As I suggested in response to a related MO question, one difficulty with answering this type of question is that most mathematicians outside of logic and set theory lack well-developed "positions" on the types of questions that occupy much of the attention of philosophers of mathematics.

As a preamble, let me ask this: For a mathematician X to be a fictionalist, is it necessary for X to know the meaning of the word "fictionalist" in, say, Hartry Field's sense?

Maybe the answer is no; maybe X just needs to espouse certain beliefs about mathematics to be a fictionalist, just like M. Jourdain spoke prose all his life without knowing it. But fictionalism is far more specific than prose, and it seems unlikely that X's beliefs would line up neatly with fictionalism unless X had studied fictionalism explicitly. More likely, X's beliefs would agree with fictionalism in some ways and would disagree in other ways. But if you insist that X know what the word "fictionalist" means, then you narrow the pool of candidates hugely. In any case, the only plausible way to find out is to conduct a formal survey. You might have to do this yourself if the surveys you have encountered don't already answer your question.

Having said that, I have noticed some aspects of fictionalism being espoused implicitly or explicitly by some mathematicians, but I have also noticed other aspects that seem to be almost universally rejected.

For MO readers who haven't heard of fictionalism, here's a caricature. Hartry Field draws an analogy with Oliver Twist. Did Oliver Twist travel to London? Answer: Yes, *according to a certain story*, but no, not *literally*, since Oliver Twist did not really exist. Analogously, mathematics, if we take its discourse at face value, makes assertions about abstract objects. But abstract objects are not real (Field is a *nominalist*), so mathematical theorems are true only *according to a certain story*. But wait, you say, isn't mathematics essential for doing science, and science surely deals with the real world? Field's response is to try to develop "science without numbers" by re-developing the scientifically applicable parts of mathematics, not on the basis of *abstract* objects, but on the basis of *concrete* objects, such as "regions of space."

If we accept this caricature, then I have certainly encountered mathematicians who, in one way or another, reject the "reality" of certain mathematical objects. Most commonly, I find this happening with regard to infinite set theory. You'll probably be able to find plenty of readers right here on MO who would agree with something like this: "Does the cardinality of the natural numbers differ from the cardinality of the real numbers? Yes, *according to a certain story*; but no, not *literally*, because infinite sets aren't real." But said readers are more likely to call themselves *formalists* than *fictionalists*, if they admit to being any kind of -ists at all.

On the other hand, the nominalist preoccupation with abstract versus concrete isn't something that you'll find resonating with many mathematicians. My reaction to Field's "science without numbers" is that his allegedly "concrete" objects seem just as abstract as standard mathematical objects. I think that this reaction is typical among mathematicians. Even the aforementioned "formalists" will generally agree, if you put the question to them, that symbols and finite strings are abstractions, while at the same time are "real" in a sense that (say) infinite sets are not. Replacing an abstract theory of numbers with an abstract theory of regions of space strikes mathematicians as being a pointless exercise.

anyism in the philosophy of math is already almost vanishingly small." Even mathematicians who believe an ism usually do not seriously argue for it. E.g.: Did Feferman seriously argue for predicativism? He usually stopped at: this is viable, you don't need more. Did Mac Lane seriously argue for functionalism or structuralism? He usually stopped at: this is a good organizing idea. I don't think either argued that their philosophy was the one correct option, or disputed formalism. $\endgroup$31more comments