Why are some equations non-analytic? Is it a deficit in our mathematics?

If a number exists and satisfies an equation, that fact doesn't change just because you can't deduce the number's value using certain methods.

>>9099662
Yes - so is it just a deficit in method, which can be overcome (hypothetically)?

>>9099656
Unsolvability of polynomial equations above degree 4.

Three-body problem.

Arc length of an ellipse.

Maybe our questions and models aren't compatible.

Taylor series are pretty good at approximating things, but they might not be the most "natural" representation.

Physicists have names for the first four terms:
Position, Velocity, Acceleration, Jerk.

If you use Fourier series you have:
DC component, 1Hz, 2Hz, 3Hz,...

Both of these representations can be used to approximate things very well, but each one is more useful for different types of questions.

>>9099687
What's your point?

>>9099749
The language of the model makes some questions easy to answer and some difficult to answer.

These arbitrary-precision approximations might be too good at approximating anything and may be the wrong "language" to use for certain questions.

>>9099687
>Position, Velocity, Acceleration, Jerk
That last item has been called "Shock" for more than a century, you jerk.