Archive for the 'Mathematics' Category

21
Mar
08

Israeli Solves Math Code

After 38 Years, Israeli Solves Math Code
JERUSALEM — A mathematical puzzle that baffled the top minds in the esoteric field of symbolic dynamics for nearly four decades has been cracked — by a 63-year-old immigrant who once had to work as a security guard.

Avraham Trahtman, a mathematician who also toiled as a laborer after moving to Israel from Russia, succeeded where dozens failed, solving the elusive ”Road Coloring Problem.”

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The conjecture essentially assumed it’s possible to create a ”universal map” that can direct people to arrive at a certain destination, at the same time, regardless of starting point. Experts say the proposition could have real-life applications in mapping and computer science.

The ”Road Coloring Problem” was first posed in 1970 by Benjamin Weiss, an Israeli-American mathematician, and a colleague, Roy Adler, who worked at IBM at the time.

For eight years, Weiss tried to prove his theory. Over the next 30 years, some 100 other scientists attempted as well. All failed, until Trahtman came along and, in eight short pages, jotted the solution down in pencil last year.

”The solution is not that complicated. It’s hard, but it is not that complicated,” Trahtman said in heavily accented Hebrew. ”Some people think they need to be complicated. I think they need to be nice and simple.”

Weiss said it gave him great joy to see someone solve his problem.

Stuart Margolis, a mathematician who recruited Trahtman to teach at Bar Ilan University near Tel Aviv, called the solution one of the ”beautiful results.” But he said what makes the result especially remarkable is Trahtman’s age and background.

”Math is usually a younger person’s game, like music and the arts,” Margolis said. ”Usually you do your better work in your mid 20s and early 30s. He certainly came up with a good one at age 63.”

Adding to the excitement is Trahtman’s personal triumph in finally finding work as a mathematician after immigrating from Russia. ”The first time I met him he was wearing a night watchman’s uniform,” Margolis said.

Originally from Yekaterinburg, Russia, Trahtman was an accomplished mathematician when he came to Israel in 1992, at age 48. But like many immigrants in the wave that followed the breakup of the Soviet Union, he struggled to find work in the Jewish state and was forced into stints working maintenance and security before landing a teaching position at Bar Ilan in 1995.

The soft-spoken Trahtman declined to talk about his odyssey, calling that the ”old days.” He said he felt ”lucky” to be recognized for his solution, and played down the achievement as a ”matter for mathematicians,” saying it hasn’t changed him a bit.

The puzzle tackled by Trahtman wasn’t the longest-standing open problem to be solved recently. In 1994, British mathematician Andrew Wiles solved Fermat’s last theorem, which had been open for more than 300 years. Trahtman’s solution is available on the Internet and is to be published soon in the Israel Journal of Mathematics.

Joel Friedman, a math professor at the University of British Columbia, said probably everyone in the field of symbolic dynamics had tried to solve the problem at some point, including himself. He said people in the related disciplines of graph theory, discrete math and theoretical computer science also tried.

”The solution to this problem has definitely generated excitement in the mathematical community,” he said in an e-mail.

Margolis said the solution could have many applications.

”Say you’ve lost an e-mail and you want to get it back — it would be guaranteed,” he said. ”Let’s say you are lost in a town you have never been in before and you have to get to a friend’s house and there are no street signs — the directions will work no matter what.”

© 2008 The Associated Press

28
Jan
08

The God Equation..

“With or without religion, good people will do good, and evil people will do evil. But for good people to do evil, that takes religion.” –Steven Weinberg

Pl read stimulating article by Ron Csillag published in Toronto Star..

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Actually, since Pythagoras the relationship between men of numbers and the Deity has been more along the lines of love-hate, but it’s a rich vein

Which math-phobic among us has not beseeched God for help with another colon-clenching algebra or calculus exam? Had we heeded the words of the German mathematician Leopold Kronecker, perhaps we would have realized we’ve been talking to the wrong person: “God made the integers; all else is the work of man.”

Pythagoras, who gave us his eponymous theorem on right-angled triangles, headed a cult of number worshippers who believed God was a mathematician. “All is number,” they would intone.

The 17th-century Jewish philosopher Baruch Spinoza echoed the Platonic idea that mathematical law and the harmony of nature are aspects of the divine. Spinoza, too, posited that God’s activities in the universe were simply a description of mathematical and physical laws. For that and other heretical views, he was excommunicated by Amsterdam’s Jewish community.

