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> any such model also exists in reality Yes, but there's still a distinction between the model and the thing modeled. A map of your city is not the same as the
by pdonis 3y ago
> any such model also exists in reality
Yes, but there's still a distinction between the model and the thing modeled. A map of your city is not the same as the city, even though the map is a real object in the same reality as the city.
Similarly, real numbers are a model that is instantiated in the brains of mathematicians, papers in the literature, etc. Those things are of course real objects. But real numbers are used to model lots of other real things.
- xanderlewis 3y agoOf course there’s a distinction between a map and the thing it’s a map of. But they’re both parts of reality.
- samus 3y agoIt's also perfectly possible to draw maps of places that don't exist in reality, i.e., that only exist in somebody's brain.
- xanderlewis 3y agoA good point. So the real numbers exist in the same sense that anything else they might describe does. My original point was really a comment on the way that the subject is often taught: the real number system is not introduced as some arbitrary set of rules that is as good as any other (though of course it is); it is introduced as if it were as incontrovertible an idea as, say… integer addition. I don’t think it is. It can perfectly well be argued that it makes sense and is useful, but its naturality is another matter. I personally find the leap from R to C less controversial than that from Q to R.
- samus 3y agoYes, I agree. Real numbers are a quite nebulous construct for which the need is a bit hard to see. Complex numbers are basically a hack to solve quadratic equations in all cases.