Hi Alex,

You seem to have partially agreed with my claims while rejecting my arguments :-( However I do not yet understand why you rejected them. So let's get a little more precise.

"Avior argues that Euclid's axioms admit various interpretations. Some apply to the sphere, some do not. Which shows a way of salvaging Euclid's proof. Brodie argues that Euclid had only one interpretation in mind: the one that follows the common planar intuition supported by the diagramatic view. All interpretations could be looked at as the models for Eulcid's logical argument. And a valid argument must be correct in all interpretations. Euclid's fails this criterion."

I'm not sure whether my own arguments relied on any specific interpretation of Euclid's. There is the (marginal in the present context, as I see it) issue of the plane-separation axiom. Euclid did not provide such an axiom, but I think that we can, in contexts such as the present one, interpret him as if he did hold such an axiom implicitly. I rely here on a familiar, plausible and important principle of interpretation: the principle of charity (see e.g. http://en.wikipedia.org/wiki/Principle_of_charity). This principle dictates that one ought to try and interpret the other so as to make the other as much rational as possible, and in particular so as to make the other's statements as much true as possible (by the interpreter's standards). The philosopher Donald Davidson famously argued in his writings that the principle of charity is not an option, but a necessity, if an understanding of another is to be achieved at all.

On the other hand I find the "formalistic" approach:

"All interpretations could be looked at as the models for Eulcid's logical argument. And a valid argument must be correct in all interpretations. Euclid's fails this criterion"

odd. It would mean that Euclid's argument is to be judged as a purely syntactic one, which is then subjected to many semantic interpretations. But this is surely not what Euclid himself meant (the clean separation of syntax and semantics wasn't seriously conceived before the 20th century). And surely there are better interpretations of any text, and less fortunate ones. We never demand that any text ought to satisfy all interpretations.

Again, substantially, I think that the above is a marginal issue in the present context. Because I don't see that Scott's counterexample against the EAT describes a situtation where a plane-separation axiom does not hold. Instead, it relied on the occurrence of antipodal points, which connect many different lines. So to criticize Euclid for not providing such an axiom is not strictly relevant.

Next, I suggested that Scott's spherical-triangle counterexample, even after making it a "skinny" one, still relies on the feature of antipodal points. I asked for a counterexample with an overall short triangle, i.e whose every side is less than 1/4 the circumference (of the sphere). I indicated that I don't see such an example. You replied:

"Why? Open up - if only a little - Scott's 90-90-90 triangle to make it 90-91-90, and then lift the equator slightly so that the base angles do not change by more than .25°. Surely you can do that by continuity."

I don't see how this gives what I asked for. Scott's 90-90-90 triangle has three sides whose length is 1/4 the circumference. If we "open it up" - widen it - we make one of the sides - the equatorial one - even longer. If we then "lift" that side, its length stays the same. So how is this supposed to describe a triangle whose every side is less than 1/4 the circumference?

The question can also be raised from the other side (so to speak), as I put it in the previous post: where exactly does Euclid's proof (construction) fail for this triangle? If its failure depends on antipodal points, then it will confirm my argument.

So I stiil don't see what justifies the judgment that Euclid's proof the the EAT is flawed. Scott's counterexamples seem to rely on a feature (antipodal points) which we can surely reasonably say the Euclid assumed was impossible, even if his express formulations were not tight enough so as to block every stray interpretation, in that regard. And invoking the issue of a plane-separation axiom seems irrelevant, because the spherical-triangle counterexamples do not make use of a situation where such an axiom does not apply. Also, we anyway seem to have to attribute such an implicit axiom to Euclid, based of the principle of charity.


What do you think?


Ram