quote:Originally posted by JonHecht: Raymond, is part of the definition of a square "not a circle". Is part of the definition of a circle "not a square"? If not, then it is conceivable that in some possible world there can be a square circle, as it doesn't contradict analytic nature of the shapes.
Aren't circles 1 sided by definition? This would, as far as I know, preclude them from also being a polygon.
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I think it depends on how you want to define it. I think the official definition is the "set of points equadistant from a particular point." It also happens that that set is one sided (or infinite sided, depending on how you look at it). If you included that in your definition, then yes, I'd say that precludes them from being squares.
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It also occurs to me that rather than sitting here arguing about the semantics of squares and circles, I could just say "square hexagons" which is a much more blatant contradiction. I'm sure there's a universe out there where things can have four and six sides at the same time, but I do not think it's appropriate to call those things either squares or hexagons.
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I don't think that you can have a geometry in which a given shape can be said to have 2 different numbers of sides, though I'll admit that I don't have a proof handy at the moment.
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A large part of the problem is that there are several different sense of "square", "circle", and especially "meaning" that are being used.
If we're just talking linguistically, of course they could be overlap. But that's boring.
If we're talking perceptually (to someone using definitions from this reality, but able to perceive as necessary to be in the other reality), then I could certainly see the argument, and there can almost certainly be no counter-proof.
If we're talking mathematically, we'll need to narrow in on what we mean by "square" and "circle", as those are not precise concepts in mathematics. There are sensible mathematical definitions by which it would be impossible for a square and a circle to ever be the same thing, and I'm pretty sure sensible mathematical definitions by which it would be easy.
I've noticed several of the people in this conversation working from different senses along the lines of the above, and so have not been surprised at people talking past each other. If you don't even vaguely agree on what context the comparison will take place in, you aren't likely to agree about the landscape of possible comparison results.
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I have no idea how such a reality might be feasible, but given how bizarre our own universe has turned out to be sometimes, I'm not going to assume anything about what might be possible in a completely different universe.
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quote:If we're talking mathematically, we'll need to narrow in on what we mean by "square" and "circle", as those are not precise concepts in mathematics. There are sensible mathematical definitions by which it would be impossible for a square and a circle to ever be the same thing, and I'm pretty sure sensible mathematical definitions by which it would be easy.
I've noticed several of the people in this conversation working from different senses along the lines of the above, and so have not been surprised at people talking past each other. If you don't even vaguely agree on what context the comparison will take place in, you aren't likely to agree about the landscape of possible comparison results.
I agree, which is part of why I switched the Squares and Hexagons since there's much less room of interpretation.
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