Number theory

On a recent vacation, I finished a book on infinity by David Foster Wallace.

The book takes a close look at the idea of infinity. Infinity is a funny creature. For instance, there are an infinite number of numbers between 1 and 2. Similarly, there are an infinite number of numbers between 0.1 and 0.2. Are both these infinities the same?

Do numbers exist as abstractions? Arithmetic says yes. This yes is predicated on the realities of the physical world. We all agree that 2 oranges plus 3 oranges yields 5 oranges. The general rule that 2+3=5 is independent of oranges and has been codified as abstract arithmetic but underlying it is the understanding that addition makes sense only in the context of some "thing."

This would imply that numbers don't exist by themselves. They exist to quantify and reason about the physical world.

For a second, let's consider that numbers also exist as pure abstractions. To me, it isn't clear why 2+3=5 in such a world. Even more fundamentally, it isn't clear to me that the numbers 1 and 2 and 3 should appear one after another on the number line. A number line, in a world of pure numbers, could equally have 1 following 17 and preceding -400.

That we put a 1 before a 2 and after a 0 carries an implicit assumption within it: it is a consequence of our understanding of the physical world where producing one widget is harder than producing no widgets and easier than producing two widgets.

Seen in this way, the number line for pure numbers seems arbitrary and any reasoning done on the basis of it will have built in failings like the questions about infinity.

To me, maths seems to be an invention. It's a language to describe the physical world in non-woozy, unambiguous terms which no one can confuse. It is no different than C++ and should be treated as such.

Edit: in fact, we have a truly abstract set of symbols which don't have any intrinsic ordering and you are reading it. Apart from an artificial alphabetical order created just to satisfy bookkeeping needs, is there a real justification for saying the letter A is less than the letter B? If you would, consider the 'space' and '?' characters. Is the space less or more than the question mark? Now extrapolate to 1 and 2.