Applicability of Maths to the Universe

[polymath257]

There is an even deeper level. Alan Turing defined the concept of a 'computable real number'. In essence, it is one for which some computer program can compute the digits in the decimal expansion of that number.

So, for example, pi *is* a computable number: we have programs that will give as many digits of pi as you want. SImilarly for e and sqrt(2), and pretty much any number you have ever heard of.

But what Turing discovered (and it is not difficult to prove) is that there are real numbers (in the usual axiom system) that are not computable: no computer program can compute their decimal expansions. And, in fact, 'most' real numbers are not computable.

In what sense do these uncomputable real numbers exist in some Platonic realm?

This brings me to a somewhat related question I've had on my mind for a while. Why is it that such special numbers "found in nature" such as pi and e are irrational (and seemingly "arbitrary") as opposed to "clean" rational numbers? Is it to do with the particular number system being used (the decimal)?

Rationality is not affected by the number system used. A number is either rational or irrational. It is rational if it is a 'ratio' of two integers and irrational if not.

So, sqrt(2) is irrational. You cannot write it as ratio of two integers. The ancient Greeks knew this well before the decimal system for *describing* numbers.

Now, pi and e are more than just irrational; they are transcendental. In other words, they are not solutions of any polynomial with integer coefficients.

Well, it turns out that there are only countably many numbers that are *not* transcendental and uncountably many that are. So, in a sense, the vast majority of real numbers are transcendental. An 'arbitrary' real number is likely to be transcendental (and hence, irrational).