Differentiate between empty, trivial, and nontrivial solutions to problems. From a category-theoretic point of view (or maybe just mathematics/logic historically), one has that empty solutions are specifically correlated with functions mapping to 0, while trivial solutions are specifically correlated with functions mapping to 1 (or "identity"). But further differentiate quantitative from qualitative triviality:
- A set of solutions is maximally quantitatively trivial if the set is absolutely infinite.
- A set of solutions is maximally qualitatively trivial if there is no filter on types of solutions in the set.
These conditions are not perfectly distinct; we'll try to show why below. Now, temporarily suspending (1) and (2), consider a generic statement:
- It is trivial that it is trivial that X.
I don't know that (3) is always true, i.e. I don't know that:
- If it is trivial that X, it is trivial that it is trivial that X.
For example, take powerset questions in ZFC. On the one hand, there is a quantitatively trivial set of forcing solutions to these questions in that one can force 2κ to equal an absolutely infinite number of things (or, as they put it, there is a proper class of κ+a that 2κ can be forced to equal). However, there are a few qualitative restrictions: cf(2κ) > cf(κ), so for example 2ℵ0 ≠ ℵω or ℵω+ω, etc. Moreover, in ZFC, the natural powerset can't be inflated to the size of a proper class. Now, there are a proper class of transfinite cardinals that are cofinal with the zeroth aleph, so ZFC does filter out a proper class of forcing solutions to 2ℵ0 = κ, and generally there are class-many solutions to other such equations that are filtered out for reasons of cofinality.
So, in one sense, if one "works in" a specific model of ZFC where 2ℵ0 is set to ℵ2, say, it seems possibly trivial (in the model) that 2ℵ0 = ℵ2. However, it doesn't seem trivial outside the model that the natural powerset can be forced to be so many things. So it doesn't seem as though triviality must always iterate over itself.
But then is triviality relative? Or, we might differentiate between:
- It is quantitatively trivial that it is quantitatively trivial that X.
- It is qualitatively trivial that it is qualitatively trivial that X.
- It is quantitatively trivial that it is qualitatively trivial that X.
- It is qualitatively trivial that it is quantitatively trivial that X.
And so on and on. I would be tempted to think, per the "too simple to be simple" theme in category theory, that triviality does end up being relative, although that seems to conflict with the idea that a solution-set is trivial when it is absolutely infinite (c.f. the understanding of the explosion argument as one based on avoiding "trivialism," when every proposition is true without distinction).
Notably, Hawkins does contain the phrase "it is trivial that it is trivial" in relation to reflection on Fregeanism, and there is a MathSE post where someone says that some fact about intersections is trivially trivial. The Wikipedia article on mathematical triviality testifies to the relativity of the concept. However, so far as I so far know, distinctions like "quantitative vs. qualitative triviality" do not seem to often, if ever, be explicitly made in mathematics (a Google search's first-page results linked to essays in other disciplines than mathematics, except with respect to something called a "Lipschitz flow-box").
Kantian postscript: I suppose the question is at least similar to asking about the following:
- If it is analytically true that X, is it analytically true that it is analytically true that X?
- If it is synthetically true that X, is it analytically true that it is synthetically true that X?