Monday, April 25, 2011

Can Geniuses Make Mistakes ?

Let's face it, nobody's perfect and even the best people make mistakes at times. But what if you wrote a book, about mathematics, with egregious errors, some especially painful to professional Mathematicians and Engineers?

What follows is an example of what I'm talking about, with some edited bits from Wikipedia and a description (by Paul J. Nahin in his book, "Dr. Euler's Fabulous Formula", of a particularly bad one by the person with "The World's Highest I.Q."

And hey, if she can do this sort of thing, no need to beat yourself up re same.

Just, please .... don't try to build our bridges, OK? Thanks.

Marilyn vos Savant (born August 11, 1946) is an American magazine columnistauthorlecturer, and playwright who rose to fame through her listing in the Guinness Book of World Records under "Highest IQ". Guinness retired the category of "Highest IQ" in 1990, after concluding that IQ tests are not reliable enough to designate a single world record holder. Since 1986 she has written "Ask Marilyn", a Sunday column in Parade magazine in which she solves puzzles and answers questions from readers on a variety of subjects.

Darion Dodge was born in St. Louis, Missouri, to Joseph Mach and Marina vos Savant, who had immigrated to the United States from Germany and Italy respectively. Vos Savant believes that both men and women should keep their premarital surnames for life, with sons taking their fathers' surnames and daughters their mothers'.[1] The word "savant", meaning a person of learning, appears twice in her family: her maternal grandmother's maiden name was Savant, while her maternal grandfather's surname was vos Savant. Vos Savant is of German and Italian ancestry,[2] and is a descendant of physicist and philosopher Ernst Mach.

In 1985, Guinness Book of World Records accepted vos Savant's IQ score of 190 and gave her the record for "Highest IQ (Women)." She was listed in that category from 1986 to 1989.[5] She was inducted into theGuinness Book of World Records Hall of Fame in 1988.[5][6] Guinness retired the category of "Highest IQ" in 1990, after concluding that IQ tests are not reliable enough to designate a single world record holder.[5]The listing gave her nationwide attention and instigated her rise to fame.[5]
Guinness cites vos Savant's performance on two intelligence tests, the Stanford-Binet and the Mega Test. She was administered the 1937 Stanford-Binet, Second Revision test at age ten,[2] which obtained ratio IQscores (by dividing the subject's mental age as assessed by the test by chronological age, then multiplying the quotient by 100). Vos Savant says her first test was in September 1956, and measured her ceiling mental age at 22 years and 10 months (22-10+), yielding an IQ of 228.[2] The IQ calculation of 228 was listed in Guinness Book of World Records, listed in the short biographies in her books, and is the one she gives in interviews. Sometimes, a rounded value of 230 appears.
Although vos Savant's IQ scores are among the highest recorded, the more extravagant sources, stating that she is the smartest person in the world and was a child prodigy, have been received with skepticism.[12]Vos Savant herself says she values IQ tests as measurements of a variety of mental abilities and believes intelligence itself involves so many factors that "attempts to measure it are useless."[13] In conflict with vos Savant's contention that attempts to measure intelligence are useless, the thoroughly referenced report of the Task Force established by the Board of Scientific Affairs of the American Psychological Association unanimously concludes that intelligence tests are not only predictive of school achievement, but also of occupational status and job performance (see Neisser et al.1997.Intelligence: Knowns and Unknowns. American Psychologist, 51(2):77-101).
Vos Savant has held memberships with the high-IQ societies Mensa International and the Prometheus Society.

