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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)

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

lhf
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  • See also https://en.wikipedia.org/wiki/Proof_of_impossibility – lhf Jan 13 '20 at 22:31
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    I'd say that integrating $e^{x^2}$ and finding a closed-form solution is the most common one that people learn of during their calculus studies. – The Pointer Jan 13 '20 at 22:33
  • Sorry the question is not about problems that are proven to not be solvable , it is about problems that are proven not be impossible to solve using the analytical approach – Hassen Dhia Jan 13 '20 at 22:34
  • @The Pointer I don't understand what you mean by "closed form solution" for the integration of $e^{x^2}$. – Jean Marie Jan 13 '20 at 22:39
  • @JeanMarie https://stats.stackexchange.com/a/70850 – The Pointer Jan 13 '20 at 22:41
  • @The pointer I don't see the connection between the reference you give and the closed dorm solution of $e^{x^2}$... – Jean Marie Jan 13 '20 at 22:44
  • @lhf can't you see how the answer you have given does'nt have anything to do with the actuall question – Hassen Dhia Jan 13 '20 at 22:47
  • @JeanMarie "In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain "well-known" functions. Typically, these well-known functions are defined to be elementary functions—constants, one variable $x$, elementary operations of arithmetic ($+ - \times \div$), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions)." The integral of $e^{x^2}$ cannot be expressed analytically in terms of a finite number of elementary functions, right? – The Pointer Jan 13 '20 at 23:31
  • @The Pointer I had misunderstood your first remark as "the antiderivative of $e^{x^2}$ can have such a closed form" (read it again...). – Jean Marie Jan 14 '20 at 07:04
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    @JeanMarie Oh, no. I meant that, with regards to integration, this concept is usually introduced as an attempt to integrate $e^{x^2}$ and find a closed-form solution, and when the student realises (through their attempts) or are told that they cannot integrate such an expression, then the concept of not all antiderivatives having a closed-form solution is introduced to them. So it is typically introduced as finding a closed-form solution for the integration of $e^{x^2}$. – The Pointer Jan 14 '20 at 07:36
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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 .)