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Have there been any successful mathematicians that also happen to be mathematical fictionalists? Let's say success is defined by at least one article published in a non-pay journal.

I ask because this seems like a very extreme position for a working mathematician to have. Also, fictionalism is a very recent position.

I think it would be interesting to hear their point of view, if they exist.

YCor
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Paul Burchett
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    How do you determine whether someone is a fictionalist? –  Oct 24 '21 at 07:05
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    https://plato.stanford.edu/entries/fictionalism-mathematics/

    Mathematical fictionalism denies the existence of abstract mathematical objects. It also denies the truth of all mathematical statements in any sense, not just the Platonic one.

     – Paul Burchett Oct 24 '21 at 07:09
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    This question doesn't seem to be about math research. But neither is this one. – WhatsUp Oct 24 '21 at 07:36
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    It is research on the sociology of the mathematical community. I've run across quite a few surveys that ask working mathematicians questions about their particular philosophy behind the mathematics. Arguments are based on questions like these. For example, Platonism is often held to be the most popular position, and there's research that backs this up. This serves as an argument for Platonism. One could make a sociological argument against fictionalism if one finds no one willing to accept the position. Such an argument seems relevant to me, even though it is a soft question. – Paul Burchett Oct 24 '21 at 07:44
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    What does ‘existence’ mean? – Martin Hairer Oct 24 '21 at 07:54
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    @MartinHairer Existence can be thought of as a way of living. That's one of the definitions anyway, and we're referring to mathematical fictionalism as lived by a working mathematician. – Paul Burchett Oct 24 '21 at 07:59
  • While there are varying degrees of belief, I'm more curious whether anyone that is a working mathematician would seriously argue for the belief. In other words, their belief isn't casual and is deeply held. – Paul Burchett Oct 24 '21 at 08:12
  • When you say "at least one article published in a non-pay journal," do you mean a mathematical journal? Because if not, then the people mentioned in your Stanford link would count. – JRN Oct 24 '21 at 08:13
  • @JoelReyesNoche I thought it would be a given that I meant a mathematics journal considering working mathematicians are the subject matter. – Paul Burchett Oct 24 '21 at 08:14
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    What I was suggesting is that whether a mathematician considers themselves a platonicist or a fictionalist may well hinge on a difference of interpretation of the word ‘existence’ as pertaining to mathematical objects, rather than an actual difference on belief. – Martin Hairer Oct 24 '21 at 08:17
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    I spent a few minutes combing through all articles on MathSciNet which cite Hartry Field's books and articles which propose fictionalism. I looked for authors who have published articles in areas of mathematics outside of formal logic and foundations of mathematics, since these are on the boundary with philosophy, and perhaps not what the OP's question asks for. The only author I found, fitting these criteria, was the well-known Dana Scott, who was a coauthor of "Can modalities save naive set theory?", a 2018 article which cites Field and considers (but does not endorse) fictionalism. –  Oct 24 '21 at 08:22
  • what is truth, is mine the same as your's? – Manfred Weis Oct 24 '21 at 08:22
  • @ManfredWeis I get the punchline. If we truly embrace fictionalism one has problems with questions such as yours. However, that is not what I'm looking for in this thread. – Paul Burchett Oct 24 '21 at 08:32
  • @MattF. I'd think that mathematicians would be concerned with consistency. There's something to be said for holding a position within a certain camp. On the surface, at least, such distinctions provide consistency.

    Are you able to provide a sociological study that tests your claim? I know Platonism is alive and well, and I'd venture to say a large number of mathematicians believe in some form of Platonism.

