A week ago I came across this video online and have not been able to stop thinking about it since seeing it.  The title of Dangerous Knowledge was indeed intriguing in itself, but I almost turned it off when I realized it was a video about Mathematicians.  The video explores the paths of four Mathematicians that were literally driven to insanity by their pursuit in solving the paradoxes that consumed them.  I’ll be taking the four, Cantor, Boltzmann, Gödel, and Turing one at a time for a four part series exploration of the questions that plagued them.  Enjoy the video below and then we can then jump into Cantor and his Continuum hypothesis.

Georg Cantor’s problem is my favorite of the four because I think it is the easiest for me to think about and share with others in the same type of wonder and spirit Cantor did.  Cantor is fascinated by the concept of infinity, and I think the best visual example of the paradox of infinity is Galileo’s circles.  Galileo defines a circle as a shape with a series of infinite sides, then suggests the idea of drawing infinitely small lines from the center of a circle to the infinite points of the circle.  The problem comes when you draw a larger circle around the original circle and extend the lines to the edges of the new larger circle, there are now gaps between these lines.  How can the small circle have an infinite number of points if those same points are not enough to create a larger circle? 

In mathematic terms the paradox can be asked: which is larger the set of all real numbers between 0 and 10 or the set of all real numbers between 0 and 100?  On one hand you can say well 100 is ten times larger than 10 therefore it is the larger set.  However it is also possible to say well both sets are infinite and therefore equal in size.  The first explanation suggests that since one is larger than the other by a definite size, than neither is really infinite.  Yet the later explanation suggests that infinity has a single definite size which defies the principle you are trying to explain.  As you can already start to see there is no good explanation for the question at hand.

It gets more complicated, lets look at the set of all integers and see how far we can push this idea of infinity.  So you have an infinite set of numbers, now add to that set the number pi, how large is your set of numbers now?  You could certainly answer ∞+1, but wouldn’t ∞ by itself describe the number you are looking at as well?  Is it possible to prove or disprove that ∞+1=∞?  I’m just getting started here, go back to your original set of all integers, now this time subtract any one integer from the set.  The new set of numbers is still infinite in size!  If you subtract billions upon trillions of integers from the set, the set remains infinite in size.  How many integers do you have to subtract from the set of all integers to have zero numbers remaining in the set?  Is it possible to subtract an infinite number of integers from your set without running out of numbers?  Sure, just remove every odd integer and you still have an infinite set of even numbers. 

It is no wonder Cantor went mad, in the matter of a couple instantaneous thoughts you can convince yourself that ∞-∞=0 then switch to ∞-∞=∞.  Then in your mind you can look at the set of all integers and the set of all odd integers and say the set of all integers is twice the size of all odd integers.  Cantor tried to quantify different sizes of infinity and it is easy to see how ∞+∞=2∞.  On the other hand, how do you really distinguish between 2∞ and ∞?  At some point you have to ask how am I going to define the size of one ∞, and unfortunately the moment you put a definition on the concept of infinity the whole thing falls apart.  This is the battle that has been going on in my head for just one week, imagine doing it for a lifetime.

The final thing that I just want to touch upon because I found it interesting was Cantor’s religious views and God pushing him to explore infinity.  In mathematics infinity is certainly the concept that mirrors the religious concept of God the best.  It is easy to see how the pursuit of defining infinity became a religious quest to perhaps become closer to God for Cantor.  This idea will probably come up again when we get to Turing, who when dealing with the same issue of infinity convinced him of humans shortcomings.

Next up in the Brain Smash section will be Boltzmann and his uncertainty…