I Am a Strange Loop - Douglas R. Hofstadter [43]
When set theory turned out to allow self-contradictory entities like this, Russell’s dream of solidly grounding mathematics came crashing down on him. This trauma instilled in him a terror of theories that permitted loops of self-containment or of self-reference, since he attributed the intellectual devastation he had experienced to loopiness and to loopiness alone.
In trying to recover, then, Russell, working with his old mentor and new colleague Whitehead, invented a novel kind of set theory in which a definition of a set could never invoke that set, and moreover, in which a strict linguistic hierarchy was set up, rigidly preventing any sentence from referring to itself. In Principia Mathematica, there was to be no twisting-back of sets on themselves, no turning-back of language upon itself. If some formal language had a word like “word”, that word could not refer to or apply to itself, but only to entities on the levels below itself.
When I read about this “theory of types”, it struck me as a pathological retreat from common sense, as well as from the fascination of loops. What on earth could be wrong with the word “word” being a member of the category “word”? What could be wrong with such innocent sentences as “I started writing this book in a picturesque village in the Italian Dolomites”, “The main typeface in this chapter is Baskerville”, or “This carton is made of recyclable cardboard”? Do such declarations put anyone or anything in danger? I can’t see how.
What about “This sentence contains eleven syllables” or “The last word in this sentence is a four-letter noun”? They are both very easy to understand, they are clearly true, and certainly they are not paradoxical. Even silly sentences such as “The ninth word in this sentence contains ten letters” or “The tenth word in this sentence contains nine letters” are no more problematical than the sentence “Two plus two equals five”. All three are false or at worst meaningless assertions (the second one refers to something that doesn’t exist), but there is nothing paradoxical about any of them. Categorically banishing all loops of reference struck me as such a paranoid maneuver that I was disappointed for a lifetime with the oncebitten twice-shy mind of Bertrand Russell.
Intellectuals Who Dread Feedback Loops
Many years thereafter, when I was writing a monthly column called “Metamagical Themas” for Scientific American magazine, I devoted a couple of my pieces to the topic of self-reference in language, and in them I featured a cornucopia of sentences invented by myself, a few friends, and quite a few readers, including some remarkable and provocative flights of fancy, such as these:
If the meanings of “true” and “false” were switched, this sentence wouldn’t be false.
I am going two-level with you.
The following sentence is totally identical with this one, except that the words “following” and “preceding” have been exchanged, as have the words “except” and “in”, and the phrases “identical with” and “different from”.
The preceding sentence is totally different from this one, in that the words “preceding” and “following” have been exchanged, as have the words “in” and “except”, and the phrases “different from” and “identical with”.
This analogy is like lifting yourself up by your own bootstraps.
Thit sentence it not self-referential because “thit” it not a word.
If wishes were horses, the antecedent clause in this conditional sentence would be true.
This sentence every third, but it still comprehensible.
If you think this sentence is confusing, then change one pig.
How come this noun phrase doesn’t denote the same thing as this noun phrase does?
I eee oai o ooa a e ooi eee o oe.
Ths sntnc cntns