Sunday, February 24, 2013

Russell’s Paradox & Frege’s Logicism

Bertrand Russell thought up a deep and perplexing paradox when reading about Gottlob Frege's system of logic. Frege thought that he could define all mathematical concepts and prove all mathematical truths solely from principles of logic. The view that mathematics can be reduced to logic in this way is called logicism. Had Frege demonstrated the truth of logicism, it would have been one of the greatest achievements in the history of philosophy. But his version of logicism was not successful. One of the logical principles used to prove the existence of numbers, functions, and other mathematical objects is: for every predicate, "is F (P)" there is a collection of things that are F. Two examples are: "is a prime number" determines the collection of numbers {2, 3, 5, 7, 11…. } and "is a collection" determines the collection of all collections. In 1903 Russell showed that (P) is self-contradictory with the following argument. Consider the predicate "is not a member of itself." With (P) there is a collection -call it R- of collections that are not members of themselves. Is R a member of itself? If it is then it isn't, and if it isn't then it is. A contradiction! This was a devastating blow to Frege and to logicism.

The collection of all collections that are not members of themselves is itself both a member of itself and not a member of itself.

Additionally: Here is a paradox involving reasoning similar to Russell's: "There is a barber who shaves all those and only those who don't shave themselves." If the barber shaves himself then he doesn't shave himself, and if he doesn't then he does. This paradox is easy to solve, simply by accepting that there cannot be such a barber. Frege couldn't accept the analogous way out for collections, since he used his principle to prove the existence of collections required by mathematics.

30-Second Philosophies by Barry Loewer

Tuesday, February 19, 2013

Aristotle’s Syllogisms

More than 23 hundred years ago, Aristotle noticed that in certain inferences it is impossible for their premises to be true and their conclusions false. An example is the inference from "All men are mortals" and "All mortals fear death" to "All men fear death."

In modern logic, such inferences are said to be deductively valid. Aristotle discovered that the validity of an inference depends not on its subject matter, but only on the form of the premises and conclusion. All inferences of the form "All Fs are Gs, and All Gs are Hs, therefore All Fs are Hs" are valid. He described a number of such forms, which are called "syllogisms."

Until the 19th century, the subject of logic pretty much consisted of Aristotle's syllogisms. But syllogisms are only a small portion of all valid inferences, and do not include many of the patterns of valid inference that are employed in science and mathematics. In 1879 Gottlob Frege devised a much more general characterization of valid inference that is sufficient for representing mathematical and scientific reasoning. A descendant of Frege's system, called "First Order Logic with Identity,“ is now generally thought to be capable of representing mathematical theories and proofs, and is taught to all philosophy students.

Remember: An inference (or argument) is valid when it is impossible for its premises to be true and its conclusion false.

Additionally: In the 20th century two great mathematical results were proved concerning first order logic: it is complete, and it is undecidable. Kurt Godel demonstrated that it is possible to program a computer to list all the valid inferences (completeness), and Alonzo Church demonstrated that it is impossible to program a computer to determine whether or not every inference is valid (undecidability).

30-Second Philosophies by Barry Loewer

Sunday, February 10, 2013

Worst Bestseller

One writer discovered the key to literary greatness: awful writing.
Everyone knows you can't judge a book by its cover. But the aphorism got an extra dose of validity in 1969, when Penelope Ashe, a bored Long Island housewife, wrote the trashy sensation Naked Came the Stranger. As part of her book tour, Ashe appeared on talk shows and made the bookstore rounds. But Ashe wasn't what her book jacket claimed. The author was as fictional as the novel she supposedly wrote-and both were the work of Mike McGrady, a Newsday columnist disgusted with the lurid state of the modern bestseller. Instead of complaining, he decided to expose the problem by writing a book of zero redeeming social value and even less literary merit. He enlisted the help of 24 Newsday colleagues, tasking each with a chapter, and instructed them that there should be "an unremitting emphasis on sex." He also warned that "true excellence in writing will be quickly blue-penciled into oblivion." Once McGrady had the smutty chapters in hand (which included acrobatic trysts in tollbooths, encounters with progressive rabbis, and cameos by Shetland ponies), he painstakingly edited the prose to make it worse. In 1969, an independent publisher released the first edition of Naked Came the Stranger, with the part of Penelope Ashe played by McGrady's sister-in-law.

To the journalist's dismay, his cynical ploy worked. The media was all too fascinated with the salacious daydreams of a "demure housewife" author. And though The New York Times wrote, "In the category of erotic fantasy, this one rates about a C," the public didn't mind. By the time McGrady revealed his hoax a few months later, the novel had already moved 20,000 copies. Far from sinking the book's prospects, the press pushed sales even higher. By the end of the year, there were more than 100,000 copies in print, and the novel had spent 13 weeks on the Times' bestseller list. As of 2012, the tome had sold nearly 400,000 copies, mostly to readers who were in on the joke. But in 1990, McGrady told Newsday he couldn't stop thinking about those first sales: "What has always worried me are the 20,000 people who bought it before the hoax was exposed."

Putting the Zen in Zzz's

If you've achieved enlightenment, chances are you're a sound sleeper.  So who better to offer sleep tips that Buddha?  Idea number one: Don't close your eyes. It's the same principle that made you bleary-eyed in your high school English class.  By tiring your eyes naturally - say, by reading until your eyelids droop - you fall asleep more peacefully, says Joseph Emet, author of Buddha's Book of Sleep.  Still can't sleep? Concentrate on your breathing to shut out other thoughts, or try a gratitude exercise: Count your blessings in addition to counting sheep.
 - Meghan Bogardus

Friday, February 01, 2013

Academic Success is More Important Now than Ever!

This isn’t a second half; it’s a whole new ballgame
We all know academic success is important (duh, that’s why we go to school). It gets talked about a lot in August as people gear up for the fall semester and a “new year.” But did you know that on average, college students earn lower grades in the spring semester than in the fall semester? This semester is a fresh new slate, so treat it just like the beginning! Here’s how:
1. Go to Class
This sounds ridiculously obvious. However, going to class is terribly underrated. It’s much easier to learn by actually attending and participating than just reading (which you may or may not have done).
2. Meet With an Academic Advisor
There are staff members on your campus that are paid to help you achieve academic success by making sure you take the classes to put you on the path to achieve your career goals. Why don’t you use them? Make sure you schedule an appointment with them at least once a semester to make sure you are in the right classes (some may not be in your major or college) to help you achieve your goals after college.
3. Have a Plan
We see so many cheesy tips about how everyone should have a study buddy or schedule time to study. But everyone learns differently, so catch-all tips are just nonsensical. It works much better to have a plan and know what works for you and create some
accountability to make sure you do it.