In "A Panorama of Pure Mathematics" by Dieudonne, he said

The history of mathematics shows that a theory almost always originates in efforts to solve a specific problem (for example, the duplication of the cube in Greek mathematics). It may happen that these efforts are fruitless, and we have our first category of problems:

(I) Stillborn problems (examples: the determination of Fermat primes, or the irrationality of Euler's constant).

A second possibility is that the problem is solved but does not lead to progress on any other problem. This gives a second class:

(II) Problems without issue (this class includes many problems arising from "combinatorics").

A more favorable situation is one in which an examination of the techniques used to solve the original problem enables one to apply them (perhaps by making them considerably more complicated) to other similar or more difficult problems, without necessarily feeling that one really understands why they work. We may call these

(III) Problems that beget a method (analytic number theory and the theory of finite groups provide many examples).

In a few rather rare cases the study of the problem ultimately (and perhaps only after a long time) reveals the existence of unsuspected underlying structures that not only illuminate the original question but also provide powerful general methods for elucidating a host of other problems in other areas; thus we have

(IV) Problems that belong to an active and fertile general theory (the theory of Lie groups and algebraic topology are typical examples at the present time).

However, as Hilbert emphasized, a mathematical theory cannot flourish without a constant influx of new problems. It has often happened that once the problems that are of the greatest importance for their consequences and their connections with other branches of mathematics have been solved, the theory tends to concentrate more and more on special and isolated questions (possibly very difficult ones). Hence we have yet another category:

(V) Theories in decline (at least for the time being: invariant theory, for example, has passed through this phase several times).

Finally, if a happy choice of axioms, motivated by specific problems, has led to the development of techniques of great efficacy in many areas of mathematics, it may happen that attempts are made with no apparent motive to modify these axioms somewhat arbitrarily, in the hope of repeating the success of the original theory. This hope is usually in vain, and thus we have, in the phrase of Polya and Szego

(VI) Theories in a state of dilution (following the example of these authors, we shall cite no instances of this).

What could such theories be? Dieudonne unfortunately didn't give any examples.

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    $\begingroup$ For a good reason. "Group theory has concentrated ideas which formerly were found scattered in algebra, number theory, geometry and analysis and which appeared to be very different. Examples of generalisation by dilution would be still easier to quote, but this would be at the risk of offending sensibilities", Pólya and Szegö: Problems and Theorems in Analysis. One can think of many ways the concept of group was diluted, but people working on semigroups would indeed take offense. The same goes for multiple variations on ZFC. $\endgroup$ – Conifold Feb 9 at 12:35
  • $\begingroup$ In addition to Conifold's comment, by "state of dilution" one can mean areas of mathematics where one uses some standard (sometimes, powerful) methods, in order to tackle artificial/uninteresting problems, e.g. in analysis, papers "on yet another class of differential equations." $\endgroup$ – Moishe Kohan Feb 9 at 14:09

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