A.J. Ayer on Geometry and Instrumentalism

[lennybound]

Comments

[awwh_snap]

Hmm. While Kant certainly held Euclidean geometry as a firm science of space, I think Ayer, like many, misunderstands the transcendental argument Kant gives for the priority of space, forming not merely logical knowledge, but content-filled knowledge. Hence, non-Euclidean geometries still pose a lot of problems for a non-logical cognizance. It's incredibly hard to sensibly imagine a non-Euclidean space. Moreover, the minute you unshackle geometry from Euclidean space, you have to essentially just say what Ayer says: that it is "a purely logical system," it no longer applies to what we can sensibly recognize and understand, except perhaps haphazardly. The consequences of all that unshackling are rife and unresolved. And it certainly doesn't conclusively determine the notion that geometry is purely analytic either. For Kant, if geometry was to be a firm science at all it had to apply to experience synthetically and it had to be able to do so necessarily based on the formal conditions of our sensibility.

[lennybound]

My knowledge of philosophy of math (and Kant in general) is not nearly as developed as it probably should be. With that said, I don’t really understand why geometry must be synthetic and "based on the formal conditions of our sensibility" in order to be considered a "firm science." I don’t see why we can’t accept the apparent a priori necessity found in formal logic and mathematics, while also denying that they express anything more than extremely complex tautologies entirely devoid of any factual content. Like Ayer says, both Euclidian and non-Euclidian geometries are internally consistent with themselves. However, one happens to map onto the physical world much better than the other, and so is preferable if we desire to effectively explain, predict, and manipulate the natural world. I imagine Ayer would take the position that logic and mathematics are purely deductive systems that are not truth-apt, but are rather instrumental tools to be used for pragmatic benefit and theoretical excellence.

Forgive me if none of that makes any