Chapter 1 The Real Numbers
“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.”“Beloved God made the whole numbers; everything else is the work of human hands.”―Leopoold Kronecker, 1886
What is a number? is a question that you may not have given much thought to. If you’re like me, you were taught early on something like the following typology:
- natural numbers
- denoted \(\mathbb{N}\text{,}\) are the numbers 1, 2, 3, etc.
- whole numbers
- denoted \(\mathbb{W}\) or \(\mathbb{J}\text{,}\) are the natural numbers, together with 0.
- integers
- denoted \(\mathbb{Z}\) are the natural numbers together with their negatives.
- rational numbers
- denoted \(\mathbb{Q}\) are of the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q\neq 0\text{.}\)
- irrational numbers
- are numbers which can’t be written like \(\frac{p}{q}\text{.}\)
- real numbers
- denoted \(\mathbb{R}\) are the rationals and the irrationals together.
\(\mathbb{N}\text{,}\) \(\mathbb{Z}\text{,}\) and \(\mathbb{Q}\) are reasonable enough (though there’s a subtlety in the definition of \(\mathbb{Q}\) because, for example, we treat \(\frac{6}{9}\) and \(\frac{2}{3}\) as being ``the same number’’ even though they’re obviously not the same fraction). But the defintions of the irrationals and the reals are different. Notice that the definition above of ``irrational’’ sort of assumes the number in question already exists, and we’re just checking whether it’s rational or not. More fundamentally, people say things like ``the irrationals are the reals which aren’t rational’’ and ``the reals are the rationals together with the irrationals’’. If you think about it for a minute, you will realize that these two statements are equivalent, and also that they give us no way to say what a real number actually is.
So the question of what counts as ``a number’’ is at least worthy of deeper thought.
The context of the above quote in the philosophy of mathematics is an interesting one. Kronecker’s view of what counts as a number ended up being on the losing side of some really interesting arguments in the late 19th and early 20th centuries, and by including his quote I am emphatically not endorsing his entire position (which is known as intuitionist finitism if you want to read about it). But Kronecker is certainly right that the whole numbers (and we might as well throw in the integers and the rationals) have a kind of basic naturalness to them. I doubt Kronecker was actually making a religious claim by invoking ``der liebe Gott’’, but even if we translate that as ``Nature’’ or some other non-theistic word, his point does hold up. Whatever real numbers are, they’re going to be something we create.
