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Many mathematicians prefer to have (at least) two equivalent and elegant(!), but different definitions of the same notion/topic/... (such definitions either both exist or one formulates one of them or even both and proves the equivalence).

One of the reasons might be that one definition may look as weak as possible, while the other one may look as strong as possible.

Then when you want to prove that a construction/theory/... is an example/model/... of the given theory, use the seemingly weak definition.

But when you want to prove hard theorems that follow from the definition then it'd be much easier to apply the strong/advanced definition.

There is intellectual energy stored between the weak and the strong definitions.

First Example: there are easily a dozen and more elementary axiomatizations of the so-called Euclidean plane geometry. They often aim at elegance understood as making the axioms as weak as possible. Given an algebraic model as Cartesian $\ \mathbb R^2\ $ it's easy to prove such weak axioms.

However, an excellent definition of such plane geometry is as follows: the study of metric invariants of the complex plane $\ \mathbb C.\ $ Suddenly you can truly prove powerful theorems in an algebraic clean way, and often even without relying on ingenuity as it is invariably in the case of the classical approach. You also get a clear understanding of the border (degenerated) cases. Finally, "elementary" proofs consider several cases due to the different situations w.r. to the "in-between" relation. This, however, is not any problem in the case of algebraic proves based on the complex plane.

PS. While the Cartesian plane is adequate for the affine geometry of the plane (and for the classical mechanics -- classical mechanics is a natural and mild extension of the affine geometry), it is the complex plane $\ \mathbb C\ $ that handles also angles.

Many mathematicians prefer to have (at least) two equivalent and elegant(!), but different definitions of the same notion/topic/... (such definitions either both exist or one formulates one of them or even both and proves the equivalence).

One of the reasons might be that one definition may look as weak as possible, while the other one may look as strong as possible.

Then when you want to prove that a construction/theory/... is an example/model/... of the given theory, use the seemingly weak definition.

But when you want to prove hard theorems that follow from the definition then it'd be much easier to apply the strong/advanced definition.

There is intellectual energy stored between the weak and the strong definitions.

First Example: there are easily a dozen and more elementary axiomatizations of the so-called Euclidean plane geometry. They often aim at elegance understood as making the axioms as weak as possible. Given an algebraic model as Cartesian $\ \mathbb R^2\ $ it's easy to prove such weak axioms.

However, an excellent definition of such plane geometry is as follows: the study of metric invariants of the complex plane $\ \mathbb C.\ $ Suddenly you can truly prove powerful theorems in an algebraic clean way, and often even without relying on ingenuity as it is invariably in the case of the classical approach. You also get a clear understanding of the border (degenerated) cases. Finally, "elementary" proofs consider several cases due to the different situations w.r. to the "in-between" relation. This, however, is not any problem in the case of algebraic proves based on the complex plane.

PS. While the Cartesian plane is adequate for the affine geometry of the plane (and for the classical mechanics -- classical mechanics is a natural and mild extension of the affine geometry), it is the complex plane $\ \mathbb C\ $ that handles also angles.

Second Example: Elementary definition of determinant is elementary but the axiomatic definition is easier to apply and provides understanding. For instance, it's natural to use the axiomatic definition to derive the formula for the derivative of a square matrix when the entries are differentiable functions.

PS. The said axioms of the determinant of a matrix are formulated as a linear function of its columns, alternative, and normalized -- the determinant of $\ \mathbb I_n:=1$.

Many mathematicians prefer to have (at least) two equivalent and elegant(!), but different definitions of the same notion/topic/... (such definitions either both exist or one formulates one of them or even both and proves the equivalence).

One of the reasons might be that one definition may look as weak as possible, while the other one may look as strong as possible.

Then when you want to prove that a construction/theory/... is an example/model/... of the given theory, use the seemingly weak definition.

But when you want to prove hard theorems that follow from the definition then it'd be much easier to apply the strong/advanced definition.

There is intellectual energy stored between the weak and the strong definitions.

