Computer proof ‘blows up’ centuries-old fluid equations

So, a question is, can we improve our ability to make such approximations and know something about the accuracy of the solutions we will get? E.g., for the Euler equations, will that approximation of an "incompressible" fluid ever work in practice and, if so, when and, there, how accurate can/will it be?

Or, what about, hmm, just to be picky and pick something, friction on the side of the tube? What if the tube is not a perfect tube?

A few grains of dirt: What if the liquid is water but, like most real water, has some solids floating around in it? Right, we can say, so there are a few grains of dirt floating around in the water, and they won't matter -- to be picky, that's an approximation, and we are likely correct, but where is an actual math theorem that says we are correct or how correct, i.e., accurate, are we? Right, a few grains of dirt -- we don't much care. But that's practical judgment and not really theorem/proof math.

And similarly for other approximations we get as we throw out, modify, or honor real features?

So, as stated, this is too difficult as a pure/applied math research direction. Okay, ..., then, is there anything at all in that direction that might be not absurdly difficult as a research direction?

Or, to be simplistic, we work hard and get a numerical solution to a boundary value problem. Now someone tweaks the boundary. Can we say that our numerical solution is only tweaked? Or, when can we say that small changes in the problem statement will result in only small changes in the solution? Right, we are into some topology and looking for a case of continuity .... Hmm .... If we had some linearity ...!!! Right, the two pillars of analysis are continuity and linearity ...! But here with Euler we were considering nonlinear partial differential equations!

Again I ask, is there any hope we can do anything for some corresponding math??

As I recall, there was a paper by Robert Tarjan where he observed that most paradoxes are from self referencing. Soooo, rule out self referencing and will rule out most paradoxes!

Mathematicians have split opinions about computer assisted proof. On one hand, there doesn't seem to be a real difference between one page of checkable inequalities vs 500 pages of checkable inequalities. On other other hand, computer assisted proof do not help humans gain clarity on the logical process. The fear is that the computer result will just be an "one and done" result, which people believe is true, but cannot build upon because they don't fully understand it.

I think you're conflating universality and objectivity. Those terms you list all have objective definitions, but the specific characteristics they have in any given logic may differ. That means they are not universal, but that doesn't make them non-objective. Objective typically means "mind independent".

Your example of equality already demonstrates you understand equality's objective definition: you implicitly operate on the notion that "equality" means some form of equivalence, some ability to substitute B for C in a specific context that results in no observable/expressible change. That is an informal but objective understanding of equality.

What you're recognizing is that equality can have different logical properties in different contexts, where "context" can be understood as the formal language we're using, ie. it's not universal. But it's role in any given logic is always the same and not dependent on the provers mind state or his surrounding culture, ie. it is objective.

Godel showed that there is no such thing as a universal logic in our current approach to formal systems, but that didn't suddenly make logic non-objective. It simply means that there is no Ur-logic that can subsume all other logics (which is why most assert that Godel ended Hilbert's program).

So what logics a culture or species may use or find interesting, and the process by which they explore these systems are socially contextual, but the structures themselves and their internal consistency is not socially constructed. A culture can certainly believe a formal system they use to be logically consistent, but that's no more interesting a statement than that some cultures believed that Thor caused lightning. In other words, they could just be wrong about the consistency of their arguments.