Topics Mathematics

Question I find the notion of fictionalism in mathematics utterly perplexing. From what I understand of it, it seems that fictionalism is the thesis that mathematics is a created fiction, and that there is no mathematical truth separate from the relevant fiction. On this view, it seems, mathematical statements -- such as 2 + 2 = 4 -- are analogous to statements like “Humbert Humbert is infatuated by Dolores Haze.” But how can this be right? Does this mean I can construct a mathematical fiction in which, e.g., 2 + 2 = 5? On the fictionalist account, I can’t see why we ought to prefer, say, a mathematics in which 2 + 2 = 4 over one where 2 + 2 = 5 unless the former captures some inherent truth that the latter misses.

Accepted:
April 8, 2010

Comments

You aren't the only one who finds mathematical fictionalism puzzling. But the nature of the analogy between mathematics and fiction needs to be spelled out carefully and, once it has been, I think a sensible fictionalist will have the resources to deny that there is an equally good fiction in which 2+2=5.

Simple equations like this one are in fact a good case for fictionalism, because there is a clear sense in which their content can be reduced to pure logic. Fictionalists ask us to think about the application of such statements. So how is "2+2=4" applied? Well, if you have two apples, and you have two oranges, and no other fruit, then you have four pieces of fruit. More generally, if you have two Fs, and you have two Gs, and none of the Fs are G, then you have four things that are either F or G. (That can be written out in logical notation fairly easily.) The thought, then, is that our talk of numbers as objects is a fiction that we build on top of these sorts of simple equational facts, one that makes them easier to handle. But the basis, the equational facts themselves, are grounded in pure logic.

There are other questions that can be asked about fictionalism, though. As I said, the case of simple arithmetic is a good one for fictionalism. The case of real analysis, let alone of set theory, is less good, and it can be harder to see what constrains those theories the way equational facts are supposed to constrain arithmetic. There are also questions about the attitude a fictionalist ought to have toward "undecidable" propositions, in the sense of Gödel; these tend to push fictionalism in the direction of various forms of finitism. All of that adds up to a worry that fictionalism gives a poor account of pure mathematics. Yet other questions concern how helpful, and accurate, the analogy with fiction is and whether, ultimately, it has any real content. There are plenty of people who think such questions have good answers. I just happen not to be one of them.