Thursday, December 20, 2007
Neither one, however, is actually the square root of 2. You can never write down all the digits for it; they are infinite. What our calculator gives us is a coarse approximation. Thirty-two digits might not seem like a coarse approximation, since five digits would probably get you safely to the moon and back, but our 32 digits only represent 0% (zero percent) of all the digits required to represent it exactly. If the digits of √2 are infinite, we can never write it as a ratio of two integers. We can approximate it as 7/5 (accurate to 2 digits) or as the ratio 25712973861329/18181818181818 (accurate to 14 digits), but it's still an approximation. The square root of two is incommensurable. It cannot be represented as the ratio of two integers, which is the definition of an irrational number (i.e. "irrational" literally means you can't represent it with a ratio).
The Pythagoreans believed that the universe was made of integers. Their religious dogma said god only created natural numbers (i.e. countable positive integers). When Hippasus, 2500 years ago, was able to prove that a number (√2) existed that couldn't be constructed from integers, the story is he was either exiled or drowned as a heretic.
There are still modern mathematicians who believe something similar to the old Pythagorean views. Amongst other things, some believe that in order for a mathematical concept to "exist", it must be physically constructable in our universe. They argue that Hippasus and Chaitin's arguments suffer the same flaw; proving that a number cannot be made in a particular way does not prove that it can be made at all (i.e. that it can exist). If I can prove that John Travolta isn't made of green jello, that doesn't mean that John Travolta must exist and be made of something else, it only proves that there is not a green jello John Travolta.
How can √2 be physically "constructable"? Let's say you have a right-triangle with two sides of length 3 and 4. The Hypotenuse of that triangle (from the Pythagorean theroem) is √(32 +42) = √25 = 5. The Pythagoreans were OK with this because 3, 4 and 5 are all nice integers and this triangle can be "constructed" and shown to actually exist. However, if you have a similar triangle with two sides both of length 1, the hypotenuse is √(12 +12) = √2 = 1.414....
Try to make this triangle. To construct it from equal integer sides, you keep running either too long or too short on the hypotenuse. If you had 10 units on each side, the hypotenuse would be 14 and a bit; an integer value would not complete the triangle. If you tried to get a little finer grained and make it with 100 units on a side, the hypotenuse would be 141 units and a bit. You could continue this Sisyphean task forever, but no matter how big you made the integer sides, a hypotenuse side made with an integer would never quite finish the triangle.
This is an important question because so far, to the best of our knowledge, our universe is integer. At one time the smallest particles were atoms, then protons, neutrons and electrons, then quarks, now maybe strings. Unless there is an infinite parade of smaller and smaller particles, the universe is necessarily made up of "smallest particles". There is no evidence of a infinitesimally small particles (particles that approach zero dimension), and as far as I know, nobody is seriously considering it.
That means that regardless of how many "smallest particles" with which we attempt to construct the triangle, the number is always finite. The hypotenuse never works out to be a single integer, and so the triangle can't be constructed in our universe.
The only way √2 could be physically constructed would be if the universe were infinite in size and you could line up the smallest particles across an infinite universe (assuming you had infinite particles in that universe) to make an infinitely sized triangle. That's mighty big.
So, in order for √2 to physically exist in this universe, we need one of two conditions: either the universe is infinite in size, or the universe is infinitesimally divisible. If one of those conditions aren't met, √2 is an asymptote, an unreal mathematical abstraction. You should no more believe that √2 actually exists in any real sense than you believe Thor or Santa Claus exists, no matter what your calculator says. I'm surprised that people don't come door to door with pamphlets trying to get me to believe in √2. Maybe I've found something to do on weekend mornings.
Do you believe in √2?
Burton MacKenZie http://www.burtonmackenzie.com/