10
$\begingroup$

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.

$\endgroup$
30
  • 5
    $\begingroup$ 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. $\endgroup$ Oct 24 at 7:44
  • 6
    $\begingroup$ What does ‘existence’ mean? $\endgroup$ Oct 24 at 7:54
  • 6
    $\begingroup$ 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. $\endgroup$ Oct 24 at 8:17
  • 7
    $\begingroup$ 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. $\endgroup$
    – Matt F.
    Oct 24 at 8:58
  • 8
    $\begingroup$ I don't understand the negative attitude in all these comments, it looks like a perfectly reasonable question to me. $\endgroup$
    – Nik Weaver
    Oct 24 at 14:56
25
$\begingroup$

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.

$\endgroup$
10
  • 1
    $\begingroup$ 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. $\endgroup$ Oct 25 at 14:16
  • 2
    $\begingroup$ @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. $\endgroup$ Oct 26 at 12:31
  • $\begingroup$ I'll see what I can make of it. Thanks for the comment. $\endgroup$ Oct 26 at 12:52
  • $\begingroup$ 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)? $\endgroup$ Nov 22 at 21:44
  • $\begingroup$ @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. $\endgroup$ Nov 22 at 22:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.