First Example: there are easily a dozen and more elementary axiomatizations of the so-called Euclidean plane geometry. They often aim at elegance understood as making the axioms as weak as possible. Given an algebraic model as Cartesian $\ \mathbb R^2\ $ it's easy to prove such weak axioms.

However, an excellent definition of such plane geometry is as follows: the study of metric invariants of the complex plane $\ \mathbb C.\ $ Suddenly you can truly prove powerful theorems in an algebraic clean way, and often even without relying on ingenuity as it invariably the case of the classical approach. You also get a clear understanding of the border (degenerated) cases.

While the Cartesian plane is adequate for the affine geometry of the plane (and for the classical mechanics -- classical mechanics is a natural and mild extension of the affine geometry), it is the complex plane $\ \mathbb C\ $ that handles also angles.

Many mathematicians prefer to have (at least) two equivalent and elegant(!), but different definitions of the same notion/topic/... (such definitions either both exist or one formulates one of them or even both and proves the equivalence).

One of the reasons might be that one definition may look as weak as possible, while the other one may look as strong as possible.

Then when you want to prove that a construction/theory/... is an example/model/... of the given theory, use the seemingly weak definition.

But when you want to prove hard theorems that follow from the definition then it'd be much easier to apply the strong/advanced definition.

There is intellectual energy stored between the weak and the strong definitions.

First Example: there are easily a dozen and more elementary axiomatizations of the so-called Euclidean plane geometry. They often aim at elegance understood as making the axioms as weak as possible. Given an algebraic model as Cartesian $\ \mathbb R^2\ $ it's easy to prove such weak axioms.

However, an excellent definition of such plane geometry is as follows: the study of metric invariants of the complex plane $\ \mathbb C.\ $ Suddenly you can truly prove powerful theorems in an algebraic clean way, and often even without relying on ingenuity as it is invariably in the case of the classical approach. You also get a clear understanding of the border (degenerated) cases. Finally, "elementary" proofs consider several cases due to the different situations w.r. to the "in-between" relation. This, however, is not any problem in the case of algebraic proves based on the complex plane.

PS. While the Cartesian plane is adequate for the affine geometry of the plane (and for the classical mechanics -- classical mechanics is a natural and mild extension of the affine geometry), it is the complex plane $\ \mathbb C\ $ that handles also angles.

Many mathematicians prefer to have (at least) two equivalent and elegant(!) but different definitions of the same notion/topic/... (such definitions either both exist or one proves one of them or even both).

One of the reasons might be that one definition may look as weak as possible, while the other one may look as strong as possible.

Then when you want to prove that a construction/theory/... is an example/model/... of the given theory then you use the seemingly weak definition.

But when you want to prove hard theorems that follow from the definition then it'd be much easier to apply the strong/advanced definition.

There is intellectual energy stored between the weak and the strong definitions.

Many mathematicians prefer to have (at least) two equivalent and elegant(!), but different definitions of the same notion/topic/... (such definitions either both exist or one formulates one of them or even both and proves the equivalence).

One of the reasons might be that one definition may look as weak as possible, while the other one may look as strong as possible.

Then when you want to prove that a construction/theory/... is an example/model/... of the given theory, use the seemingly weak definition.

But when you want to prove hard theorems that follow from the definition then it'd be much easier to apply the strong/advanced definition.

There is intellectual energy stored between the weak and the strong definitions.

First Example: there are easily a dozen and more elementary axiomatizations of the so-called Euclidean plane geometry. They often aim at elegance understood as making the axioms as weak as possible. Given an algebraic model as Cartesian $\ \mathbb R^2\ $ it's easy to prove such weak axioms.

However, an excellent definition of such plane geometry is as follows: the study of metric invariants of the complex plane $\ \mathbb C.\ $ Suddenly you can truly prove powerful theorems in an algebraic clean way, and often even without relying on ingenuity as it invariably the case of the classical approach. You also get a clear understanding of the border (degenerated) cases.

While the Cartesian plane is adequate for the affine geometry of the plane (and for the classical mechanics -- classical mechanics is a natural and mild extension of the affine geometry), it is the complex plane $\ \mathbb C\ $ that handles also angles.