’s paradox, Gödel’s sentence says Like Zeno’s paradox, Gödel’s sentence says of itself, ‘I am undecidable’, and within that of itself, ‘I am undecidable’, and within that domain there is no decidability; however, we domain there is no decidability; however, we can add another theorem to that domain and can add another theorem to that domain and thus render it decidable; hence the thus render it decidable; hence the ‘paradox’, testifying to its ‘inconsistency’, ‘paradox’, testifying to its ‘inconsistency’, was merely ‘incomplete’, and it was we who was merely ‘incomplete’, and it was we who imperiously ‘completed’ it; but now, the imperiously ‘completed’ it; but now, the domain finds itself caught up in the same domain finds itself caught up in the same dilemma anew.dilemma anew.

Quixote says ‘Giant!’; Sancho says ‘Windmill!’, Quixote says ‘Giant!’; Sancho says ‘Windmill!’, and within the domain Cervantes offers us there and within the domain Cervantes offers us there is no solution. However we ‘suspend disbelief’ in is no solution. However we ‘suspend disbelief’ in the fictive world and give Quixote the benefit of the fictive world and give Quixote the benefit of the doubt, or, we place stock in Sancho’s view. the doubt, or, we place stock in Sancho’s view. Yet, within the fictive domain undecidability ruled. Yet, within the fictive domain undecidability ruled. Outside that domain, we would ordinarily opt for Outside that domain, we would ordinarily opt for Sancho’s reality and Quixote’s dementia; yet, Sancho’s reality and Quixote’s dementia; yet, interpreted from another perspective, by interpreted from another perspective, by ‘suspending our ordinary belief in our level-‘suspending our ordinary belief in our level-headed reason and logic’, we become aware of a headed reason and logic’, we become aware of a little bit of Sancho and a little bit of Quixote in all little bit of Sancho and a little bit of Quixote in all of us, so the paradox remains.of us, so the paradox remains.

How do we resolve How do we resolve this apparent this apparent dilemma?dilemma?

The Gödel sentence is like a statement The Gödel sentence is like a statement made by a character in a play about made by a character in a play about himself and at the same time about real himself and at the same time about real people. Suppose Clint Eastwood says, people. Suppose Clint Eastwood says, playing a role in a movie, “In this situation playing a role in a movie, “In this situation even Clint Eastwood couldn’t keep a even Clint Eastwood couldn’t keep a straight face”. This statement is part of the straight face”. This statement is part of the movie, not of real life. Yet the meaning and movie, not of real life. Yet the meaning and the truth of that statement have to be the truth of that statement have to be judged by reference to real persons, in this judged by reference to real persons, in this case Clint Eastwood, the actor. case Clint Eastwood, the actor.

The truth or falsity of the The truth or falsity of the fictive-character/real-life-person can only fictive-character/real-life-person can only be decided by examining when the actual be decided by examining when the actual real-life person would smile in a certain real-life person would smile in a certain situation, not when the fictive-character in situation, not when the fictive-character in a movie might break out in a grin. a movie might break out in a grin. Likewise, the Gödelian sentence is a part of Likewise, the Gödelian sentence is a part of a self-referential play, and its truth or a self-referential play, and its truth or provability must be judged as if it were an provability must be judged as if it were an ordinary arithmetical statement. ordinary arithmetical statement.

Peirceanly speaking, once again:Peirceanly speaking, once again:1. A sentence may be considered true within a local 1. A sentence may be considered true within a local

perspective (i.e. ‘Atoms are indivisible spheres’), perspective (i.e. ‘Atoms are indivisible spheres’), but its truth can’t be absolutely provable. From but its truth can’t be absolutely provable. From the global perspective it is the global perspective it is neitherneither true true nornor false, false, for at some future timespace juncture, there is no for at some future timespace juncture, there is no knowing which of the range of possible alternate knowing which of the range of possible alternate sentences may be considered a more viable sentences may be considered a more viable candidate (i.e. ‘Atoms are solid indivisible candidate (i.e. ‘Atoms are solid indivisible spheres, they are vortices, they are like a solar spheres, they are vortices, they are like a solar system, they are a probability amplitude, etc.).system, they are a probability amplitude, etc.).

2. A 2. A particularparticular sentence, strictly within the sentence, strictly within the norms of a local context, can be norms of a local context, can be considered considered eithereither true true oror false, with the false, with the stipulation that the sentence may turn stipulation that the sentence may turn out to be out to be eithereither inconsistent inconsistent oror incomplete or incomplete or bothboth within that local within that local context. context.

3. The range of pure, unactualized 3. The range of pure, unactualized possibilitiespossibilities holds the promise that, holds the promise that, however however inconsistentinconsistent and/or and/or incompleteincomplete the sentences within a the sentences within a particularparticular context context that have been given that have been given generalitygenerality status status may be, some alternate sentence will be may be, some alternate sentence will be available for selection at some future available for selection at some future timespace context.timespace context.