The appearance of robust and expansive mathematical knowledge with fictionalist undertones might be symptomatic of the average mathematician's temperament: as a historical community, they are less minded towards competition, in part because getting oneself into the spirit of analyzing the outputs of bizarre (or at least esoteric) functions and sequences of functions means putting other people out of mind enough to where amour-propre is not triggered so much as in the often toxic fandoms of narrative fiction.
Still, rancor occurs: Kroenecker's vitriol vs. Cantor, Cantor's vitriol vs. the whole notion of infinitesimals, Brouwer's vitriol towards classical logic. One mathematician decades back expressed some dread regarding appeals to intuition like so:
A popular view is that axioms are self-evident truths concerning their domain. The difficulty with this view is that there is wide disagreement in the foundations of mathematics as to which statements are self-evident and if we restrict ourselves to the intersection of the statements that all mathematicians would regard as self-evident then the result will be quite limited in reach, perhaps coinciding with Q or slightly more.[4] Markov had a similar complaint with the employment of the related notion of being “intuitively clear”:
I can in no way agree to taking ‘intuitively clear’ as a criterion of truth in mathematics, for this criterion would mean the complete triumph of subjectivism and would lead to a break with the understanding of science as a form of social activity. (Markov (1962))
And Lawvere echoed such sentiments years later in his own way:
In ‘Diagonal arguments and Cartesian closed categories’ (Lawvere 69) we demystified the incompleteness theorem of Gödel and the truth-definition theory of Tarski by showing that both are consequences of some very simple algebra in the Cartesian-closed setting. It was always hard for many to comprehend how Cantor’s mathematical theorem could be re-christened as a “paradox” by Russell and how Gödel’s theorem could be so often declared to be the most significant result of the 20th century. There was always the suspicion among scientists that such extra-mathematical publicity movements concealed an agenda for re-establishing belief as a substitute for science. Now, one hundred years after Gödel’s birth, the organized attempts to harness his great mathematical work to such an agenda have become explicit.
But so another thing about all this, about mathematics overall, is how relatively trivial or ethereal much of the "knowledge" we have therein turns out to be. Fiddling with obscure and subtle parameters usually results in only obscure and subtle variations (many functions do not as such "compound" their outputs, so to speak). It is often easy enough to accept insubstantial "facts" without too much ado, especially if the facts are internally conditional/hypothetical in form ("If there is a proper class of measurables, then Setop doesn't have a small dense subcategory," can be innocuously admitted even by someone otherwise allergic to the higher infinite or even transfinite numbers generally, say).
On the flip side are those genres of narrative fiction where commitments to "verisimilitude," "realism," or at least detail are a mainstay of successful contributors. Fictionalism about various scientific terms intersects fictionalism about the objects of science fiction, but we still find consensus and the appearance of knowledge in science and so we have the distinction between soft and hard science fiction proper. Fantasy worlds that obey "Magic A is Magic A" also leave room for readers to "figure out" how certain themes will ultimately be fulfilled (in the Wheel of Time series, for example, it was possible to pay close attention to the oracles of a side character and yet then deduce a major component of the central protagonist's victory over the final enemy). Before I ever set foot on the PhilosophySE, I actually became acquainted with the SE format via a subsection of the 17th Shard, an analysis site devoted to the works of Brandon Sanderson, which testifies to the "factoid-based" nature of those works. Sanderson himself explains how his "science fantasy" operates:
For a while now, I’ve been working on various theories regarding magic systems. There’s a lot to consider here. As a writer, I want a system that is fun to write. As a reader, I want something that is something fun to read. As a storyteller, I want a setting element that is narratively sound and which offers room for mystery and discovery. A good magic system should both visually appealing and should work to enhance the mood of a story. It should facilitate the narrative, and provide a source of conflict.
For example, in many fantasy stories, the protagonist has a crucially powerful implement at hand, but either shouldn't or can't use it very often, if ever. The magic doesn't so much as solve problems as it is the problem. Then, depending on the skill of the storyteller, describing how the characters actually solve those problems either comes across as meaningful/"realistic" (at least psychologically) or as, of course worse, quite contrived. Whether, "X is a good/optimal solution to narrative problem A," counts as fictionalistic knowledge, I'm not keen to say; there are apparently undecided, or undecidable, questions that can be posed relative to various stories, like, "How many times did Frodo smile over his lifetime?" even if there are also questions that are decided explicitly in the given text or which could be decided by inference from what the direct author or authoritative readership interpreted their own texts as implicitly saying. (But then how much of such a difference in decidability is there between, "How many inaccessible cardinals are there?" and, "How many hairs were there on Galadriel's head?")
One domain of discourse that seems to involve precise fictions with relatively stable, if internal, implications is the law. There is the phrase/concept of "legal fictions," for example, yet notwithstanding the pseudo-reality of entities like corporations or imputable guilt, theories about what some or another law is seem able to "knowledgeably" proceed on those entities' basis. At least, lawyers, judge, etc. often spend years upon years learning about precedents and other factor that inform their legal strategies and conclusions; whether their "knowledge" as such is on a par with the ocean of mathematics seems doubtful to me, but I haven't studied legal reasoning very much (or exercised it much myself), so I would hesitate before claiming that lawyers and judges "don't know" what laws are or mean (if I claimed that, it would be modulo a claim that there aren't enough "real" laws to know about in the required way).
