So from that point of view, where the real numbers exist in the same way infinity exists -- they're not really numbers but ideas about numbers. Pi isn't a number but an idea about the circle, which isn't a shape but an idea about shapes. Is that where constructive math takes you? This is all quite fascinating but totally unfamiliar to me.

They can be numbers and abstractions. There's not really any definitive singular notion of what a number is, so it wouldn't be very useful to say something like pi isn't a number. (If you can do arithmetic on it, chances are it is a number, regardless of what else it represents)

Indeed, numbers are abstractions. Even the number ‘5’ is an abstraction, let alone the numbers that cannot even be measured precisely - not even in principle.

>idea about a circle

As if a circle is a real thing. (Circles do not exist, either - or, rather, they ‘exist’ in a very particular sense of the word, as they are themselves abstractions; round things do.)

> As if a circle is a real thing. (Circles do not exist, either - or, rather, they ‘exist’ in a very particular sense of the word, as they are themselves abstractions; round things do.)

Well, yes, exactly. The thing that intrigues me about this discussion is that it seems like mathematicians have a much clearer idea than I do about what the differences between these kinds of abstractions are.