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Dangerous Minds: Cantor, Boltzmann, Godel, Turing And The Limits Of Mathematics And Logic.

David Malone looks at the revolutionary discoveries and troubled mental lives of four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing - who struggled against the accepted ideas of their time.

The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century’s mathematics. He believed he was God’s messenger and was eventually driven insane trying to prove his theories of infinity.


Ludwig Boltzmann’s struggle to prove the existence of atoms and probability eventually drove him to suicide. Kurt Gödel, the introverted confidant of Einstein, proved that there would always be problems which were outside human logic. His life ended in a sanatorium where he starved himself to death.

Finally, Alan Turing, the great Bletchley Park code breaker, father of computer science and convicted homosexual, died trying to prove that some things are fundamentally unprovable.

12:35 pm, by jamreilly7 notes Comments

Good documentary about the proving of Fermat’s Last Theorem. (1 of 5) via ContraWagner

09:27 pm, by jamreilly Comments

The Ulam spiral.  - Black pixels show prime numbers.
"An infinitude of prime numbers exists, as demonstrated by Euclid around 300BC. .. Given the fact that there is an infinity of primes, it is natural to seek for patterns or irregularities in the distribution of primes… The occurrence of individual prime numbers among the natural numbers is (so far) unpredictable, even though there are laws (such as the prime number theorem and Bertrand’s postulate) that govern their average distribution. Leonhard Euler commented :

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate."
In a 1975 lecture, Don Zagier commented:
 “There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.” “http://en.wikipedia.org/wiki/Prime_number#Distribution

The Ulam spiral.  - Black pixels show prime numbers.

"An infinitude of prime numbers exists, as demonstrated by Euclid around 300BC. .. Given the fact that there is an infinity of primes, it is natural to seek for patterns or irregularities in the distribution of primes… The occurrence of individual prime numbers among the natural numbers is (so far) unpredictable, even though there are laws (such as the prime number theorem and Bertrand’s postulate) that govern their average distribution. Leonhard Euler commented :

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate."

In a 1975 lecture, Don Zagier commented:

“There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.” “

http://en.wikipedia.org/wiki/Prime_number#Distribution

09:44 pm, by jamreilly1 note Comments