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My understanding is that many "intuitively obvious" statements about continuous functions, like the intermediate value theorem, are only true in the real number
by vladTheInhaler 8y ago
My understanding is that many "intuitively obvious" statements about continuous functions, like the intermediate value theorem, are only true in the real numbers, not the rational numbers. So you might say that we defined the reals so that two intersecting lines actually have to touch somewhere. That seems like a reasonable motivation to me.
- acjohnson55 8y agoAren't you presupposing that lines are themselves a continuum? It's a reasonable abstraction, but as far as we know, it's not physical real. If a line were just a collection of discrete points related by some formula, there's no reason two lines that aren't parallel should have to intersect, from the perspective of sharing a point.
- vladTheInhaler 8y agoYou're absolutely right that it might not reflect physical reality. I don't think it's known whether space is fundamentally discrete or not. But I can definitely say that continuous space meshes a lot more with my experience as a human being, and that because of that, the real numbers are motivated by more than just taking square roots. They model the human experience of space.