.... "If the brain was so simple that we could understand it, then we would be so simple that we couldn't." -- Emerson M. Pugh


Web mindshavings.blogspot.com

Tuesday, August 12, 2008

Further Recursion Excursion

Some more on recursive or self-referential sentences today. A simple example first: "This sentence has five words." Of course, "This sentence has four words," or even "This sentence has thirty-two words" would be equally good (and self-referential) English sentences, but they would be FALSE, whereas the initial sentence is TRUE. Combining self-reference with 'truth' or accuracy is what can sometimes be tricky. Or self-referential sentences can also lead to irresolvable paradoxes, as in this famous pairing:
"The following sentence is true. The preceding sentence is false."

Douglas Hofstadter recounted many thoughts regarding self-referential sentences in his volume "Metamagical Themas" (much of the work having been previously published in his Scientific American column of that time). Lee Sallows is responsible for generating many of the cleverest examples. The most interesting sentences are "self-enumerating" ones which accurately report the number of specific letters or words within the sentence itself. Many examples are recorded here (as well as in Hofstadter's book):


Here are some key examples:

1. In this sentence the word AND occurs twice, the word EIGHT occurs twice, the word FOUR occurs twice, the word FOURTEEN occurs four times, the word IN occurs twice, the word OCCURS occurs fourteen times, the word SENTENCE occurs twice, the word SEVEN occurs twice, the word THE occurs fourteen times, the word THIS occurs twice, the word TIMES occurs seven times, the word TWICE occurs eight times, and the word WORD occurs fourteen times.

2. This pangram has five a's, one b, one c, two d's, twenty-eight e's, five f's, three g's, seven h's, ten i's, one j, one k, one l, two m's, twenty n's, thirteen o's, two p's, one q, five r's, twenty-three s's, twenty t's, one u, six v's, nine w's, two x's, five y's, and one z.

[a "pangram," by the way, is a sentence that contains at least one instance of every letter of the alphabet]

and here a favorite example of Hofstadter's, from Lee Sallows, which remarkably enumerates both letters AND punctuation:

3. Only the fool would take trouble to verify that his sentence was composed of ten a's, three b's, four c's, four d's, forty-six e's, sixteen f's, four g's, thirteen h's, fifteen i's, two k's, nine l's, four m's, twenty-five n's, twenty-four o's, five p's, sixteen r's, forty-one s's, thirty-seven t's, ten u's, eight v's, eight w's, four x's, eleven y's, twenty-seven commas, twenty-three apostrophes, seven hyphens and, last but not least, a single !

(Again, obviously, many similar FALSE sentences could be easily constructed, but the above sentences are all TRUE, and yet their truth is not established 'til the very completion of the sentence! --- i.e., if just the last couple words were altered in any one of these sentences, it would become false; the initial words 'anticipate,' in a sense, what is yet to come, as if foreseeing the future.) [ BTW, computer programs have been written that generate certain types of these sentences. And such sentences have, of course, been generated in other languages as well.]

What does all this tell us about the human mind...? I'm not sure, except that it combines aspects of language (letters and semantics), math (counting), logic (truth), and even temporal awareness in a peculiar way... that no other animal is capable of. Even studying these sentences to see what they may tell us, is itself a kind of recursive process --- analyzing a process we have ourselves created to begin with. Hofstadter's most recent book, "I Am A Strange Loop" concerns some of these issues, though I don't find him altogether successful at resolving or describing them. Indeed there is some question whether it is even possible for the human mind to be turned upon itself in a manner self-revealing of its own workings, or do we, in trying, merely enter an endless feedback loop of no return? Is the level of complexity of the brain or consciousness always inherently one step above what the brain itself is capable of comprehending?

Well, enough on recursion for now... almost gives me a headache just thinking about it, or... thinking about thinking about it.

No comments: