Friday, January 31, 2014

Nicholas Cusanus and Squaring the Circle

I was looking up something or other on Bl. Nicholas of Cusa the other day and started reading about his works on squaring the circle. One of the things that I noticed was that a lot of websites simply stated that he attempted to square the circle -- which, of course, is mathematically impossible. Now, this, and particularly the dispute with Regiomontanus on squaring the circle, which the websites mostly had in view, is not something I am deeply familiar with, but I have read more than a bit of Cusanus, and one thing I am quite sure of is that his position on the subject is not that straightforward. The fact of the matter is that you can find arguments both for and against the possibility of squaring the circle in his works; part of the reason for this being that he often is attempting to summarize the dispute. He also very famously has a tendency to slip out of subjunctive mood into indicative mood in talking about things that he does not necessarily believe, which sometimes makes it difficult to determine whether he is putting forward something as his own view or as a reasonable conclusion given some assumption he might not actually hold. And he uses examples and analogies in a way that takes some getting used to. Thus, for instance, it is often said that he held that a straight line was a circle of infinite diameter although it's not actually clear that he thought that this was literally true. Interpretation is further complicated by the fact that he seems to have thought that the dispute between those who thought the circle could be squared and those who thought it could not be was based on equivocation, and that the two sides were not using the same definition of equality.

Nonetheless, it is definitely true that he here and there gives theological arguments, in which he uses squaring the circle as an example, that seem very clearly to presuppose that it is not possible to square the circle. What is more, he occasionally uses it as an analogy in discussing one of his most important philosophical positions, that we are not able to achieve exact, precise truth, but only an approximation to it. (As he says at one point, the human intellect is to truth as a many-sided polygon to a circle.) So it seems more likely to be his view that it is not, in fact, possible to square the circle, in the proper sense, but that we can 'square' the circle in the sense of getting something like it by approximation.

Again, the whole is quite complicated, and I don't pretend to have an understanding of his full view; it's possible that there is some nuance I am missing. But I am, again, quite sure that it is an error simply to claim that he held that the circle could be squared when one can find both explicit statements in his work denying that it is possible and importantly placed analogies that seem to assume that it is not; his view is certainly more complicated.

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