The 3n+1 Phenomenon

Here is an interesting experiment: Pick any number. If it is odd, then multiply it by three and add one. If it is even, then divide it by two. Repeat this procedure until you obtain one and then stop. For instance, if you had picked the number 7, you would have obtained 3x7+1=22, 22/2=11, 11x3+1=34, 34/2=17, 17x3+1=52, 52/2=26, 26/2=13, 13x3+1=40, 40/2=20, 20/2=10, 10/2=5, 3x5+1=16, 16/2=8, 8/2=4, 4/2=2, 2/2=1.

Computers have verified that such a procedure will stop at one for all numbers up to about 4.79 x 10^17 - see the website http://www.ericr.nl/wondrous/index.html. But no one knows for certain that this procedure will stop at one for all numbers. And no one will ever be certain. Why?

Because in order to be certain of such, it is necessary to test the procedure out on all numbers. And since there are infinitely many numbers, doing such must take an infinite amount of time. Since we humans are finite beings, we don't have an infinite amount of time, so if the procedure always halts at one, we can never know this with absolute certainty - only G-d can know this. (For a rigorous mathematical proof of this fact, see http://arxiv.org/abs/math.GM/0312309.)

So this famous mathematics problem, which is known as the 3n+1 problem, is a great illustration of our limitations as human beings to comprehend our universe - it is a problem that is so simple that even a second grader can comprehend it, yet so complex that all of the mathematicians in the world will never be able to prove deductively that the procedure will always stop at one.

According to the ancient Greek humanistic view of the world, reason and logic are what make the world go around. According to the Jewish Torah view of the world, G-d is what makes the world go around. The 3n+1 problem demonstrates to us all that if the 3n+1 procedure stops at one for all possible numbers, then it is not because reason and logic dictate that it must do such; it is because G-d, Who created mathematics, decided that the 3n+1 procedure must stop at one for all possible numbers. From this simple mathematics problem, we all can see clearly how the ancient Greek humanistic view of the world is seriously flawed.