.... "If the brain was so simple that we could understand it, then we would be so simple that we couldn't." -- Emerson M. Pugh
Sunday, August 31, 2008
Saturday, August 30, 2008
Friday, August 29, 2008
The phaonmneel pweor of the hmuan mnid: I cdnuolt blveiee taht I cluod aulaclty uesdnatnrd waht I was rdeanig. Aoccdrnig to a rscheearcr at Cmagbride Uinervtisy, it deosn't mttaer in waht oredr the ltteers of a wrod are, the olny iprmoatnt tihng is taht the frist and lsat ltteer be in the rghit pclae.
The rset can be a taotl mses and you can sitll raed it wouthit a porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe. Petrty amzanig huh? And I awlyas tghuhot slpeling was ipmorantt!
Thursday, August 28, 2008
Initially, do this calculation IN YOUR HEAD ONLY -- NOT on paper.
Take 1000 to start and to it add:
What is the new total???
If you said 5000, you're wrong; check the correct answer (4100) on a calculator or by hand, to realize where you went wrong.
Wednesday, August 27, 2008
Tuesday, August 26, 2008
Monday, August 25, 2008
Sunday, August 24, 2008
Saturday, August 23, 2008
Friday, August 22, 2008
Name "the smallest possible integer NOT definable by fewer than twelve words".
It's easy to imagine examples that DON'T work:
the speed of light in meters per second [only 8 words]
the number of inches in a foot [7 words]
the number of stars in the universe [7 words]
one hundred thirty million, sixty seven thousand, three hundred and thirteen [11 words]
The problem arises however (if it isn't already obvious), that if you did concoct a sentence of 12 or more words to define some integer, and it IS the smallest such definable integer, IT can then be accurately designated (defined) by the original 11-word sentence above ("the smallest possible integer not definable by fewer than twelve words") -- thus a self-referential contradiction!
Another example of where mixing language/semantics with numbers/mathematics proves vexing, throwing light on illogical ambiguity within language.
More on the Berry paradox at Wikipedia:
Thursday, August 21, 2008
Wednesday, August 20, 2008
Tuesday, August 19, 2008
Sunday, August 17, 2008
Saturday, August 16, 2008
peace of mind
piece of my mind
John beat Mary up every morning.
John beat up Mary every morning.
The book was read and reviewed.
The book was red and blue.
Tom hit the man with a stick.
Tom hit the man with a mustache.
On the wall were tin cans.
On the wall were ten cans.
A rival of John's brother was late.
Arrival of John's brother was late.
The wood in baseball bats comes from a lumberyard in St. Louis.
The wooden baseball bats come from a lumberyard in St. Louis.
Friday, August 15, 2008
Thursday, August 14, 2008
It includes pages for learning French and Spanish (just two of the many languages Daniel knows), and an interesting art portrait of the first 20 digits of pi (3.1415926535897932384) as Daniel perceives them in his own mind (as color-scapes). For anyone not familiar with Daniel's story, one of his unfathomable claims to fame is accurately reciting from memory the first 22,514 decimal places (that's NOT a typo) of pi in a 5 hour time period.
Some more on savantism here.
And savantism video clips from "60 Minutes" here.
Tuesday, August 12, 2008
"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.
Saturday, August 9, 2008
The opening header-quote from Emerson M. Pugh represents a form of recursion (as does the above post title). In this case a kind of inescapable paradoxical loop. The ability to think recursively is likely one of the things which most separates us cerebrally from other animals. We can use the human brain to study the human brain... we can think about thinking... In fact we can think about thinking about thinking... and so it goes.
Recursion occurs in mathematics, language, computer programming, and the physical world as well (the childhood fun of placing two mirrors face-to-face and observing the receding reflections back-and-forth is an example of recursion --- magazine covers have occasionally dabbled in recursive depictions of the covers themselves). Often recursion involves either iteration or self-reference.
In language the recursive element is what (theoretically) allows for infinitely long and infinitely many new sentences, by the embedding of phrases:
1. Jack ate the pie.
2. Jack, the clarinetist, ate the pie.
3. Jack, the clarinetist, who wore a beret, ate the pie.
4. Jack, the clarinetist, who wore a beret, that was made of Scottish wool, ate the pie.
5. Jack, the clarinetist, who wore a beret, that was made of Scottish wool, bought from a shepherd who raised several herds of sheep, that once belonged to Schlomo, who was the best juggler in town, and previously worked for the circus when he was a young and clever lad in the Slobovian hillside, ate the pie.
Here's another odd example of a sort of recursion in language (in which the meaning of the sentence suddenly changes just as we get to the very end):
"The shooting of the young and handsome doctor, mystified all his friends and colleagues, most of whom had always thought him to be an excellent marksman."
Yes, the human mind is a pretty incredible lil' instrument.
In real estate they say it's all about location, location, location; or in stock trading, it's timing, timing, timing. I'm not so sure but that in human cognition it isn't recursion, recursion, recursion.
Now, I have to go shampoo my hair --- which, if I was TOO anally recursive about it could take quite awhile:
1. lather, 2. rinse, 3. repeat ;-)