Patricia and I were talking last night about math and the words used describe things mathematical, like elegant and charmed.

I have always been struck that one of the simplest and most basic relations in mathematics results in a truly irrational number. The circle is arguably the earliest and by far the easiest form to draw. Pick a point, stretch a string and draw a circle around the point. Yet, in our mathematics, the relationship between the length of the string and the circumfrence of the circle is pi. Mathmeticians have computed the value of pi to a gazillion places, and still no pattern emerges. It is a random number.

This leads me to two possible conclusions. The first is that our system of mathematics is inherently flawed since the measurement of a fundamental relationship results in randomness.

The second is that our system of mathematics is correct and thereby recognizes that a full description of the universe depends on the intertwining of the rational and the irrational. It is a marriage of understanding and idicates the a priori necessity of both. The inability to measure art is no worse than the inability to measure pi.

Bruce responds:

I hope you find the following comment on one of your recent posts interesting:  

it is not accurate to call Pi a random number. irrational and random are two different things; different concepts entirely. an irrational number is a number that cannot be expressed as a fraction where the numerator and denominator are integers (p/q).

it IS true that the digits of pi can be used as a simple random number generator. I suppose one could say that any given digit of Pi is random (actually, even this in not correct, as obviously there is a method for calculating it, and the 575th digit will always be the same no matter how many times you calculate it), but the number itself is not random at all!

many of the most important numbers are irrational numbers, for example, e (which is the base for what is called the "natural" logarithms and is perhaps the most important number in all of mathematics, with the possible exception of 0 and Pi: http://mathworld.wolfram.com/e.html), and numbers such as 2^1/2 (square root of 2 -- I don't feel like rooting around for the square root sign), 3^1/2, etc. if memory  serves me correctly e is about 2.718... all irrationals have the same lack of pattern as you extend out to more decimal places, the feature of pi that you were noting.

the rationals (numbers that CAN be expressed as p/q, where p and q are both integers) and the irrationals together constitute the reals. then again, there are an infinite number of numbers that are NOT real, the most well-known of these are the complex numbers and the imaginary numbers. a complex number has an imaginary coordinate or part and a real coordinate/part. the imaginary part is simply a real number multiplied by a number known as i, which is the square root of -1. you can probably see why  hese numbers are called imaginary!

to make it even odder, if memory serves me correctly, while both the rational numbers and the irrational numbers are infinite in number, I think it has been proven that there are MORE irrational numbers than there are rationals, i.e. one group (actually, set) is MORE infinite than the other, or more precisely, one infinite set is larger than another infinite set.

while much of mathematics is odd and counter-intuitive, there really can't be much doubt that it is correct.