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I think, the Cosmonut mentioning of Stokes' theorem (as a response to Gowers' specification of the answer) must be generalized to a wider statement which can't be bypassed here: in fact, it was exactly the impossibility to make Calculus rigorous that led to the appearance can be considered as a cause of why Analysis appeared.

In modern language the problem is the following. One could expect that the operations over what is called "elementary functions" in Calculus can be axiomatized independently from the axioms of real numbers, so that one gets a closed purely algebraic system, where the equalities of elementary functions are derived from the list of "axiomatic identities" between $x^y$, $\sin x$, etc., and the operations like derivative and integral are conceived in purely algebraic way - the formulation of the problem can be found at page 197 in my textbook in arxiv: http://arxiv.org/abs/1010.0824 (unfortunately, in Russian).

But as it turns out, this is impossible, at least for the complete list of elementary functions: even the equality of elementary functions can't be defined axiomatically. And, what is amazing, this is not a classical result, it is quite new. However, I can't give a reference, what I write here is what Sergei Soloviev from Toulouse told me not long ago.

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I think, the Cosmonut mentioning of Stokes' theorem (as a response to Gowers' specification of the answer) must be generalized to a wider statement which can't be bypassed here: in fact, it was exactly the impossibility to make Calculus rigorous that led to the appearance of Analysis.

In modern language the problem is the following. One could expect that the operations over what is called "elementary functions" in Calculus can be axiomatized independently from the axioms of real numbers, so that one gets a closed purely algebraic system, where the equalities of elementary functions are derived from the list of "axiomatic identities" between $x^y$, $\sin x$, etc., and the operations like derivative and integral are conceived in purely algebraic way - the formulation of the problem can be found at page 197 in my textbook in arxiv: http://arxiv.org/abs/1010.0824 (unfortunately, in Russian).

But as it turns out, this is impossible, at least for the complete list of elementary functions: even the equality of elementary functions can'be can't be defined axiomatically. And, what is amazing, this is not a classical result, it is quite new. However, I can't give a reference, what I write here is what Sergei Soloviev from Toulouse told me not long ago.

In modern language the problem is the following. One could expect that the operations over what is called "elementary functions" in Calculus can be axiomatized independently from the axioms of real numbers, so that one gets a closed purely algebraic system, where the equalities of elementary functions are derived from the list of "axiomatic identities" between $x^y$, $\sin x$, etc., and the operations like derivative and integral are conceived in purely algebraic way - the formulation of the problem can be found at page 197 in my textbook in arxiv: http://arxiv.org/abs/1010.0824 (unfortunately, in Russian).