Why "Stable" in Stable vector bundle and Stable homotopy theory? Why/what/how does stable mean in the

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*Stable vector bundle?


*Stable homotopy theory?
Of course, it is described in the Wiki texts in a formal way. But I wonder what are the intuitions and concepts behind both stables in the text.
Another way to phrase it may be that when we remove stable, what makes the differences for vector bundle and homotopy theory?
 A: Any good answer to this would be very long, so perhaps some people can answer with alternative perspectives.
Essentially stability removes "pathologies" that only happen in low dimensions. For example, there is only one tangent bundle to a manifold, but many normal bundles. However, if we make the codimension of our embedding large enough all normal bundles become isomorphic, so stabilization here allows us to talk about "the" stable normal bundle. There are other reasons to care about stable bundles, for example, most of our characteristic classes care only about the stable isomorphism type. And perhaps the most important reason is that stable bundles (or the slightly different virtual bundles), form a group rather than a monoid.
When it comes to homotopy theory, stable homotopy theory tends to be more algebraic than homotopy theory. This can be seen at the level of spaces. There is no natural group structure on $[X,Y]$ but if $X$ is a suspension, then this naturally has a group structure. Moreover, if $X$ is a double suspension, this is even nicer than a group, it is an abelian group.
One really concrete way of seeing the algebraic difference between spaces and spectra is through the following example: Quillen proved the remarkable result that (subject to a small connectivity assumption) spaces up to rational equivalence have the exact same homotopy theory as rational differential graded commutative algebras. The proof of this is rather involved.
Of course we can ask what spectra up to rational equivalence are? The answer is just rational chain complexes, and it is very easy to see. It follows from the fact that rationally every spectrum is equivalent to the generalized Eilenberg-MacLane spectrum with the same homotopy groups.
To me, chain complexes are entirely algebraic, but products on a chain complex can encode a lot of geometry (i.e. cup products on singular cochain complexes or de Rham wedge products, etc.)
Now what "pathology" is this a result of? I haven't thought about this in a while, but I think it is due to the fact that in the stable world the rational Eilenberg-MacLane space is equivalent to the rational Moore space (the rational sphere). However, unstably they are not equivalent. This is due to an unstable obstruction in the cohomology of the even dimensional Eilenberg-MacLane space, the cup square, so passing to the stable world this vanishes.
This will imply that Postnikov towers for rational spectra are all trivial, but for rational spaces they could have nontrivial k invariants. Of course, there is so much more that can be said, so I hope others, who assuredly know a lot more stable homotopy theory than me, can contribute
