Suppose that clarity occurs for two representations when one attends to what makes the representations different. (This is, to my knowledge, a somewhat common or accepted "definition" of clarity.) When a representation is complex, this differentiation can be accomplished by differentiating between differences themselves, and so on. However, wouldn't this "run out" such as to leave us with ostensively defined terms whose clarity is ostensible, if apparent at all? Yet Peter Koellner quotes a certain Markov such that:
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)) [emphasis added]
Are e.g. individual shades of red "clear"? This would be intuitively, it seems (or "perceptually"), yet Markov testifies on behalf of a social/intersubjective factor, here, instead. All I can think of are:
- The practice of translating one formal system into another (of navigating between a given set theory, model theory moreover, multiple styles of logic, Gentzen's proof structures, category theory, epistemic graph theory, etc.). Clarity is achieved in understanding how this or that subset of a system corresponds via translation to the subset of another system.
- Roughly the same as (1), except the presenter switches from one natural language to another in the course of the presentation. I noticed that Immanuel Kant and Hannah Arendt do this to a remarkably extent (and if I remember correctly, John Rawls somewhere in A Theory of Justice leaves an entire quoted section of some other text, in the original (French, I believe) language). This so as to conform to the "social" standard of clarity, ideally (to appeal to multiple linguistic frameworks in the expression of concepts).
Now, all that being said, offhand none of that seems especially "clear" in a "pre-theoretic"/"intuitive" manner. Imagine, too, that there was a difference between clear and unclear sets; then if there were a set of all clear sets, and this set were strictly well-founded, then this set would not itself be clear, i.e. it would be unclear what the domain of clarity is as a whole. (Alternatively, a set of all unclear sets would be clear, which would be surprising, I suppose, except that I suspect that a set of all unclear sets must not be very well-founded.) Is the clear/unclear distinction itself clear, or does it go beyond itself and is not, therefore, absolutely absolute (but only absolute for its subsumed domain of applicable discourse)?
Sidebar: this question is somewhat akin to the question on this SE about the comparative rigorousness of mathematics vs. philosophy, although I am uncertain as to whether rigorousness and clarity should be knit quite so tightly as to have the questions coincide. I can think of e.g. a sample of rigorously programmed software, with hundreds (or thousands, or...) of lines of code, which is then quite precise but to an outsider will seem potentially quite unclear even so. (If rigor and clarity go together more closely, perhaps we have a paradox for these concepts instead, though.)
