What is a math problem that is proven to not be expressable analytically/algebraically and what is that proof?

Question

There is an intuitive feeling the one may encounter, that is: "All math problems can be expressed analytically, and solved algebraically"

Some problems may be harder to be expressed analytically ( i can imagine graph problems, for instance, harder to express anatically than say geometric problems).

But is there a problem that is proven to be impossible to express analytically, and what is that proof?

This question is NOT about problems that are proven to not be solvable as the first answer is trying to answer, it is about problems that are proven to impossible to express in an algebratic/analytic form

I am asking because I don't want my sense of problem-solving to be based on bare feeling and intuition but based on proof seeking ( i can imagine how the famous mathematician Srinivasa Ramanujan may disagree in this part as he is the father of intuitive approach in math src: https://www.bbvaopenmind.com/en/science/leading-figures/ramanujan-the-man-who-saw-the-number-pi-in-dreams/ )

(Execuse my off-topic tags as i am taging fields that i expect to contain such non-anatical problems)

Answer 1

There are many mathematical problems that have been proved not to be solvable, given a precise definition of solvable. Here are a few:

• Halting problem

• Solving general polynomial equations using radicals

• Doubling the cube with rule and compass

• Trisecting angles with rule and compass

• Squaring the circle with rule and compass

• Integrating $e^{x^2}$ in elementary terms

Answer 2

The closest thing that I (sort of) know of, to what I think you are saying, is an attempt to give concrete examples in certain set theory or measure theory contexts where existence of an example is proven using Axiom of Choice/Zorn's Lemma. For example: a concrete example of a non-(Lebesgue)-measurable subset of real numbers. See: https://en.wikipedia.org/wiki/Non-measurable_set .

Another example are computable numbers: https://en.wikipedia.org/wiki/Computable_number . We know that there are countably many, which means that almost all real numbers are not computable, but it is impossible (in principle) to give a concrete example of a non-computable number. (See however https://en.wikipedia.org/wiki/Specker_sequence .)