Are there different forms of rigor, and if so, are some forms of rigor more rigorous than others?

Question

Is there only one kind of rigor? Or does rigor come in different forms, like mathematical rigor, philosophical rigor, and scientific rigor? And if it does, are some forms of rigor more rigorous than others? Personally, I would rank mathematical rigor above philosophical rigor, and, in turn philosophical rigor above scientific rigor. I don't know what tags this question should have, there was no tag for "rigor", so I just put other tags on this question.

Answer 1

The SEP article on vagueness includes these two passages (early on):

Inquiry resistance recurses. For in addition to the unclarity of the borderline case, there is normally unclarity as to where the unclarity begins. Twilight governs times that are borderline between day and night. But we are equally uncertain as to when twilight begins. This suggests there are borderline cases of borderline cases of ‘day’. Consequently, ‘borderline case’ has borderline cases. This higher order vagueness seems to show that ‘vague’ is vague (Hu 2017). We are slow to recognize higher order vagueness because we are under a norm to assert only what we know. When we discuss cases of twilight, we focus on definite cases of this borderline between day and night, not the marginal cases between definite day and definite twilight.

The vagueness of ‘vague’ would have two destructive consequences. First, Gottlob Frege could no longer coherently characterize vague predicates as incoherent. For this attribution of incoherence uses ‘vague’. Frege’s ideal of precision is itself vague because ‘precise’ is the complement of ‘vague’. Second, the vagueness of ‘vague’ dooms efforts to avoid a sharp line between true and false with a buffer zone that is neither true nor false. If the line is not drawn between the true and the false, then it will be between the true and the intermediate state. Any finite number of intermediates just delays the inevitable.

Our first interpretation of the question as to commensurable or incommensurable standards of rigor then is, more concretely, the question as to whether there is a sorites problem for technical systems modulo the concept of rigor. Let us quote the SEP again (this time, the article on the philosophy of mathematics in general):

Kreisel pointed out long ago that even if a thesis cannot be formally proved, it may still be possible to obtain intrinsic evidence for it from a rigorous but informal analysis of intuitive notions (Kreisel 1967). Kreisel calls these exercises in informal rigour. Detailed scholarship by Sieg revealed that the seminal article (Turing 1936) constitutes an exquisite example of just this sort of analysis of the intuitive concept of algorithmic computability (Sieg 1994).

Insofar as rigor is usually coupled with formality/formalistic behavior (in the inscription of notation), we then ask if there is a sorites problem for formality, which seems evident enough in light of the initial quote. Accordingly, the vagueness of informal rigor will be a function, at first (at least), of the vagueness of its informality.

Now, some systems lend themselves to more austere "technical writing" than others; of philosophical interest is the special precision of musical notation, which mediates not only the mathematical presence of musical sequences in abstracto but also concrete series of actions (on the part of singers, instrument players, and so on). (There was a reason, after all, that betimes they used to have drummers accompany people marching in war.) Is musical rigor commensurable with mathmetical, philosophical, scientific, or informal rigor—or programming rigor, among the newer, and stricter, domains of this discourse? For consider that programming rigor must pass the test of being run on sometimes-touchy hardware: one cannot let gaps and conflicts in the sequences run amok or the computer might go haywire, too.

On the flipside, people write stable code all the time, so it is not quite difficult to maintain extensive standards of programming rigor. Literary rigor, a species of informal rigor perchance (tied to audience critiques of narrative continuities), is paradoxically much harder to achieve the longer one's story goes on. (C.f. the rigor of Immanuel Kant's Critique of Pure Reason, which was no small feat.^R) If a level of difficulty as such, is not equated with commensurable rigorousness, then we have no such grounds for claiming that informal, but abstract, rigor is "more rigorous" than formal-and-abstract precision à la mathematics. Moreover, there are works of literature that vaguely occupy the difference between normal fiction and technical nonfiction (among other such differences); there are works of fiction in which the audience is asked to make choices about how to go through the text; there is asemic writing and presumably, therefore, asemic rigor (see also about conlanging, AKA constructed/quasi-natural language production); there is an intuition, to some extent, that if there is such a thing, it could be easily misguided to judge its given degree of rigorousness as commensurate with the range of degrees for rather other kinds of writing. (Differences in mathematical style are not irrelevant factors, here, either.)

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^ROr, for that matter, neither was John Rawls' achievement with A Theory of Justice dimly lit, but it exemplified informal nonfictional rigor and informal poetic rigor par excellence: maybe even what we might call moral or aretaic or even deontic rigor. Or worse(!), if moral language games are already sort of fictionalistic, then Rawls yet achieved narrative rigor to a great degree, there, too. (Moreover, if there is a norm of rigor, if rigorousness carries normativity in its nature, then there is a prototype of moral rigor proper anyway.)