German mathematician Georg Cantor’s work on infinity and numbers beyond infinity (the mystical “transfinite”) was denounced by theologians who saw it as a challenge to God’s infiniteness. Cantor’s obsession with mathematical infinity and God’s transcendence eventually landed him in an insane asylum.

For the Hindu math genius Ramanujan, an uneducated clerk from Madras who wowed early 20th-century Cambridge, an equation “had no meaning unless it expresses a thought of God.” Though an agnostic, the prolific Hungarian mathematician Paul Erdos imagined a heavenly book in which God has inscribed the most elegant and yet unknown mathematical proofs.

And famously, Albert Einstein said God “does not play dice” with the universe.

What is it with God and mathematics? Even as science and religion have quarrelled for centuries and are only recently exploring ways to kiss and make up, mathematicians have been saying for millennia that no truer expression of the divine can be found than in an ethereally beautiful equation, formula or proof.

Witness, for example, such transcendent numbers as phi (not to be confused with pi), often called the Divine Proportion or the Golden Ratio. At 1.618, it describes the spirals of seashells, pine cones and symmetries found throughout nature. Other mysterious constants like alpha (one-137th) and gamma (0.5772…) pop up in enough odd places to suggest to some that they are an expression of the underlying beauty of mathematics, and to others that someone or something planned it that way.

But does that translate into actual belief ?

The New York Times reported recently that mathematicians believe in God at a rate 2 1/2 times that of biologists, quoting a survey of the National Academy of Sciences. Admittedly, that’s not saying much: Only 14.6 per cent of mathematicians embraced the God hypothesis, versus 5.5 per cent of biologists (versus some 80 per cent of Canadians who believe in a supreme being).

Count John Allen Paulos among the non-believers. A mathematician who teaches at Temple University in Philadelphia and who has popularized his subject in bestselling books such as Innumeracy and A Mathematician Reads the Newspaper, Paulos’s latest offering is a slim but explosive volume whose title is self-explanatory: Irreligion: A Mathematician Explains Why the Arguments for God Just Don’t Add Up (Hill & Wang).

This newest addition to the neo-atheist field crowded by the likes of Richard Dawkins, Christopher Hitchens, Sam Harris and others emboldened by the recent transformation of non-belief from a 97-pound weakling into a he-man, Paulos thankfully employs little math, preferring to see things, as he tells us, in the stark light of “logic and probability.”

Deploying “a lightly heretical touch,” he dissects a playlist of “golden oldies” that includes the first-cause argument (sometimes tweaked as the cosmological argument, which hinges on the Big Bang), the argument for intelligent design, the ontological argument (crudely, that if we can conceive of God, then God exists), the argument from the anthropic principle (that the universe is “fine-tuned” to allow us to exist), the moral universality argument, and others.

The famous Pascal’s wager – that it’s in our self-interest to believe in God because we lose nothing in case He does exist – is upended as logically flawed, based on what statisticians call Type I and Type II errors.

Lord knows Paulos isn’t the first mathematician to proclaim his lack of religious faith. Cambridge’s famous wunderkind G.H. Hardy loudly and proudly adjudged God to be his enemy. To Erdos, God, if He existed, was “the supreme fascist.”

Even as Paulos works to refute the classical arguments for God’s existence, he does something too few of his mindset do: Chide non-believers for unsportsmanlike conduct.

“It’s repellent for atheists or agnostics,” he admonishes, “to personally and aggressively question others’ faith or pejoratively label it as benighted flapdoodle or something worse. Those who do are rightfully seen as arrogant and overbearing.”

That doesn’t prevent him from doffing the gloves. The ontological argument is “logical abracadabra.” The design, or teleological argument, is a “creationist Ponzi scheme” that “quickly leads to metaphysical bankruptcy.”

Much of theology is “a kind of verbal magic show.” A claim that a holy book is inerrant because the book itself says so is another logical black hole.

However, math, specifically something called Ramsey theory, which studies the conditions under which order must appear, can account for the illusion of divine order arising from chaos.

Paulos provides a nice counterpoint to theoretical physicist Stephen Unwin’s 2003 book The Probability of God, which calculated the likelihood of God’s existence at 67 per cent, and to Oxford philosopher Richard Swinburne’s use of a probability formula known as Bayes’ theorem to put the odds of Christ’s resurrection at 97 per cent.

