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.)
The fact that a bijection can show you something in a new way that suddenly makes the answer to the question so obvious, is the basis of the jokes on www.theproofistrivial.com.
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.
Or here’s an example from calc 101: u-substitution. You’re essentially saying “Instead of solving this integral, how about if I solve a different one which is exactly equivalent?” The
→ƒ in the top diagram is the u-substitution itself. The “main verb” is doing the integral. U-substituters avoid doing the hard integral, go the long way, and end up doing something much easier.
Or in physics—like tensors and Schrödinger solving and stuff.
Physicists look for substitutions that make the computation they have to do more tractable. Try solving a Schrödinger PDE for hydrogen’s first electron
xyz coordinates (square grid)—then try solving it in spherical coordinates (longitude & latitude on expanding shells). Since the natural symmetry of the
s¹ orbital is spherical, changing basis to polar coords makes life much easier.
Likewise one of the goals of tensor analysis is to not be tied to any particular basis—so long as the basis doesn’t trip over itself, you should be free to switch between bases to get different jobs done. Terry Tao talks about something like this under the keyword “spending symmetry”—if you use up your basis isomorphism, you need to give it back before you can use it again.
"Going the long way" can be easier than trying to solve a problem directly.