> Well, let's see: in Alice in Wonderland, Humpty Dumpty fell off a
> wall. This is true, isn't it? It is certainly true independent
> of our minds. Indeed, it is true in such a way that even when
> all humans have died, this universe will have a contained a life-form
> which produced an author who wrote a book in which Humpty Dumpty
> fell off a wall. But neither Humpty Dumpty nor the fact that he
> fell off a wall were ever true in this universe - only that this
> story was written, and that many people read about it and could
> converse about it.
> So if you believe that numbers have an independent existence, then you
> would definitely also have to believe that Humpty Dumpty exists.
> Both are products of the mind. Either both exist, or both don't
> (other than as brain patterns).
> As much as I would like Humpty Dumpty to exist, I'm afraid that
> it is not so.
The existence of numbers is not like the existence of objects, and I don't
think that most mathematical Platonists would say that it is. It is indeed
true independent of our minds that, if someone were to write a story
in which Humpty Dumpty falls off the wall, then Humpty Dumpty would fall
off the wall in that story. That is different to saying that Humpty Dumpty
actually did fall off the wall in the real world, such as it is. It is also
that given the axioms of Euclidian geometry, the angles in a triangle add
up to 180 degrees, and this is so independently of whether the angles of
a triangle in the real world add up to 180 degrees, or whether Euclid ever
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