isomorphismes

We want to take theories and turn them over and over in our hands, turn the pants inside out and look at the sewing; hold them upside down; see things from every angle; and sometimes, to quotient or equivalence-class over some property to either consider a subset of cases for which a conclusion can be drawn (e.g., “all fair economic transactions” (non-exploitive?) or “all supply-demand curveses such that how much you get paid is in proportion to how much you contributed” (how to define it? vary the S or the D and get a local proportionality of PS:TS? how to vary them?)

Consider abstractly a set like {a, b, c, d}. 4! ways to rearrange the letters. Since sets are unordered we could call it as well the quotient of all rearangements of quadruples of once-and-yes-used letters (b,d,c,a). /p>

Descartes’ concept of a mapping is “to assign” (although it’s not specified who is doing the assigning; just some categorical/universal ellipsis of agency) members of one set to members of another set.

• For example the Hash Map of programming.
  

  {
   '_why' => 'famous programmer',
   'North Dakota' => 'cold place',
   ... }

• Or to round up ⌈num⌉: not injective because many decimals are written onto the same integer.
  
  
  
  
  
• Or to “multiply by zero” i.e. “erase” or “throw everything away”:

Old programmers don’t die, they’re piped to /dev/null.

— protea (@isomorphisms) January 2, 2013

Linear maps with det=0 :: - | /dev/null :: matter into a black hole :: (or, on Earth, a trash grinder)

— protea (@isomorphisms) January 2, 2013

In this sense a bijection from the same domain to itself is simply a different—but equivalent—way of looking at the same thing. I could rename A=1,B=2,C=3,D=4 or rename A='Elsa',B='Baobab',C=√5,D=Hypathia and end with the same conclusion or “same structure”. For example. But beyond renamings we are also interested in different ways of fitting the puzzle pieces together. The green triangle of the wooden block puzzle could fit in three rotations (or is it six rotations? or infinity right-or-left-rotations?) into the same hole.

By considering all such mappings, dividing them up, focussing on the easier classes; classifying the types at all; finding (or imposing) order|pattern on what seems too chaotic or hard to predict (viz, economics) more clarity or at least less stupidity might be found.

The hope isn’t completely without support either: Quine explained what is a number with an equivalence class of sets; Tymoczko described the space of musical chords with a quotient of a manifold; PDE’s (read: practical engineering application) solved or better geometrically understood with bijections; Gauss added 1+2+3+...+99+100 in two easy steps rather than ninety-nine with a bijection; ….

It’s hard for me to speak to why we want groups and what they are both at once. Today I felt more capable of writing what they are.

So this is the concept of sameness, let’s discuss just linear planes (or, hyperplanes) and countable sets of individual things.

Leave it up to you or for me later, to enumerate the things from life or the physical world that “look like” these pure mathematical things, and are therefore amenable by metaphor and application of proved results, to the group theory.

But just as one motivating example: it doesn’t matter whether I call my coordinates in the mechanical world of physics (x,y,z) or (y,x,z). This is just a renaming or bijection from {1,2,3} onto itself.

Even more, I could orient the axis any way that I want. As long as the three are mutually perpendicular each to the other, the origin can be anywhere (invariance under an affine mapping — we can equivalence-class those together) and the rotation of the 3-D system can be anything. Stand in front of the class as the teacher, upside down, oriented so that one of the dimensions helpfully disappears as you fly straight forward (or two dimensions disappear as you run straight forward on a flat road). Which is an observation taken for granted by my 8th grade physics teacher. But in the language of group theory means we can equivalence-class over the special linear group of 3-by-3 matrices that leave volume the same. Any rotation in 3-D

Sameness-preserving Groups partition into:

• permutation groups, or rearrangements of countable things, and
• linear groups, or “trivial” “unimportant” “invariant” changes to continua (such as rescaling—if we added a “0” to the end of all your currency nothing would change)
  
• conjunctions of smaller groups

The linear groups—get ready for it—can all be represented as matrices! This is why matrices are considered mathematically “important”. Because we have already conceived this huge logical primitive that (in part) explains the Universe (groups) — or at least allows us to quotient away large classes of phenomena — and it’s reducible to something that’s completely understood! Namely, matrices with entries coming from corpora (fields).

So if you can classify (bonus if human beings can understand the classification in intuitive ways) all the qualitatively different types of Matrices,

then you not only know where your engineering numerical computation is going, but you have understood something fundamental about the logical primitives of the Universe!

Aaaaaand, matrices can be computed on this fantastic invention called a computer!

unf

• “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 material-isomorphic.

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=x is 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