""" For lack of a better (or more ironic, since Euler went blind) dichotomy lever, the operational conception might be called Lagrangian and the latter Eulerian. From a Lagrangian perspective any point in an open ball is infinitely far from the outer bound as long as our operations are functions of that outer bound. But from an Eulerian perspective, it's trivial to see the boundary is just fuzzy and all we need do is take constant steps to leave the ball. That renders the koan a simple fallacy of ambiguity, hinged on the conception of "center". """
I am unsure that I can address everything here, and I feel like my reasoning has been sh!t lately, but here is an attempt to work out your suggestion of a "fallacy of ambiguity"[⊬]: a) the term shared by the two premisses b) the subject of the conclusion c) the predicate of the conclusion a) Perhaps here you are talking about *center of a ball* as the shared concept between a Lagrangian and an Eulerian perspective? Or maybe the *center of a ball* as shared between the p-adic and euclidean conceptions of space? In this last case, maybe the "missing premise" is that we are making an analogy to balls and centers when we move to non-archimedean norms? The fallacy might then appear as: "All Euclidean balls have a single center. x is the center of a p-adic ball. Thus x is the single center of the p-adic ball." b) I am not sure how to finagle this one a wholes and parts argument, but let me try. Maybe it is that centers are things that balls have but are not properties of points? That attributing center to a point becomes a category error? c) Perhaps you feel that *center of a ball* fails operationally, that centers of balls are singular by their very nature, privileging insights of Archimedean experiences? I haven't worked it out, but I (perhaps falsely) assume that any one of these centers can be handled as a center of mass, a barycenter, maximally situated away from the ball's closure. Here, I suppose, is where your Gordian step is apt? From each "center" it takes the same number of constant steps to leave. All of this is to say that I would like to better understand the *fallacy*. In terms of the larger metaphor, I like the image of many individuals, all within the scope of one another, granted the center of their shared milieux. [⊬] https://www2.palomar.edu/users/bthompson/Fallacies%20of%20Ambiguity.html
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