Morris on Kuhn

Errol Morris is writing a highly engaging and entertaining series about Thomas Kuhn's notion of "paradigms" and "incommensurability" in the Times. While I highly recommend each of the three parts published so far, a lot of interesting points emerge in the post on Hippasus of Metapontum, the poor fellow who, as legend has it, had the misfortune of discovering the square root of two...or rather, that some lengths are incommensurable.

Incommensurability is an interesting notion that Kuhn has doing quite a lot of heavy lifting in his most well-known book, The Structure of Scientific Revolutions. Mathematically, the notion of incommensurability amounts to the idea that two lengths, or more generally, quantities, have no common measure, and hence there is no way to express the relation of the one to the other using a rational number. No matter how you slice one of the quantities up, so long as you do so evenly, the divisions won't be able to exactly map onto the other quantity. Kuhn extended this basic idea to the realm of concepts and scientific theories, arguing that periods of "revolutionary" science resulted in new "paradigms," or what the Germans might call "Weltanschauungen;" whole new inter-nested sets of concepts with which to understand the world. More provocatively (and I would say less persuasively), Kuhn argued (or at least seemed to in some instances) that different paradigms were incommensurable, in the sense that the concepts from any one paradigm could not be fully and faithfully carried over to either its successor or its predecessor paradigms.

(This is far from my specialty, but it would be interesting to see how much cross pollination went on between Kuhn's work and the Quine-Duhem thesis that no scientific hypothesis is testable in iosolation. Maybe next go around I'll specialize in something like that!)

Anyhow, Morris raises a lot of interesting points. For my money, though, the most interesting are those concerning the reliability of the evidence, and the various conjectures about how we interpret the Hippasus legend. He winds up phoning up Walter Burkert, the world's foremost scholar on Ancient Pythagoreanism, who is appropriately skeptical of much of the tradition about, well, both Pythagoras and Hippasus.

All that to the side, though, there's a kind of dark irony to the whole detour into ancient Greek conceptions of incommensurability. Morris' mission seems to rest on a mistake. Noting that in a late interview, Kuhn calls the concept of "incommensurability" a metaphor, he neglects to note that Kuhn is referring to his metaphorical extension of the core mathematical notion. Instead, he intereprets Kuhn's remark as a remark about the mathematical notion, and this error starts him down the odd detour on poor old Hippasus. Worse, there's simply no reason to think that Kuhn was in any way inspired by the Hippasus story. So, in presenting his own historical reconstruction about the history of Kuhn's notion of incommensurability, along with a number of insightful lessons about the possible pitfalls of taking historical reconstructions at face value, Morris very likely gives us a specimen of the kind of work he's warned us against.

Ain't life grand?!