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On the Academia site, there is a recent question that asks about obtaining reviewers for a "new theory". I'm only an amateur mathematician, not a professional, and the question got me wondering what would be considered to constitute a new theory in mathematics, as opposed to (merely) a new theorem.

It would be helpful if you could use an historical example, rather than something at the cutting edge of modern mathematical research.

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    $\begingroup$ Examples that come to mind are non-Euclidean geometry, Cantor's transfinite numbers, Frechet's introduction of metric space ideas as a way of conceptualizing the analogies between analytic geometric notions and the early work in functional analytic methods, quaternions, and summability theory. $\endgroup$ Mar 12, 2023 at 10:40

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The Riemann integral is very useful for most of the basic needs of integration. However, it became clear around the end of the XIXth century that it wasn't general enough. That's why Lebesgue developed his integral. Every Riemann-integrable function is also Lebesgue-integrable, but there are Lebesgue-integrable functions which are not Riemann-integrable, such as$$\begin{array}{rccc}f\colon&[0,1]&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}1&\text{ if }x\in\Bbb Q\\0&\text{otherwise;}\end{cases}\end{array}$$it is not Riemann-integrable, but, with respect to the Lebesgue integral, $\int_0^1f(x)\,\mathrm dx=0$.

So, Lebesgue did not just provide a new theorem about the Riemann integral. He developed a new theory of integration.

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  • $\begingroup$ Nice answer, but do you really want the domain to be in $\mathbb Q$? $\endgroup$ Mar 12, 2023 at 17:32
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    $\begingroup$ @J.W.Tanner Thank you for calling my attention to that. It as a mistake. $\endgroup$ Mar 12, 2023 at 17:52
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The distinction is not formal. A new theory usually involves some new set of definitions, new methods and several theorems. Examples are abundant. Theory of groups, theory of distributions (generalized functions), theory of integration in the complex domain (Cauchy theory), etc.

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