Yesterday, I came across this amazing documentary entitled: Dangerous Knowledge. It's been passed around the internet quite a bit, so there's a possibility you may have seen it pop up since it's debut yesterday. The website, Best Free Documentaries, has got it right here:

http://bestdocumentaries.blogspot.com/2007/09/dangerous-knowledge-full-documentary.html

Dangerous Knowledge is about 4 amazing mathematicians who descended into madness trying to solve some impossible questions..

Canter, a German mathematician, is the first subject of this documentary born 1845 died 1918. His birth of madness began with trying to understand the infinite. He could not accept that one infinite space is equal to another infinite space..

for example: a 1 by 1 square contains an infinite amount of points. well, a 2 by 1 square also contains an infinite amount of points. Through Mathematics, both infinities are equal. Well this can't be! The 2 by 1 square has twice the area as the 1 by 1 square. Surely the infinite space of the 2 by 1 is at least twice the size as the 1 by 1 square. But Mathematics says No! It's the same.

Logically this makes no sense, so Canter sought to proves that there are an infinite amount of infinities, which became known as Canter's Theorem. Well.. doing so totally unraveled the thought of mathematics. Simple things became untrue. This doesn't make any sense either.. So he began working at it the other way.. Since these simple basics of mathematics are true, then they shall disprove his infinite infinites.. Well he found out to also be impossible.

Day by Day he worked on this until his son died.. and with his son's death, he lost a part of himself. He killed himself because of what he lost because of his failure to prove or disprove his theorem. There's also some religious mumbo-jumbo in there too.

Boltzmann is our second case and lived in the same time frame, born 1844 died 1906. Boltzmann was physicist and hypothesized that time is irreversible. Time is not something that can be defined on two way plane, it is an ever-driving forward concept that can never be stopped.

Today, we already know this. In fact, this is what modern physics is based on today. Boltzmann, unfortunately, discovered this fact 20 years too early, before later physicists adopted the truth. But because he lived in a time of unquestionable obedience to world leaders, such a massive concept was utterly opposed because of what it implicated.

The severe oppositions to Boltzmann's discovery drove him into a deep depression that he escaped when he committed suicide.

The final two cases are of Kurt Godel and Alan Turing, who both saw Canter's work, believed to have understood it, and began to solve it in their own ways. Both men were born around the times of the Boltzmann's and Canter's deaths.

Godel was an Austrian mathematician who fell into the same madness that Canter had suffered from, but began looking at the problem a little differently. Through trying to understand why infinite was undefinable, he began to understand certainty and completeness. He set forth to define these abstract topics..

Out of these 4, Alan Turing is the most I was familiar with prior to the film. Alan Turing is particularly famous in the field of Artificial Intelligence and modern Computer Science as well. He figured that if this theorem was actually solvable then a Computer could surely figure it out.

This where Turing had developed one of his most famous Theorems: the halting problem. The basic thought of the theorem is how do you tell if a problem is unsolvable.. In AI class back in Platteville, Professor Hasker put this problem in terms of finding infinite loops in your code. Is there ever a way to determine whether your program is stuck in an infinite loop, or does it just take a really long time to run? How do you determine this way code?

//paraphrased from old notes from Hasker's class
//http://www.uwplatt.edu/csse/courses/prev/s07/cs303/notes/303n01.html
public void work(String input)
{
   if( terminatesOnInput(input) )
   {
      //statement #1
      sysout("Program can operate on passed input");
   }
   else
   {
      //statement #2
      sysout("Program is caught in an infinite loop while processing input");
   }
}

There no possible way statement #2 will ever be called, because the terminatesOnInput will be stuck in an infinite loop, BUT how do we know that the input passed is such that it just takes a while. Are you going to wait a week to see if the input takes a week to process? Or is it just unsolvable?

In the real world, if something takes more than a minute or two, there's something wrong.. but in the research field, in philosophy and mathematics, how long is too long?

How could you not go crazy thinking about this? Turing began to think of Canter's continuum theorem as the difference between humans and machines. Machines are irrefutable logic devices which can solve anything, so us, as humans, most possess an extra attribute beyond the capacity of a computer to go above and beyond. He thought of this as our intuition, but he figured that machines had the capability to intuitively seek out answers and grow..

He absorbed himself so much in this problem that he could not tell the difference between Human and Machine. What was it to be human, and how could this NOT be replicated with machine..

Anywho, to Turing's defense, foundationally it wasn't the halting problem or the continuum theorem that ultimately led to his death. He was homosexual, in a time when homosexuality was a mental deficiency. He was very open about it as well, which didn't take too long for "concerned" citizens to commit him to an insane asylum.

The treatment for homosexuality was pumping him full of estrogen, which doesn't make any sense at all early-mid 1900 doctors. Turing began to grow breasts and other womanly features.. Turing began to contradict himself as he contradicted humanity.. and killed himself as a result of overwhelming uncertainty.

Pretty crazy stuff, eh? Definitely check out the full hour and a half commercial free presentation linked above at Best Free Documentaries. Well worth the watch or listen to. Especially since they refer to Math as Maths.

I should really check out more mathematics documentaries.. That one we watched for Math History about Fermat's Last Theorem was an equally interesting watch.