     – Paul Burchett Oct 24 '21 at 08:45
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    My claim was: "The number of mathematicians in the past 40 years who have deeply held and seriously argued for any ism 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. –  Oct 24 '21 at 08:58
  • @MattF So I might need to soften my criteria. Perhaps a deeply held belief that could either be argued or not, but is believed in a deep way. It could be a private belief, but a strong one. – Paul Burchett Oct 24 '21 at 09:03
  • @MattF. Your way of viewing camps as having the sole value as organization principles itself suggests a philosophy. Your view is very much akin to pragmatism. While I have no problems with pragmatism, when you attempt to dispel the deeper truth of "isms", you introduce us to yet another "ism" hidden there in your argument. – Paul Burchett Oct 24 '21 at 09:19
  • @A.S. I gave you an upvote. While not perfect, this was more along the lines of what I was looking for. There are many that want to muddy the waters here when I don't think this much was necessary. – Paul Burchett Oct 24 '21 at 09:31
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    The pragmatism was already explicit in Feferman and Mac Lane! This is why Feferman talked about "Working Foundations" and why Mac Lane made a comment on foundations that "our present assumption of 'one universe' is an adequate stopgap, not a prediction for the future". I think their philosophical pragmatism is typical of the working and successful mathematicians in the question. –  Oct 24 '21 at 10:15
  • @MattF. You stated there were no deeply held "isms" anymore in the mathematical community. None. This should include pragmatism by your rule. Yet hear you are arguing for a form of pragmatism. – Paul Burchett Oct 24 '21 at 10:25
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    @MattF. Maybe Mac Lane was responsible for emergence of a significant such ism after all. It might have become negligible by now but still. For example, I've heard from a devoted follower of Mac Lane (maybe more devoted than Mac Lane himself was) that Category Theory is to Mathematics what Mathematics is to Physics. – მამუკა ჯიბლაძე Oct 24 '21 at 10:36
  • @PaulBurchett I'm not sure that not having many mathematicians or even any mathematicians who are fictionalists is a good argument against ficitonalism. It may be that fictionalism is not an attitude which is compatible with doing good math whether or not it is true. Alternatively, the sort of person who is strongly a fictionalist may just not be inclined to into mathematics. (Would one take that a lot of people who like music are convinced it is a universal among sentient life be anything but the very weakest evidence for that?) – JoshuaZ Oct 24 '21 at 11:46
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    I had never heard about fictionalism before. Now I have looked at its "main argument" at https://plato.stanford.edu/entries/fictionalism-mathematics/#MaiArg, where I have found this: "If sentences like ‘4 is even’ should be read at face value, and if moreover they are true, then there must actually exist objects of the kinds that they’re about". But this certainly looks like a fallacy. Cf. e.g. this statement: "Every integer in the interval $(0,1)$ is the square of another integer". The statement is obviously true, even though no integer in the interval $(0,1)$ exists in any imaginable sense. – Iosif Pinelis Oct 24 '21 at 12:45
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    I don't understand the negative attitude in all these comments, it looks like a perfectly reasonable question to me. – Nik Weaver Oct 24 '21 at 14:56
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    @NikWeaver I'm inclined to agree with you. Question seems on topic (although I'm not sure it will give the information that Paul Burchett thinks it might). – JoshuaZ Oct 24 '21 at 15:27
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    @IosifPinelis Well, if I get it right, most of the argument there is actually redundant. The main premises are (4) and (5), the rest is unimportant. – მამუკა ჯიბლაძე Oct 24 '21 at 19:13
  • @მამუკაჯიბლაძე, the last sentence of the post is "I think it would be interesting to hear their point of view, if they exist." I think the OP thought that the question "What does 'existence' mean?" referred to the existence of mathematical fictionalists. – JRN Oct 25 '21 at 02:21
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    @JoelReyesNoche Oh I see, thanks for explaining – მამუკა ჯიბლაძე Oct 25 '21 at 04:21
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    Uninteresting as this particular question is to me, nonetheless there should be room for it and for many other non research-level math-related questions in this forum, which otherwise might be too technical and dull to attract many of the research mathematicians active on it. – Yaakov Baruch Oct 25 '21 at 13:01
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    I think this is question is very interesting. When people like Connes, Scholtze and Borcherds saying they are strongly Platonist, it seems to me that Platonism is actually Pragmatist in the field of mathematics (in the sense that Platonism is a philosophy that makes progress). I would find it very interesting if there are any high-performing fictionalists who can persevere in the face of the constant barrage of mathematical setbacks... – Jon Bannon Oct 25 '21 at 18:51
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    One professor of philosophy told me that Platonism is the proposition that universals exist independently of particulars. A universal is anything that can be predicated of a particular. Thus "That animal is a horse," predicates of one particular animal the property of being a horse. The property of being a horse is then said to exist independently of all particular horses. That property is a "Platonic form." – Michael Hardy Nov 08 '23 at 18:03
  • @MichaelHardy I Believe you have the basics of Platonism down with your explanation. – Paul Burchett Nov 08 '23 at 18:13
  • Leibniz had at least one article published in a non-pay journal: Nova Methodus. He was a fictionalist: Two-track depictions of Leibniz's fictions. – Mikhail Katz Nov 08 '23 at 16:40
  • @Mikhail Katz The article cited seems interesting. I'm unable to access, however. None the less, I did pick up that Leibniz viewed imaginary roots, negatives, and infinitesimals as useful fictions - or so the paper argues. What about the natural numbers? Does the paper also argue that Leibniz took those as fiction?

    I ask because fictionalism is typically considered more recent than Leibniz. It seems to me that as a rationalist he'd have to take at least part of mathematical thought as real in some way - even though he took at least part of mathematics as fictional.

