The fabula of mathematics

This here is an interesting post by Matt Weiner on the relationship between "fictionality" and the history of mathematics. His point is that much in the same way that readers are sensitive to some and only some kinds of inconsistencies in narratives, mathematics has historically been able to accomodate some inconsistency -- after all, how could it not, when the history of mathematics, until recent times, is largely that of the development of new concepts to explain anomalies? Think of transcendental numbers or the development of the axiom of choice. Without these concepts, people had a whole universe of mathematics to work with, but some corners were stubbornly inaccessible. As long as you weren't dealing with questions about pi and e, polynomal numbers were enough; the vast majority of proofs don't need even the concept of the axiom of choice.

But maybe the issue isn't really that of inconsistency, just incompleteness. The project of works like the Principia Mathematica was to build up a consistent, complete mathematics from first principles. Gödel famously demonstrated that you can't have both.*

Mathematics has to be able to tolerate some inconsistency or some imcompleteness (or both!); but I don't know that this means mathematicians do. They're on the hunt, after all, for just those kinds of gaps. It's clever to find one, and cleverer, of course, to fill it. I guess the question is whether the simple fact that people can go on for centuries without noticing a certain gap means that they're "tolerating" the lack. Probably it does.

The other thing is that mathematics is a group project in a way that literature isn't.** Individual pieces of fiction get read and used by some slew of readers -- some of them may "read over" a given inconsistency; others may not; the degree to which it interferes with their pleasure in the text or their ability to use it for whatever purpose may vary as well.

But the result is not, usually, even when the inconsistency is not tolerated, as in the example Matt gives of the thirty-four-year-old who is celebrating his Leap Day birthday (which by the way I suspect is actually an example of the very kind of inconsistency that most readers do utterly fail to notice), the response is not to publish a peer-reviewed paper that explains why the book needs to be revised, thereby causing it to be revised. And even if it were, that still wouldn't be a parallel situation -- if literature were like math, we wouldn't be talking about inconsistencies in a piece of fiction, we'd be talking about inconsistencies in Fiction, or Literature, in general.

Also, I'm not entirely down with the way this discussion (which, by the way, has its roots in and presumably takes its terms from Jerrold Katz's Realistic Rationalism, which I haven't read) talks about Fiction and Mathematics as the things which are or are not tolerating inconsistency. To what degree is it a question of whether these systems in themselves do or don't tolerate it, and to what degree is it a question of whether experts/people in general tolerate inconsistency in the systems they encounter? There seems to be a lot of free play back and forth between these different construals of the question, with fuzzy results.

*See me flirt daringly with the risk of making myself one of those profoundly irritating literature people who throw Gödel's name around with wild and untutored abandon!

**The linked discussion mentions this point, too, though not all of it -- "So it's important that there be consensus in mathematics, and that doesn't usually go along with inconsistency. But I think you might be able to make the case that that's a matter of the purpose of mathematical inquiry rather than a question of its having more objective reality than fiction." Yeah, absolutely, and it's cool to call mathematics "a group-authored fiction," but I think there's also a non-trivial point to be made about the fact that we have many many fictions and one (big, complex, in the ideal) mathematics. Am I wrong?

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