#isomorphisms

[cauchyfeelings]

Me breaking up with someone: It appears that our internal representational geometries are non-isomorphic. I wish you the best in your local neighborhood.

[lawsofemotion]

A blog about mathematics.

Updated Rings and Modules: Quotient Groups and Homomorphisms!

[isomorphismes]

[lawsofemotion]

A blog about mathematics.

Updated Chapter 9: Isomorphisms!

[isomorphismes]

Galileo observed that the set of positive integers can be put into 1‒1 correspondence with the set of square integers, even though the set of nonsquares seems more numerous than the squares. He deduced from this that "the attributes =, >, < are not applicable to infinite quantities".

Stan Wagon

[dsrandois]

If daydreams are walks on the beach

Poems are the footsteps in the sand.

[studyingwithjessie]

More on Isomorphisms

[isomorphismes]

It's a good thing that finite-dimensional vector spaces are not canonically isomorphic to their duals. Such an iso. would amount to choosing an inner product (when the field is appropriate), and that would mean that vector spaces come with some intrinsic geometry, which is a terrifying notion. Related to that: the non-degeneracy of symplectic structures picks an isomorphism between the tangent and cotangent spaces at a point, which is what leads to one being able to define a hamiltonian as a vector field, leading to time evolution being the flow of a vector field. so basically there not being a canonical isomorphism leads right to classical mechanics,lol. That leads to the question: what does it mean in terms of classical mechanics when two symplectic forms are in the same cohomology class?

John Paprocki (@jpoprox)