Again, I think you may be misreading the question, which I think OP tried to head off at the pass. Yes, there are many, including many in positions of political power in the community, who would question whether e.g. category theory is a worthwhile thing to be doing. That's something different from whether the field uses the recognized traditional standards of how we know or accept something is true in mathematics, or mathematical methods of precise definitions. The Jaffe-Quinn paper is all about that, for example. For my money, the answers by Tao and Chow are hitting it on the head.

I won't say it's a bad answer, but it's long and takes a while to get into, on a topic which maybe few have exposure to, and it may not be very clear what exactly is surprising about it. As opposed to, say, the link between the Riemann Hypothesis and random matrix theory.

This answer has nothing to do with epistemic status in my opinion (both areas are universally accepted as rigorous mathematics by the general community). It may be deleted soon.

And I'm trying to sketch a general reason: roughly speaking, if you start with a cartesian closed category, and if the objects of certain full subcategory are definable as those objects c that satisfy certain limit properties (like Segal conditions), then exponentials c^x will also satisfy those conditions since (-)^x preserves limits. (I guess you also need that the full embedding preserves products. This is true for the nerve functor, for instance.)

Pos can piggyback on Cat because (like the Segal conditions) what it means for a category to be a preorder or poset can be described by certain limit conditions. One could say that Cat is an "exponential ideal" of s-sets (if C is a category and X is an s-set, then the s-set C^X is a category), and likewise, Pos is an exponential ideal of Cat.

It's hard to know what sort of answer would satisfy you. I'm going to guess that you could grind through a proof if you had to, so that what you want is some sort of conceptual reason. Here's one attempt: via the nerve functor N, Cat embeds fully faithfully into simplicial sets, and are characterized as s-sets satisfying "Segal conditions". Meanwhile, s-sets form a cartesian closed category (since, e.g., it's a topos). It then suffices to check that for a category C and simplicial set X, that NC^X is also the nerve of a category, by showing it too satisfies the Segal conditions.

Yes, that's right.

Something looks wrong with $S_7$: the top coefficient should be $1/8$, whereas you have $3/90$.

It's metaphorical, but sometimes smell is invoked for mathematical intuitions that are difficult to describe. "That smells like a theorem we proved about five years ago." C.F.J. Ross said Jung hypothesized that intuition may have been the remnant of the faculty of smell that was well-developed in many mammals, but underdeveloped in the human cortex compared to the other cortical lobes (Coll. Wks 18, p. 326). In an interview toward the end of his life, Jung stated ‘The intuitive is a type that doesn’t see the stumbling block before his feet, but smells a rat for ten miles’ (Evans 1976, p. 103).

Bernard Morin, who was among the first to show that "a sphere may be inverted", had lost his sight as a small boy. Bill Thurston commented that the visual sense could even sometimes get in the way, and that apprehension of such things might be partly kinesthetic. (See his interview in More Mathematical People.) Von Neumann said his dominant sense in mathematics was hearing (and I don't think developing a fine-tuned sense of logical structure is primarily a visual affair, even if we use symbols as an aid in logic).

@AlexM. I thought you would have been automatically pinged since you were the first commenter. Anyway, thanks, and no worries at all. Do you mind if I clean up this thread?