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I don't know if I really have enough mathematical background to give a profound answer, but there are some things I would like to mention here since I have also pondered a lot about questions when and why finiteness conditions are so important.

Basically the answer to the question "why are finiteness conditions so important?" is very, very simple: Because they make it possible to do mathematics. A mathematical theory which tries to overcome natural finiteness conditions tends to be isolated and narrow. In contrast to that, when you impose good finiteness conditions, the theory becomes rich, very beautiful (which is, of course, subjective). Also "which finiteness conditions" has a very simple answer: Exactly the ones which you need to do the mathematics you want to develope or, at least, imagine. There is no general recipe to produce a good finiteness condition, except that it should fit best to your situation. I've choosen the word "exactly" in order to exclude too restrictive resp. strong finiteness conditions here. On the other hand, we don't always have to look for the most general finitness conditions, unless for some application, we really need more general ones.

For example there is nothing wrong with Hartshorne's book in the definition of coherent sheaves when we restrict ourselves to noetherian schemes - everything works out nicely. But if we jump, some day, to non-noetherian schemes, then we have to reconsider the notions of "coherent", "of finite type", "of finite presentation", etc. In the affine case, this also motivates the definition of noetherian rings: I also agree that the definition involving increasing chaings of ideals might not be the most natural one, but what about the equivalent one which Pete has mentioned: Every submodule of a module of finite type is, again of finite type. Actually exactly this property is ofted needed and maybe it has motivated the definition of noetherian rings. No obscure chains. Besides, from a more modern perspective, it is a relative condition, which talks about objects "over" the ring.

Also quite useful in practice (for example when surjectivity comes from an abstract argument and the injectivity just does not work out): A surjective endomorphism of a noetherian ring is an automorphism. Well I expect that you can list thousands of nice properties here. Remark that this property illstruates that often the finiteness condition is used to conclude something which says nothing at all about finitness. Another example: If $X$ is a compact topological space, then for every topological space $Y$ the map $X \times Y \to Y$ is a closed map. But of course this fits well since in the proof we want to use a finite intersection of open subsets etc., and the definition of a topology with this restricted intersection property is again based on basic examples out of which this notion was developed. So this fits together very well. Also, the property above characterizes compact topological spaces and probably has motivated the corresponding notion of proper schemes in algebraic geometry.

Some general remarks about finitness: One of the most natural object of our mathematical universe is the set of natural numbers $\mathbb{N}$ (I hope no one already here objects and wants to generalize everything to regular cardinals), and the most basic proof involving natural numbers is induction (by the definiton of $\mathbb{N}$ as the smallest inductive set). In order to use induction in more sophisticated situations, we have to give our mathematical objects a measure in $\mathbb{N}$, for example dimension, length, depth, height, etc.. One of the most beautiful and basic examples for this is Grothendieck's vanishing result in sheaf cohomology for finite dimensional topological spaces. So basically you induct on the complexity of the topological space, which you cannot do for arbitrary topological spaces.

Finally I have to admit that my oppinion on finiteness conditions has changed in the last months. For years, I wanted to generalize every notion, theorem or even theory in order to avoid all the occuring finiteness conditions. See this MO question for a very clear example: What about infinite tensor products of vector spaces? We can write them down and prove some basic stuff, but in the end there is nothing interesting which we can do with them and there are no useful connections or applications. So let's just forget about them! :-) The same goes for schemes which are not quasi-separated (link, link).

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I don't know if I really have enough mathematical background to give a profound answer, but there are some things I would like to mention here since I have also pondered a lot about questions when and why finiteness conditions are so important.

Basically the answer to the question "why are finiteness conditions so important?" is very, very simple: Because they make it possible to do mathematics. A mathematical theory which tries to overcome natural finiteness conditions tends to be isolated and narrow. In contrast to that, when you impose good finiteness conditions, the theory becomes rich, very beautiful (which is, of course, subjective). Also "which finiteness conditions" has a very simple answer: Exactly the ones which you need to do the mathematics you want to develope or, at least, imagine. There is no general recipe to produce a good finiteness condition, except that it should fit best to your situation. I've choosen the word "exactly" in order to exclude too restrictive resp. strong finiteness conditions here. On the other hand, we don't always have to look for the most general finitness conditions, unless for some application, we really need more general ones.

For example there is nothing wrong with Hartshorne's book in the definition of coherent sheaves when we restrict ourselves to noetherian schemes - everything works out nicely. But if we jump, some day, to non-noetherian schemes, then we have to reconsider the notions of "coherent", "of finite type", "of finite presentation", etc. In the affine case, this also motivates the definition of noetherian rings: I also agree that the definition involving increasing chaings of ideals might not be the most natural one, but what about the equivalent one which Pete has mentioned: Every submodule of a module of finite type is, again of finite type. Actually exactly this property is ofted needed and maybe it has motivated the definition of noetherian rings. No obscure chains. Besides, from a more modern perspective, it is a relative condition, which talks about objects "over" the ring.

Also quite useful in practice (for example when surjectivity comes from an abstract argument and the injectivity just does not work out): A surjective endomorphism of a noetherian ring is an automorphism. Well I expect that you can list thousands of nice properties here. Remark that this property illstruates that often the finiteness condition is used to conclude something which says nothing at all about finitness. Another example: If $X$ is a compact topological space, then for every topological space $Y$ the map $X \times Y \to Y$ is a closed map. But of course this fits well since in the proof we want to use a finite intersection of open subsets etc., and the definition of a topology with this restricted intersection property is again based on basic examples out of which this notion was developed. So this fits together very well.

Some general remarks about finitness: One of the most natural object of our mathematical universe is the set of natural numbers $\mathbb{N}$ (I hope no one already here objects and wants to generalize everything to regular cardinals), and the most basic proof involving natural numbers is induction (by the definiton of $\mathbb{N}$ as the smallest inductive set). In order to use induction in more sophisticated situations, we have to give our mathematical objects a measure in $\mathbb{N}$, for example dimension, length, depth, height, etc.. One of the most beautiful and basic examples for this is Grothendieck's vanishing result in sheaf cohomology for finite dimensional topological spaces. So basically you induct on the complexity of the topological space, which you cannot do for arbitrary topological spaces.

Finally I have to admit that my oppinion on finiteness conditions has changed in the last months. For years, I wanted to generalize every notion, theorem or even theory in order to avoid all the occuring finiteness conditions. See this MO question for a very clear example: What about infinite tensor products of vector spaces? We can write them down and prove some basic stuff, but in the end there is nothing interesting which we can do with them and there are no useful connections or applications. So let's just forget about them! :-) The same goes for schemes which are not quasi-separated (link, link).