     – Paul Burchett Nov 10 '23 at 07:01
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    @PaulBurchett, that's a very interesting question. I think it is possible that one can view Leibniz as a fictionalist with regard to "the" natural numbers in the following sense. Since he did envision "infinite numbers" as being part of his number system, he may have sensed a distinction between metalanguage natural numbers and object-languge (formal) natural numbers. We have written a number of articles of Leibniz; I would be interested in hearing your reaction. See https://u.math.biu.ac.il/~katzmik/leibniz.html As far as the article I linked, it is at ... – Mikhail Katz Nov 10 '23 at 11:27
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    Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M.; van Atten, M. "Two-track depictions of Leibniz's fictions." The Mathematical Intelligencer 44 (2022), no. 3, 261-266. https://doi.org/10.1007/s00283-021-10140-3, https://arxiv.org/abs/2111.00922 Of course, Leibniz would not have had the language to express the distinction between metalanguage numbers and object-language numbers. @PaulBurchett – Mikhail Katz Nov 10 '23 at 11:27

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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.

Timothy Chow
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    I appreciate the answer. I was upset when the question was closed. It might not be a fruitful question, as you point out, but it was frustrating that it was outright closed. – Paul Burchett Oct 25 '21 at 14:16
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    @PaulBurchett You may enjoy the thread "Logic/syntax versus arithmetic" here, where I criticize Mary Leng's version of nominalism. Again, mathematicians don't necessarily balk at claims that mathematical objects aren't real or don't exist, but when nominalists say, "X is abstract but Y is concrete," mathematicians will stare blankly. In Leng's case, how is logic/syntax less abstract than arithmetic? They're mutually interpretable. And it's precisely the arithmetization of syntax that enabled the breakthroughs of Goedel and others. – Timothy Chow Oct 26 '21 at 12:31
  • I'll see what I can make of it. Thanks for the comment. – Paul Burchett Oct 26 '21 at 12:52
  • I would like to know how the fictionalists listed in the SEP entry view the sequence |, ||, |||,.....(do they mention it in any of their works?). Also, if "said readers are more likely to call themselves formalists than fictionalists...", is Hilbertian formalism a type of fictionalism (consider the real/ideal dichotomy)? – Thomas Benjamin Nov 22 '21 at 21:44
  • @ThomasBenjamin Nominalists deny the existence of abstract objects, so if you are proposing that "the sequence |, ||, |||, ..." is an actually existing abstract object, they would beg to differ. But as for "how they view the sequence"...that depends on exactly what question you want to ask. Regarding Hilbertian formalism, my understanding is that Hilbert thought that finitary mathematics was unproblematically real, whereas fictionalists would not agree. – Timothy Chow Nov 22 '21 at 22:48
  • I do not. I do propose (along with Hilbert as shown in his paper, "On the Infinite"--- see th – Thomas Benjamin Nov 29 '21 at 02:33
  • (cont.) the Erna Putnam/Gerald Massey translation in Benacerraf and Putnam's compilation, Philisophy of Mathematics) that each of the "numerical symbols" 1, 11, 111, 1111,.... exist and constitute the "subject matter" of number theory, the collection constituting all of these "numerical symbols" being an "ideal element" from which theorems can be proven (leaving open the possibility that there can be ideal "numerical symbols" as the ultrafinitists believe). – Thomas Benjamin Nov 29 '21 at 03:02
  • @ThomasBenjamin Good question. Fictionalists, or nominalists, are totally committed to denying that abstract objects exist, so either the numerical symbol 1 doesn't exist, or it's not an abstract object. Neither makes sense to me. I can write a 1 on a blackboard or type it into a computer; each of these instantiations is arguably a "concrete token" and not abstract, but I want both tokens to instantiate the same symbol, so abstraction seems unavoidable. In an FOM post Alan Weir tries to sketch a response, but I'm not convinced. – Timothy Chow Nov 29 '21 at 03:33
  • I would say that the numerical symbols 1, 11, 111, 1111, ... – Thomas Benjamin Nov 29 '21 at 03:45
  • (cont.) have a dual nature, both as an existing token and as an ideal abstract object. This is both a blessing and a curse.... – Thomas Benjamin Nov 29 '21 at 03:58
  • "More likely, X's beliefs would agree with fictionalism in some ways and would disagree in other ways." I couldn't agree more. I've read a little bit about fictionalism, and some aspects of it resonate strongly with me. I might even be more of a fictionalist than a platonist. Nevertheless, I would not label myself as a fictionalist -- it seems too much like signing a contract I've not read all the way through. – Will Brian Nov 08 '23 at 17:25