Posts Tagged 'Mathematics'

XKCD

Because this says what happens to me in so many of my math classes:

Advertisements

Commutativity and Life

(Cross posted at the other site)

Sitting in a math class, and the professor announces that the next topic will be a brief study of matrices (matrix is the singular form). Then is asked a show of hands of those who have NOT had some previous experience in the topic. Up goes my hand, relieved to see that mine is not the only uncluttered mind, but saddened that there are so few of us. Those emotions are replaced when the professor announces that he will ‘go slow’ so that we midgets can keep up with the crowd. Thanks.

images.jpg

As he takes us through the steps of ever increasing arithmetic manipulation, the point is made that some properties of matrices are commutative while others are not. It is the non-commutative properties that are of interest, he observes. For those of you who have my level of understanding, note that an arithmetic operation is commutative if the order of the process returns the same result; 3 * 2 = 6 and 2 * 3 = 6.

As the link above reports:

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products.[6][7] Euclid is known to have assumed the commutative property of multiplication in his book Elements.[8] Formal uses of the commutative property arose in the late 18th and early 19th century when mathematicians began to work on a theory of functions. Today the commutative property is a well known and basic property used in most branches of mathematics. Simple versions of the commutative property are usually taught in beginning mathematics courses.

But, predictably, there is a large portion of mathematics that is not commutative. I knew it was just too good to be true. As the professor observed, there are many, many examples in life where the order of a process is very important. As examples, he pointed out that opening the window and sticking your head out of the car window are operations where the order of things is critical.

Wikipedia expands on the idea:

Noncommutative operations in everyday life

  • Washing and drying your clothes resembles a noncommutative operation, if you dry first and then wash, you get a significantly different result than if you wash first and then dry.
  • The Rubik’s Cube is noncommutative. For example, twisting the front face clockwise, the top face clockwise and the front face counterclockwise (FUF’) does not yield the same result as twisting the front face clockwise, then counterclockwise and finally twisting the top clockwise (FF’U). The twists do not commute. This is studied in group theory.

I’m confused but more impressed than ever with the nature of our existence. How can an idea as powerful as mathematics embrace contradictory behavior? Why do we think that mathematics can explain the physical world when it is riddled with inconsistency? Could it be that the nature of our existence transcends the universe of mathematics?

Am I having a metaphysical moment?

David Gale

My favorite math professor has, over the course of two semesters, introduced his classes to many prominent mathematicians through brief stories about their lives. Each recounting reminded us of the importance of the scientist and the frailty of their existence. From mild psychosis to paranoia, from greed to altruism, from lives filled with joy and happiness to those wrecked by tragedy and sadness, we have learned that the perfection of mathematics springs from the imperfection of the human condition. The question is not why, but how.

So, it is to be expected that I am alert for translations to the highest sphere…..

Via the Wall Street Journal:

David Gale (1921 – 2008)

Mathematician Who Loved Games Helped Unknot a Pairing-Up Puzzle

This spring’s medical-school graduates have just completed the nerve-wracking “match day,” in which they rank the hospitals where they would like to do their residencies and bite their fingernails until they find out where they will be placed.

Few realize that the algorithm pairing them with a teaching hospital was developed by a game-loving University of California, Berkeley, mathematics professor, David Gale.

[Gale]
David Gale

Enamored of recreations from sudoku to the roller derby, Mr. Gale, who died March 7 at age 86, was a game-theory specialist often mentioned alongside a onetime collaborator, Nobel laureate John Nash, as a giant in the field.

Mr. Gale’s best-known contribution came as a solution to the “stable-marriage problem,” the question of how best to pair up an equal number of men and women, each of whom has his or her own preferences for a mate.

In a 1962 paper written with University of California, Los Angeles, professor Lloyd Shapley, Mr. Gale proposed a multistage process beginning with each man asking his top choice whether she will have him. Women with multiple offers tell one of the suitors “maybe” and all the others “no.” The rejected men move on to make offers to other women. If a woman gets a new offer from someone she likes better, she gives him a “maybe” and tells the earlier “maybe” that he is now a “no.”

After many rounds, as the rejected men turn to women who didn’t get any offers at first, everyone has paired up. Then each woman turns to her man and says “yes.”

Although offered as an academic solution to a theoretical problem, Mr. Gale’s paper proved a remarkably fertile contribution to real-world cases of “two-sided matching” such as the medical-residency example, where hospitals are choosing students at the same time as students are choosing hospitals. A related algorithm is used by school systems in Boston and New York to allocate slots in high schools.

“David’s work will be remembered for generations to come,” says Alvin E. Roth, a professor of economics at Harvard. Mr. Roth helped design the school-choice systems and has lately been working to apply the theory to the allocation of scarce kidney donations.

Mr. Gale inspired headlines as recently as last year when he challenged studies reporting that men had more lifetime heterosexual partners than women, a situation he labeled a logical impossibility.

Mr. Gale studied math at Princeton, where he was a doctoral candidate alongside Mr. Nash. He was known for devising elegant puzzles and games. Among these was Chomp, in which players take cookies from a board until a final, poison cookie must be removed by the loser. Simple enough for a preschooler to master, the game turns out to have mathematical subtleties that have inspired dozens of academic papers.

At dinner time, says his daughter, Katharine Gale, “he would ask us if we all toasted, how many clinks would there be. He would write matrices all over napkins.”

Once, in a flash of inspiration, “he wrote all over an airplane ticket. The airline refused to honor it and he had to buy a new one,” she says.

“He thought math was beautiful, and he wanted people to understand that,” Ms. Gale says.

–Stephen Miller

 

 

Thank you, Professor Gale.


“Life’s hard, son. It’s harder when you’re stupid.” — The Duke.

Education is a companion which no misfortune can depress, no crime can destroy, no enemy can alienate,no despotism can enslave. At home, a friend, abroad, an introduction, in solitude a solace and in society an ornament.It chastens vice, it guides virtue, it gives at once grace and government to genius. Without it, what is man? A splendid slave, a reasoning savage. - Joseph Addison
The term informavore (also spelled informivore) characterizes an organism that consumes information. It is meant to be a description of human behavior in modern information society, in comparison to omnivore, as a description of humans consuming food. George A. Miller [1] coined the term in 1983 as an analogy to how organisms survive by consuming negative entropy (as suggested by Erwin Schrödinger [2]). Miller states, "Just as the body survives by ingesting negative entropy, so the mind survives by ingesting information. In a very general sense, all higher organisms are informavores." - Wikipedia

Blog Stats

  • 30,792 hits
Advertisements