Those and other efforts remind one of the story, perhaps apocryphal, of Catherine the Great’s request of the German mathematical giant Leonhard Euler to confront atheist French philosopher Denis Diderot with evidence of God. The visiting Euler agreed, and at the meeting, strode forward to proclaim to the innumerate Frenchman: “Sir, (a+bn)/n = x, hence God exists. Reply!”

Diderot was said to be so dumbfounded, he immediately returned to Paris.

To Paulos, the tale is a great example of “how easily nonsense proffered in an earnest and profound manner can browbeat someone into acquiescence.”

His arguments notwithstanding, Paulos concedes that there’s “no way to conclusively disprove the existence of God.” The reason, he notes, is a consequence of basic logic, but not one “from which theists can take much heart.”

As for the problem of good and evil, he defers to fellow atheist, the Nobel Prize-winning physicist Steven Weinberg: “With or without religion, good people will do good, and evil people will do evil. But for good people to do evil, that takes religion.”

Or as Paulos might say, no mathematician has ever deliberately flown planes into buildings.

© Copyright Toronto Star- Ron Csillag January 26, 2008

21
Nov
07

Pi Haiku

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Pi – ratio of
Around : across a circle –
An endless number ?

© e-poems.org

17
Nov
07

Anna of Arithmetic

This eloquent piece form a Book – “The Advent of the Algorithm” by David Berlinski expressing a contemplation that has prevailed in me.
Why we should we care to read literature ? it also explores underlying relationship between Art & Science..

Anna of Arithmetic

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Reading a novel with an innocent eye, students very often lose themselves in its pages, making their decision about the novel on the basis of whether they felt comfortable or at home within its world and more often than not identifying the author with his or her protagonist, every novelist receiving from time to time letters addressed to his creation — Dear Anna, don’t do it. Such is the triumph of art. But such is the triumph of illusion, as well.

After some experience, the student learns to step back, recognizing that Anna, she’s got to do what she’s got to do, and this because what she’s got to do is artistically required. No one reading Anna Karenina is quiet prepared to see her departing the novel, therapist in hand, and briskly getting her life together. A sense of literary sophistication begins when aesthetic standards are substituted for moral judgments. This makes art profoundly amoral undertaking, but a profoundly interesting one as well.

Mathematics is, among other things a form of art. Before Hilbert, mathematicians and logicians had banged around within the confines of various mathematical systems, hoping against hope to arrange the system so that it seemed entirely secure, the effort as doomed as the correlative effort to persuade Anna Karenina to undertake therapy.

Hilbert persuaded everyone to step back. Stepping back, mathematicians saw mathematics for what it might be, a formal game, the perspective cold but liberating. Thus removed from what they habitually did, mathematicians, like students of literature, were forced to ask not whether the Anna of arithmetic seemed nice, friendly, kind of snooty, confused, or otherwise irritating, but whether she made artistic or mathematical sense. A question of judgment had come to replace a question of certainty.

And with judgments come standards. They must, those standards, be chosen so as to reflect the original impulse yielding the decision to distinguish mathematics from metamathematics. And they must, those standards, be standards that can be met by proof, even if it is proof delivered in the meta language itself, for without proof, there is simply no mathematics at all.

©-David Berlinski.

11
Oct
07

The Astronomer’s Drinking Song

This is a delightful drinking song from the old Mathematical Society of London, which seems to have been sung at a meeting around 1800. It was published in “A Budget of Paradoxes” by Augustus de Morgan. (1806-1871) who was a mathematician of considerable merit, a brilliant and influential teacher, a founder, with Geroge Boole, a symbolic logic as is developed in England, a writer of many book, an indefatigable contributor to encyclopedias, magazines and learned journals.

He was an uncompromising advocate of religious liberty and free expression, an insatiable collector of curious lore, anecdotes, quaint, and perverse opinions, paradoxes, puzzles, riddles and puns; a bibliomaniac, a wit and polemicist, a detester of hypocrisy and sordid motive, an impolitic, independent, crotchety, overworked, lovable, friendly and contentious Englishman.

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Whoe’er would search the starry sky,
Its secrets to divine, sir,
Should take his glass – I mean, should try
A glass or two of wine, sir!
True virtue lies in golden mean,
And man must wet his clay, sir,
Join these two maxims, and ’tis seen
He should drink his bottle a day, sir!

