Saturday, September 19, 2009
When does a question have no answer?
Have you ever had a feeling that one of your math homework problems couldn't be answered? You're not alone. Psychologists will tell you that this is a common human trait; when something is too hard for us to do, we'll often try to convince ourselves and others that no one can do it. Mathematicians are not immune to this feeling.
Sometime in the 1600's, Pierre de Fermat claimed to have a proof that xn+yn=zn has no natural number solutions when n>2. He never demonstrated his proof though, leading other mathematicians of his day to attempt their own proofs, to no avail. This result became known as Fermat's Last Theorem.
When n=2, this equation has infinitely many natural number solutions. Can you find one?
One of the most prolific mathematicians of all time, Leonhard Euler, was said to be so frustrated by his inability to prove Fermat's Last Theorem that he had Fermat's house torn apart, looking for any scrap of paper that may contain hints of Fermat's proof. Fortunately, Fermat was long dead by the time Euler did this.
Over the centuries, many of the world's most accomplished mathematicians tried to prove Fermat's Last Theorem. It wasn't until 1995 that Andrew Wiles, working in secret for seven years, finally provided a proof. Wiles worked in secret, because he didn't know if he'd ever find a proof and he didn't want others thinking he was wasting his time. To many mathematicians, the theorem seemed unprovable. After all, the world's best tried and failed for over 300 years.
What do we mean by unprovable? We don't mean that humans are unable to do it. We mean that, even in theory, no proof can possibly be given. Is it possible that such unprovable ideas or unanswerable questions could exist? In the early 1930's, the mathematician Kurt Göedel proved that not only is it possible that some mathematical questions can't be answered, but that there definitely are unanswerable questions. Essentially, Göedel's Incompleteness Theorem says that there are mathematical hypotheses that can never be proved or disproved. The worst part is, there seems to be no way of knowing which questions have this undesirable quality.
In the late 1850's, Bernhard Riemann proposed an idea that is now called the Riemann Hypothesis. He was never able to prove his idea. In the past 150 years, many of the world's best mathematicians have attempted to provide a proof. There have been hundreds of mathematical papers and books written on the subject. Some brilliant people have devoted their entire careers to finding a proof. Yet proof still remains to be found. In fact, if someone does prove it, they can cash in on a $1,000,000 prize.
Find the complex roots, win a prize!
Some people worry that the Riemann Hypothesis may be one of Göedel's predicted mathematical enigmas. But there is no known way to decide if it is or not. So, the frustration that you feel when you encounter a difficult homework problem, is felt the world over by mathematicians facing some of the most mind-boggling questions.
Sometime in the 1600's, Pierre de Fermat claimed to have a proof that xn+yn=zn has no natural number solutions when n>2. He never demonstrated his proof though, leading other mathematicians of his day to attempt their own proofs, to no avail. This result became known as Fermat's Last Theorem.
When n=2, this equation has infinitely many natural number solutions. Can you find one?
One of the most prolific mathematicians of all time, Leonhard Euler, was said to be so frustrated by his inability to prove Fermat's Last Theorem that he had Fermat's house torn apart, looking for any scrap of paper that may contain hints of Fermat's proof. Fortunately, Fermat was long dead by the time Euler did this.
Over the centuries, many of the world's most accomplished mathematicians tried to prove Fermat's Last Theorem. It wasn't until 1995 that Andrew Wiles, working in secret for seven years, finally provided a proof. Wiles worked in secret, because he didn't know if he'd ever find a proof and he didn't want others thinking he was wasting his time. To many mathematicians, the theorem seemed unprovable. After all, the world's best tried and failed for over 300 years.
What do we mean by unprovable? We don't mean that humans are unable to do it. We mean that, even in theory, no proof can possibly be given. Is it possible that such unprovable ideas or unanswerable questions could exist? In the early 1930's, the mathematician Kurt Göedel proved that not only is it possible that some mathematical questions can't be answered, but that there definitely are unanswerable questions. Essentially, Göedel's Incompleteness Theorem says that there are mathematical hypotheses that can never be proved or disproved. The worst part is, there seems to be no way of knowing which questions have this undesirable quality.
In the late 1850's, Bernhard Riemann proposed an idea that is now called the Riemann Hypothesis. He was never able to prove his idea. In the past 150 years, many of the world's best mathematicians have attempted to provide a proof. There have been hundreds of mathematical papers and books written on the subject. Some brilliant people have devoted their entire careers to finding a proof. Yet proof still remains to be found. In fact, if someone does prove it, they can cash in on a $1,000,000 prize.
Find the complex roots, win a prize!
Some people worry that the Riemann Hypothesis may be one of Göedel's predicted mathematical enigmas. But there is no known way to decide if it is or not. So, the frustration that you feel when you encounter a difficult homework problem, is felt the world over by mathematicians facing some of the most mind-boggling questions.
Labels: Euler, fermat, goedel, homework, riemann, wiles
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