You forgot that Euclid specifies a circle as a plane figure. — fdrake
No I didn't.
I realise you're not going to accept that a great circle is not a Euclid circle, or that a circle in a plane at an angle isn't a Euclid circle without a repair of his definition — fdrake
See:
Yet perhaps it is not a torus but is nevertheless a set of coplanar points, falling on an implicit plane which possesses a spatial orientation. Is it a circle then? Not strictly speaking, because two-dimensional planes do have not a spatial orientation. — Leontiskos
I've been using the word "verbatim" to try to mean a couple of things:
A ) At face value.
B ) Using only the resources at hand in a symbolic system.
Thus Euclid's definition of a circle, verbatim, would exclude the great circle. — fdrake
But it is here illustrative that I am not familiar with the concept "great circle," especially as to its specific geometrical properties, and I did query you about the picture you posted. You thought there was a verbatim sense of "great circle," but you were mistaken. You would have to explain what you mean by it in order to achieve your contradiction, because "great circle" says very little, verbatim.
And if you want to just talk about your intuitions without recourse to formalism, I don't know if this topic of debate is even something you should concern yourself with. — fdrake
I think you're moving too fast. Formalisms have limits. What are the specific properties of lines, points, circles, great circles, two-dimensional planes, three-dimensional planes, etc.? How do they relate to each other? For example, can points be deleted or not? Is the great circle a torus, and if not is it three-dimensional at all? You're making a bunch of assumptions in all of this and drawing a fast conclusion.
But the deeper issue is that I don't see you driving anywhere. I don't particularly care whether the great circle is a Euclidean circle. If you have some property in your mind, some definition of "great circle" which excludes Euclidean circles, then your definition of a great circle excludes Euclidean circles. Who cares? Where is this getting us?
If you actually want my perspective on things, rather than trying to illustrate points from the paper: I'm very pragmatist toward truth. I prefer correct assertion as a concept over truth (in most circumstances) because different styles of description tend to evaluate claims differently. As a practical example, when I used to work studying people's eye movements, I would look at a pattern of fixation points on an image - places people were recorded to have rested their eyes for some time, and I would think "they saw this", and it would be correctly assertible. But I would also know that some subjects would not have had the focus of their vision on some single fixation points that I'd studied, and instead would have formed a coherent image over multiple ones, in which case they would not have "seen" the area associated with the fixation point principally, they would've seen some synthesis of it and neighbouring (in space and time) areas associated with fixation points (and other eye movements). So did they see it or didn't they?
So I like correctly assertible because it connotes there being norms to truth-telling, rather than truth being something the world just rawdogs into sentences regardless of how they're made. "There are 20kg of dust total in my house's carpet"... the world has apparently decided whether that's true or false already, and I find that odd. Because it's like I'm gambling when I whip that sentence out. — fdrake
Okay, thanks. And I agree with this. I am interested in knowledge—including justification—as opposed to just truth. Very often justified knowledge is precisely that which has been (correctly) logically inferred. I would define logic as that thing that gets you to (discursive) knowledge, or at least to justified assertion.
I would agree that every quantification is into a domain, and I don't think there are context independent utterances. I do not think it follows that there is no metaphysics. I'm rather fond of it in fact, but the perspective I take on it is more like modelling than spelling out the Truth of Being. I think of metaphysics as, roughly, a manner of producing narratives that has the same relation to nonfiction that writing fanfiction has to fiction. You say stuff to get a better understanding of how things work in the abstract. That might be by clarifying how mental states work, how social structures work, or doing weird concept engineering like Deleuze does. It could even include coming up with systems that relate lots of ideas together into coherent wholes! Which it does in practice obv. — fdrake
And this sounds a lot like Srap's approach. I was encouraging him to write a new thread on the topic.
Plato's phrase, "carving nature at it's joints," seems appropriate here. I would say more but in this I would prefer a new or different thread (in the Kimhi thread I proposed resuscitating the QV/Sider thread if we didn't make a new one). I don't find the OP of this thread helpful as a context for these discussions touching on metaphysics.
I would have thought it clear how it relates to logical pluralism. If you model circles in Euclid's geometry, you don't see the great circle. But if you look for models of the statement "a collection of all coplanar points equidistant around a chosen point", you'll see great circles on balls (ie spheres, if you don't limit your entire geometry to the points on the sphere surface). They thus disagree on whether the great circles on balls are circles.
If you agree that both are adequate formalisations of circlehood in different circumstances, this is a clear case of logical pluralism. — fdrake
So:
Let's suppose it is a countermodel. How does the logical pluralism arise? I can only see it arising if we say that a "circle" means both Euclid's definition and the great circle countermodel, and that these two models are incompatible. Is that what you hold? — Leontiskos
For the univocalist the two definitions are incommensurably different. For the analogical thinker there is an analogy between a great circle and a circle. I think both adhere to the definition, "A set of coplanar points equidistant around a single point," but this also involves analogical equivocity between 2D planes and 3D planes.
That also lines up just fine with my view of logic. If logical pluralism means there are incommensurably different logics which are true/correct, then I disagree. If it means there are analogically similar logics which are true/correct, then I agree. But I don't think that all true logics are isomorphic. "Incommensurably" is meant as strong incommensurability, in the sense of excluding analogical equivocity.
The taxicab example is designed as a counterexample to the circle definition "a collection of all coplanar points equidistant around a chosen point", since the points on the edge of the square in Euclidean space are equidistant in the taxicab metric on that Euclidean space. It isn't so much an equivocation as highlighting an inherent ambiguity in a definition. — fdrake
Again, I think there is an equivocation on "distant." Equidistant qua circularity pertains to straight lines. The taxicab circle is premised on an extreme redefinition of "distance" - an equivocation.
The extensional difference between all of these different formalisms are the scope of what counts as a circle. A pluralist could claim that some definitions work for some purposes but not others, a monist could not. — fdrake
Although I don't hold to logical monism, this doesn't seem right. You are claiming that for the logical monist a token such as 'circle' can mean only one thing. I don't think that's right.
The Analytic dispute between logical pluralism and monism strikes me as a superficial dispute. The deeper question is univocal vs. analogical predication. That source abandons the more interesting question as soon as it limits itself to, a "model-theoretic definition." Pluralism looks like a poor man's analogicity, like trying to draw a perfect circle with pixels. My guess is that most versions of soft pluralism and monism are not even differentiable, unless there is some precise concept of "equally correct" logics or arguments (which I highly doubt).
To put it in super blunt terms, Euclid's theory would have as a consequence that the great circle on a ball is not a circle. The equidistant coplanar criterion would prove that the great circle on a ball is a circle. Those are two different theories - consequence sets - of meaningful statements. A pluralist would get to go "wow, cool!" and choose whatever suits their purposes, a monist would not. — fdrake
If they are different theories then they define different things, i.e. different "circles." The monist can have Euclidean circles and non-Euclidean circles. He is in no way forced to say that the token "circle" can be attached to only one concept.