Old Archimedes, reverend sage!
By trump of fame renowned, sir,
Deep problems solved in every page,
And the sphere’s curved surface found, sir:
Himself he would have far outshone,
And borne a wider sway, sir,
Had he our modern secret known,
And drank a bottle a day, sir!

When Ptolemy, now long ago,
Believed the earth stood still, sir,
He never would have blundered so,
Had he but drunk his fill, sir:
He’d then have felt it circulate,
And would have learnt to say, sir,
The true way to investigate
Is to drink your bottle a day, sir!

Copernicus, that learned wight,
The glory of his nation,
With draughts of wine refreshed his sight,
And saw the earth’s rotation;
Each planet then its orb described,
The moon got under way, sir;
These truths from nature he imbibed
For he drank his bottle a day, sir!

The noble Tycho placed the stars,
Each in its due location;
He lost his nose by spite of Mars,
But that was no privation:
Had he but lost his mouth, I grant
He would have felt dismay, sir,
Bless you! he knew what he should want
To drink his bottle a day, sir!

Cold water makes no lucky hits;
On mysteries the head runs:
Small drink let Kepler time his wits
On the regular polyhedrons:
He took to wine, and it changed the chime,
His genius swept away, sir,
Through area varying as the time
At the rate of a bottle a day, sir!

Poor Galileo, forced to rat
Before the Inquisition,
E pur si muove was the pat
He gave them in addition:
He meant, whate’er you think you prove,
The earth must go its way, sirs;
Spite of your teeth I’ll make it move,
For I’ll drink my bottle a day, sirs!

Great Newton, who was never beat
Whatever fools may think, sir;
Though sometimes he forgot to eat,
He never forgot to drink, sir:
Descartes took nought but lemonade,
To conquer him was play, sir;
The first advance that Newton made
Was to drink his bottle a day, sir!

D’Alembert, Euler, and Clairaut,
Though they increased our store, sir,
Much further had been seen to go
Had they tippled a little more, sir!
Lagrange gets mellow with Laplace,
And both are wont to say, sir,
The philosophe who’s not an ass
Will drink his bottle a day, sir!

Astronomers! what can avail
Those who calumniate us;
Experiment can never fail
With such an apparatus:
Let him who’d have his merits known
Remember what I say, sir;
Fair science shines on him alone
Who drinks his bottle a day, sir!

How light we reck of those who mock
By this we’ll make to appear, sir,
We’ll dine by the sidereal clock
For one more bottle a year, sir:
But choose which pendulum you will,
You’ll never make your way, sir,
Unless you drink – and drink your fill, –
At least a bottle a day, sir!

(c) Source : The World of Mathematics by James R Newman

13
Sep
07

Golden Ratio..

As such, my blogger friend Michele sent me her brilliant composition.

Pl click Picture for larger image..

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Mathematics…would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude –

-Friedrich Neitzsche

For more about her awe-inspiring blog refer: http://micheleroohani.com/blog/

10
Sep
07

Math:Gift from God or Work of Man?

Mathematics, Religion and Evolution in School Curricula

COMMENTARY by JOHN ALLEN PAULOS Sept. 2, 2007 —

School begins again, and we read more about the intrusion of pseudoscience into school science curricula in this country, particularly into the study of biology and evolution.

The motive, despite the claims of proponents of intelligent design and other bogus “disciplines,” has been religious. Although some of the creation scientists’ arguments presented have a probabilistic flavor, the mathematics curriculum has seemed somewhat resistant to this trend. Recently a number of readers have sent me course descriptions from various schools that suggest otherwise, however.

The issue is complicated (perhaps too complicated for a column), but I’ll also briefly discuss the relevance of evolution to a more defensible, but still flawed argument relating religion and mathematics.

Religion in the Math Curriculum

Consider first a Baptist school in Texas whose description of a geometry course begins:

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Students will examine the nature of God as they progress in their understanding of mathematics. Students will understand the absolute consistency of mathematical principles and know that God was the inventor of that consistency. They will see God’s nature revealed in the order and precision they review foundational concepts while being able to demonstrate geometric thinking and spatial reasoning. The study of the basics of geometry through making and testing conjectures regarding mathematical and real-world patterns will allow the students to understand the absolute consistency of God as seen in the geometric principles he created.

Continue reading ‘Math:Gift from God or Work of Man?’




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