This is a historical post.
Did you ever hear the one about the dyslexic, agnostic, insomniac? He stayed up all night wondering if there really was a dog.
Like many people, I’m curious about the nature of reality and really do sometimes stay up all night wondering about… well, just about anything.
A Slice of PI
What I find so fascinating about logic and reason are that they do work. For example, what is the value of PI?
PI is defined at the ratio of the circumference to the diameter of a circle. In school, I was always taught to use 3.14 as the value for PI. But actually this is just an approximation. Supposedly, PI actually goes on infinitely past the decimal, apparently never even repeating. You can’t even turn PI into a fraction in the form of one whole number over another, so this makes it a classic so-called “irrational number.”
But have you ever thought to ask how they know all that? How do they even know that the ratio of the circumference to the radius of a circle is approximately 3.14?
Do they take a measuring tape and measure it and find out its 3.14? I confess, I once read a book that suggested actually trying to measure the circumference and radius of a circle to calculating PI this way. I got 3.16 as my answer. Presumably the reason I didn’t get the right answer is because my measuring tape wasn’t infinitely accurate and because my circle wasn’t a perfect circle.
Actually, true circles doesn’t even exist in physical reality. But if true circles don’t exist in real life, then how the heck can we confidently say that the ratio of the circumference to the radius of something that does not exist is 3.1416…etc?
Maybe since true circles are just imaginary anyhow that’s how we came up with it? If I decide that there is something called a barf-fat and it only exists in our minds, then can’t I pretty much make up anything about it that I want? If this is true, then why can’t we just take a vote and decide to change the value of PI to 3. Now personally if PI had a value of 3 there would be many advantages over an infinitely long number that is impossible to memorize. Think about how much easier it would be to teach children about PI if its value was 3. Plus, think of all those mathematics and physics equations that use PI that would suddenly be easier to calculate. Since circles don’t really exist except as figments of our imagination, I am going to personally start writing to my congressmen today to make sure the “PI is 3” law passes the next time congress meets.
What is Real?
So just exactly how do we know what the value of PI is if we can’t measure it?
Well, here is one way you could do it. Take a look at the following picture that I took from Mathematics for the Million: How to Master The Magic of Numbers. I love this book because it teaches math from a historical perspective, showing what problems the ancients were trying to solve as they discovered various mathematical principles.
As this picture shows, we can take our ‘imaginary’ perfect circle and pretend to split it up into boxes. The boxes actually come in two sets, those that bound our circle (i.e. the circle is inside of the boxes) and those that our circle bounds (i.e. the boxes are inside the circle.)
Now imagine that each of these boxes is exactly equal in width. Note that this means that the “inner” and “outer” boxes in a quarter section are all exactly the same except for one box. In our example we have one quarter circle with 9 boxes and one with 10. But other than that extra boxes, the boxes are identical.
Now let’s assume that our circle is a unit circle, so by definition its radius is 1. It doesn’t matter what it’s “1” of. It could be 1 foot, or 1 inch, or 1 mile. It does not matter for our purposes so long as it’s “1” of some unit. Maybe think of it being “1” on one of those number lines that you used to draw on back in high school that had no units.
Now if you remember back to your old geometry days, you might remember that the area of a circle is PI*Radius^2.  Given that our radius is known to be 1, we know that means the area of our circle is equal to PI because 1 squared is still 1 and PI X 1 = PI.
Now if we know that the boxes are exactly equal in width and we know the radius of the circle is 1, then we already know the width of each box. From here we can use geometry to figure out the area of the boxes. 
The end result is that we now have the area of two sets of boxes, one that is known to be larger than the area of the circle and one that is known to be smaller than the area of the circle. For our example picture, the result would be 3.44 for the outer boxes and 2.64 for the inner boxes. So we can now be absolutely certain that PI is less than 3.44 and greater than 2.64.
Now double the boxes. Then double them again. In fact, double them as many times as you have patience for. The end result is obvious: you are now able to calculate PI to any level of precision you want. Just keep adding boxes until the upper and lower bounds match out to as many decimal places as you wish. Throw the rest away. 
You now know at least one way to calculate PI.
What is so fascinating about it is that you know it works. In your heart, now that you understand how it’s done, you know it’s completely reliable. Legislating what PI is now seems as silly as voting on whether or not the dinosaurs once lived on the earth.
Now how is it that a purely mental concept like a circle – and remember, it does not physically exist anywhere in universe – can have such a specific and computable characteristic like PI?
I’ll tell you why. It’s because circles really do exist. I can’t get over the significance of this. Something that only exists in our minds actually and really objectively exists and we can prove it beyond doubt. Mathematics is real.
I also can’t miss the display of beauty in math and reason demonstrated by this example. I’m left with several questions worthy of further discussion:
- Why is it that made up things can really exist objectively?
- Is Math (like circles or PI) “invented” or “discovered?”
- I mentioned that this example was “beautiful” to me. Is that a subjective “in the eye of the beholder” sort of thing? Or could beauty be objectively real too?
- What do we even mean by the word “beauty” in the first place? Why is math, nature, and my wife all beautiful? Is that one word for three things or one word for one thing?
 See Mathematics for the Million, p. 217 and p. 154
 Mathematics for the Million, p. 217-218 for full details
 Actually, this algorithm to solve for PI is what we call intractable. Though in principle we can find PI to any arbitrary decimal point, in practice once you get past a certain point the procedure becomes too computationally intensive and even a super fast computer will never complete prior to the universe collapsing into a big crunch – at least not with current technology. There are better ways to find PI that aren’t so process intensive using calculus. See this link here for details.