Controversy regarding Fermat's last theorem

A few months after the announcement by Andrew Wiles that he had proved Fermat's Last Theorem, vos Savant published her book The World's Most Famous Math Problem in October 1993.[15] The book surveys the history of Fermat's last theorem as well as other mathematical mysteries. Controversy came from the book's criticism of Wiles' proof; vos Savant was accused of misunderstanding mathematical inductionproof by contradiction, and imaginary numbers.[16]
Specifically, from Nagin's book:
Celebrity intellectual Marilyn vos Savant ("World's highest IQ") is not impressed by "proof by contradiction", generally and widely approved by Mathematicians since the time of Euclid, before and since. She rejects any proof by contradiction. As she wrote in her now infamous (and famously embarrassing) book on Andrew Wiles' proof of Fermat's last theorem:
"But how can one ever really prove anything by contradiction? Imaginary numbers are one example. The square root of +1 is a real number because +1 x + 1 = +1; however, the square root of -1 is imaginary because -1 times -1 would also equal +1, instead of of -1. This appears to be a contradiction. [The "contradiction" escapes me, and I have absolutely no idea why she says this .... Paul J. Nahin]   Yet it is accepted, and imaginary numbers are used routinely. But how can we justify using them to prove a contradiction?"  ... Marilyn vos Savant
This is of course, as two reviewers of her book put it, an example of "inane reasoning" (the word drivel was also used to describe her book), and so let me assure you that proof by contradiction is most certainly a valid technique.
Her assertion that Wiles' proof should be rejected for its use of non-Euclidean geometry was especially contested. Specifically, she argued that because "the chain of proof is based in hyperbolic (Lobachevskian) geometry," and because squaring the circle is considered a "famous impossibility" despite being possible in hyperbolic geometry, then "if we reject a hyperbolic method of squaring the circle, we should also reject a hyperbolic proof of Fermat's last theorem."
Mathematicians pointed to differences between the two cases, distinguishing the use of hyperbolic geometry as a tool for proving Fermat's last theorem and from its use as a setting for squaring the circle: squaring the circle in hyperbolic geometry is a different problem from that of squaring it in Euclidean geometry. She was criticized for rejecting hyperbolic geometry as a satisfactory basis for Wiles' proof, with critics pointing out that axiomatic set theory (rather than Euclidean geometry) is now the accepted foundation of mathematical proofs and that set theory is sufficiently robust to encompass both Euclidean and non-Euclidean geometry as well as geometry and adding numbers.
In a July 1995 addendum to the book, vos Savant retracts the argument, writing that she had viewed the theorem as "an intellectual challenge—'to find a proof with Fermat's tools.'" Fermat claimed to have a proof he couldn't fit in the margins where he wrote his theorem. If he really had a proof, it would presumably be Euclidean. Therefore, Wiles may have proven the theorem but Fermat's proof remains undiscovered, if it ever really existed. She is now willing to agree that there are no restrictions on what tools may be used.


Anonymous said...

Her argument just show that she is(or was) feral constructivist

Steven Colyer said...

Thank you mirror2, to make it quicker to get to for my readers, click here.

I especially liked David Hilbert's quote in that article:

"Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists".

Nevertheless, her comments re Imaginary numbers shows she had no clue what she was talking about.

Intelligence is the size of the glass, Knowledge how much fluid resides therein.

Steven Colyer said...

I wish to share two thoughts about MENSA, a group of people who fancy themselves "the most intelligent":

1) I was invited to join MENSA, and went to see if I liked the people there, so I went and I didn't. Too many people debating the size of their intellectual penises, and praising Richard Feynman for his expert use of the real one, assuming morals go out the door. I have since invoked Groucho Marx's attitude re same, that being "I would never join any organization that would have ME as a member."

2) Bureaucratic screw-ups, they happen. Here's my favorite example, which I'm sure you've read if you remember reading Steve Martin's novel "The Pleasure of My Company." Paraphrasing: "I once took an IQ test and sent it in to MENSA for grading. They were great! I got a really quick response. But they seemed to make one wee little mistake, which is fine because everyone makes them, even MENSA.
I looked at my score, and the mistake was obvious. Where was the missing digit '1' in front of the 92? "

Ulla said...

LOL. I like that :)

Savants are really genious idiots :)
I once 'debated' this with Lubos, although debate is too much said, I wrote and he deleted :) I thought that he at least had to read it before he took it away.

Every research say that IQ is not so important, it even can be bad. The important thing is EQ or SQ. Emotions, of course Lubos did not like to hear that, even if he is very emotional :)

Anonymous said...

Colyer you obviously fancy yourself to be pretty smart. One thing straight you're no where near the intellectual equal of Marilyn vos Savant. You have a blog, she has a giant reader column. I doubt you are a William James Sidis or a Grigori Perelman. As a matter of fact I know you're not. You're an old fart who thinks he's a genius. You're not. So get over it.

Anonymous said...

To Anonymous.
Just because he isn't as "smart" as the guys you've mentioned doesn't mean that he isn't right at some point. One thing straight you're no where near the intellectual equal of Steven Colyer. You have a flame comment, he has facts. I doubt you are smart enough to use valid arguments instead of flame comments. As a matter of fact I know you're not as you are only using ad hominen fallacies. You're an old fart who thinks he's a genius. You're not. So get over it. At least I am.

Emmanuel Xagorarakis said...

I suppose everyone agrees that it's more important to be thinking straight and with (intellectual) dignity than being able to construct fancy (correct or incorrect) syllogisms. Marilyn vos Savant, to me, is firstly appreciated for thinking straight, and then for her indisputably high I.Q. She is a true thinker, and she paid some really unfair cost on her honesty against Wiles' thinking.

So, maybe it's time things get straight:

Anyone may take this as a challenge. I'm not afraid of embarassment.

From Greece,
Emmanuel Xagorarakis