Here's a formalization of an old argument between me and adam and norm. (Informally, the question is, is there any way of agreeing on where exactly "the point" is, in downtown pittsburgh, for example?)

Let L_k be the set {(x,y) | x >= k and y >= k}. Let L be the set consisting of the set difference L_0 \ L_1. That is, an infinite ell-shape that has thickness 1, to the right of the y-axis and above the x-axis.

Now, consider the set C of smooth curves (x(t),y(t)) that lie in L, such that x(t) goes to infinity as t does, and y(t) goes to infinity as t goes to minus infinity. So the curve is supposed to swoop in from above, make some sort of bend near the origin, and zoom off into large-positive-x-land.

Consider the set T of angle-preserving transformations of the plane, that is, scalings, rotations, and translations. Not every t in T will carry every c in C to a t(c) still in C. That is, the curve t(c) may not still fit inside L. However, we will say that t is good for c if t(c) is still in the set C.

The question is: is there any nontrivial* mapping f, which, for every curve c in C, picks out a subset of c, such that for any c and any t good for c, we have that f(t(c)) = t(f(c))? In short, are there any nontrivial subsets of curves that are invariant under scaling, rotation, and translation?

*by nontrivial I mean, a map which always picks out a nonempty and nontotal subset of the curve.

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Oh, damn, I think the answer to the formal question is yes, which is contrary to the side of the argument I was always on. My approach depends critically on what I thought was an incidental feature of the formalization, the fact that the curve is pinned down to these narrow infinite strips.

I think talking about smooth curves is a red herring. I need what adam called "focus invariance", too: if you just look at a smaller window around "the point", you should get the same answer. If we're only talking about smooth curves, this isn't really a restriction, since the whole curve might be predictable from its nth derivatives of any point. So I definitely want general continuous curves, also because I don't want any cheating descriptions of points of maximal curvature: out in the real world we have to choose the length of our rulers, and the rivers are fractal.

Talking to Noam, he referred me to some papers of Girard's. A common point between one of them and the part of Lakoff I just read today is a decrying of the distinction between syntax and semantics. I think I need to meditate on that idea some more.

Lakoff made another comment later on that really surprised me:

Any theory of meaning at all, model-theoretic or not, must obey the following constraint: [...] The meaning of the parts cannot be changed without changing the meaning of the whole.

This seems like a very strong strictness condition, but I can't think of any counterexamples which would not admit reasonable objections. The recurring problem, when I try to imagine a dialogue between a supporter and detractor of the claim, is that the supporter seems to always be able to weasel around about what the parts mean.

For instance, if I say that "This microwave has a thingumabob to set the temperature" has the same meaning as "This microwave has a widget to set the temperature", then the supporter says, "well, then thingumabob and widget are both nonsense words of equal standing, therefore they have the same meaning". However, I could easily say "the thingumabob is above the widget, turn the thingumabob ninety degrees to the right", and in this case they clearly mean different things.

I really hope I'm setting up a strawman, though, because Lakoff clearly can't in good conscience think that the "parts" have an objective meaning inependent of the context-providing "whole", can he?