- I cheated on you. ∄ way to restore the original pure trust of our early relationship.
- The broken glass. Even with glue we couldn’t put it back to be the same original glass.
- I got old. ∄ potion to restore my lost youth.
- Adam & Eve ate from the tree of the knowledge of good & evil. They could not unlearn what they learned.
- “Be … careful what you put in that head because you will never, ever get it out.” ― Thomas Cardinal Wolsey
- We polluted the lake with our sewage runoff. The algal blooms choked off the fish. ∄ way to restore it.
- Phase change. And the phase boundary can only be traversed one direction (or the backwards direction costs vastly more energy). The marble rolls off the table, the leg poisoned by gangrene. The father dies at war. The unkind words can’t be unsaid.
Posts tagged with bijection
going the long way
What does it mean when mathematicians talk about a bijection or homomorphism?
Imagine you want to get from
X′ but you don’t know how. Then you find a “different way of looking at the same thing” using ƒ. (Map the stuff with ƒ to another space
Y, then do something else over in
image ƒ, then take a journey over there, and then return back with ƒ ⁻¹.)
In a given category the homomorphisms
Hom ∋ ƒ preserve all the interesting properties. Linear maps, for example (except when
det=0) barely change anything—like if your government suddenly added another zero to the end of all currency denominations, just a rescaling—so they preserve most interesting properties and therefore any linear mapping to another domain could be inverted back so anything you discover over in the new domain (
image of ƒ) can be used on the original problem.
All of these fancy-sounding maps are linear:
They sound fancy because whilst they leave things technically equivalent in an objective sense, the result looks very different to people. So then we get to use intuition or insight that only works in say the spectral domain, and still technically be working on the same original problem.
Pipe the problem somewhere else, look at it from another angle, solve it there, unpipe your answer back to the original viewpoint/space.
For example: the Gaussian (normal) cumulative distribution function is monotone, hence injective (one-to-one), hence invertible.
By contrast the Gaussian probability distribution function (the “default” way of looking at a “normal Bell Curve”) fails the horizontal line test, hence is many-to-one, hence cannot be totally inverted.
So in this case, integrating once
∫[pdf] = cdf made the function “mathematically nicer” without changing its interesting qualities or altering its inherent nature.
“Going the long way” can be easier than trying to solve a problem directly.
- “X does something whilst preserving a certain structure”
- “There exist deformations of Y that preserve certain properties”
- “∃ function ƒ such that P, whilst respecting Q”
This common mathematical turn of phrase sounds vague, even when the speaker has something quite clear in mind.
Smeet Bhatt brought up this unclarity in a recent question on Quora. Following is my answer:
It depends on the category. The idea of isomorphism varies across categories. It’s like if I ask you if two things are “similar” or not.
- “Similar how?” you ask.
Think about a children’s puzzle where they are shown wooden blocks in a variety of shapes & colours. All the blocks that have the same shape are
shape-isomorphic. All the blocks of the same colour are
colour-isomorphic. All the blocks are wooden so they’re
In common mathematical abstractions, you might want to preserve a property like
- still a group
- still a vector space
- still a simplicial complex
- still a plane
- still similar to the original triangle
- still sum to the same constant (symplectic)
- deforming the path without pushing it over a singularity doesn’t change the contour integral
after some transformation
φ. It’s the same idea: “The same in what way?”
As John Baez & James Dolan pointed out, when we say two things are “equal”, we usually don’t mean they are literally the same.
x=xis the most useless expression in mathematics, whereas more interesting formulæ express an isomorphism:
- “Something is the same about the LHS and RHS”.
- “They are similar in the following sense”.
Just what the something is that’s the same, is the structure to be preserved.
A related idea is that of equivalence-class. If I make an equivalence class of all sets with cardinality 4, I’m talking about “their size is equivalent”.
Of course the set is quite different to the set in other respects. Again it’s about “What is the same?” and “What is different?” just like on Sesame Street.
Two further comments: “structure” in mathematics usually refers to a tuple or a category, both of which mean “a space” in the sense that not only is there a set with objects in it, but also the space or tuple or category has mappings relating the things together or conveying information about the things. For example a metric space is a tuple . (And: having a definition of distance implies that you also have a definition of the topology (neighbourhood relationships) and geometry (angular relationships) of the space.)
In the case of a metric space, a structure-preserving map between metric spaces would not make it impossible to speak of distance in the target space. The output should still fulfill the metric-space criteria: distance should still be a meaningful thing to talk about after the mapping is done.
I’ve got a couple drafts in my 1500-long queue of drafts expositing some more on this topic. If I’m not too lazy then at some point in the future I’ll share some drawings of structure-preserving maps (different “samenesses”) such as the ones Daniel McLaury mentioned, also on Quora:
- Category: Structure-preserving maps; Invertible, structure-preserving maps
- Groups: (group) homomorphism; (group) isomorphism
- Rings: (ring) homomorphism; (ring) isomorphism
- Vector Spaces: linear transformation, invertible linear transformation
- Topological Spaces: continuous map; homeomorphism
- Differentiable Manifolds: differentiable map; diffeomorphism
- Riemannian Manifolds: conformal map; conformal isometry