Posts tagged with equivalence class

Double integrals ∫∫ƒ(x)dA are introduced as a “little teacher’s lie” in calculus. The “real story” requires “geometric algebra”, or “the logic of length-shape-volume relationships”. Keywords

  • multilinear algebra
  • Grassmann algebra / Grassmanian
  • exterior calculus
  • Élie Cartán’s differential-forms approach to tensors

These equivalence-classes of blobs explain how

  • volumes (ahem—oriented volumes!)
  • areas (ahem—oriented areas!)
  • arrows (vectors)
  • numbers (scalars)

"should" interface with each other. That is, Clifford algebra or Grassman algebra or "exterior algebra" or "geometrical algebra" encodes how physical quantities with these dimensionalities do interface with each other.

(First the volumes are abstracted from their original context—then they can be “attached” to something else.)


EDIT:user mrfractal points out that Clifford algebras can only have dimensions of 2,4,8,16,… Yes, that’s right. This post is not totally correct. I let it fly out of the queue without editing it and it may contain other inaccuracies. I was trying to throw out a bunch of relevant keywords that go along with these motivating pictures, and relate it to equivalence-classing, one of my favourite themes within this blog. The text here is disjointed, unedited, and perhaps wrong in other ways. Mostly just wanted to share the pictures; I’ll try to fix up the text some other time. Grazie.


One of my projects in life is to (i) become “fluent in mathematics" in the sense that my intuition should incorporate the objects and relationships of 20th-century mathematical discoveries, and (ii) share that feeling with people who are interested in doing the same in a shorter timeframe.

Inspired by the theory of Plato’s Republic that “philosopher kings” should learn Geometry—pure logic or the way any universe must necessarily work—and my belief that the shapes
a covering, drawn by Robert Ghrist

and feelings thereof operate on a pre-linguistic, pre-rational “gut feeling” level, this may be a worthwhile pursuit. The commercial application would come in the sense that, once you’re in a situation where you have to make big decisions, the only tools you have, in some sense, are who you have become. (Who knows if that would work—but hey, it might! At least one historical wise guy believed the decision-makers should prepare their minds with the shapes of ultimate logic in the universe—and the topologists have told us by now of many more shapes and relations.)

To that end I owe the interested a few more blogposts on:

  • automorphisms / homomorphisms
  • the logic of shape, the shape of logic
  • breadth of functions
  • "to equivalence-class"

which I think relate mathematical discoveries to unfamiliar ways of thinking.


Today I’ll talk about the breadth of functions.

If you remember Descartes’ concept of a function, it is merely a one-to-at-least-one association. “Associate” is about as blah and general and nothing a verb as I could come up with. How could it say anything worthwhile?

The breadth of functions-as-verbs, I think, comes from which codomains you choose to associate to which domains.

The biggest contrast I can come up with is between

  1. a function that associates a non-scalar domain to a ≥0 scalar domain, and
  2. a domain to itself.

If I impose further conditions on the second kind of function, it becomes an automorphismThe conditions being surjectivity  and injectivity : coveringness ≥ 

and one-to-one-ness 
≤  ↑
successor function and square function
Monotone and antitone functions  (not over ℝ just the domain you see = 0<x<1⊂ℝ)  These are examples of invertible functions.

If I impose those two conditions then I’m talking about an isomorphism (bijection) from a space to itself, which I could also call “turning the abstract space over and around and inside out in my hands” — playing with the space. If I biject the space to another version of itself, I’m looking at the same thing in a different way.


Back to the first case, where I associate a ≥0 scalar (i.e., a “regular number” like 12.8) to an object of a complicated space, like

  • the space of possible neuron weightings;
  • the space of 2-person dynamical systems (like the “love equations”);
  • a space containing weird objects that twist in a way that’s easier to describe than to draw;
  • a space of possible things that could happen;
  • the space of paths through London that spend 90% of their time along the Thames;
  • the space of possible protein configurations;

then I could call that “assigning a size to the object”. Again I should add some more constraints to the mapping in order to really call it a “size assignment”. For example continuity, if reasonable—I would like similar things to have a similar size. Or the standard definition of a metric: dist(a,b)=dist(b,a); dist(x,x)=0; no other zeroes besides dist(self,self), and triangle law.

Since the word “size" itself could have many meanings as well, such as:

  • volume
  • angle measure
  • likelihood
  • length/height
  • correlation
  • mass
  • how long an algorithm takes to run
  • how different from the typical an observation is
  • how skewed a statistical distribution is
  • (the inverse of) how far I go until my sampling method encounters the farthest-away next observation
  • surface area
    File:Bronchial anatomy.jpg
  • density
  • number of tines (or “points” if you’re measuring a buck’s antlers)
  • how big of a suitcase you need to fit the thing in (L-∞ norm)

which would order objects differently (e.g., lungs have more surface area in less volume; fractals have more points but needn’t be large to have many points; a delicate sculpture could have small mass, small surface area, large height, and be hard to fit into a box; and osmium would look small but be very heavy—heavier than gold).


Let’s stay with the weighted-neurons example, because it’s evocative and because posets and graphs model a variety of things.



An isomorphism from graphs to graphs might be just to interchange certain wires for dots. So roads become cities and cities become roads. Weird, right? But mathematically these can be dual. I might also take an observation from depth-first versus breadth-first search from computer science (algorithm execution as trees) and apply it to a network-as-brain, if the tree-ness is sufficiently similar between the two and if trees are really a good metaphor after all for either algorithms or brains.

imageBrains sound like a wicked-hard space to think about.  It’s a tightly connected (but not totally connected) network (graph theory)  Each of the nodes’ 3-D location may be important as well (voxels)  The signals propagate through time (dynamical)

More broadly, one hopes that theorems about automorphism groups on trees (like automorphism groups on T-shirts) could evoke interesting or useful thoughts about all the tree-like things and web-like things: be they social networks, roads, or brains.


So that’s one example of a pre-linguistic “shape” that’s evoked by 20th-century mathematics. Today I feel like I could do two: so how about To Equivalence-Class.

Probably due to the invention of set theory, mathematics offers a way of bunching all alike things together. This is something people have done since at least Aristotle; it’s basically like Aristotle’s categories.

  • The set of all librarians;
  • The set of all hats;
  • The set of all sciences;
  • Quine’s (extensional) definition of the number three as “the class of all sets with cardinality three”. (Don’t try the “intensional” definition or “What is it intrinsically that makes three, three? What does three really mean?” unless you’re trying to drive yourself insane to get out of the capital punishment.)
  • The set of all cars;
  • The set of all cats;
  • The set of all computers;
    Water Computer
  • The set of all even numbers;
  • The set of all planes oriented any way in 𝔸³
  • The set of all equal-area blobs in any plane 𝔸² that’s parallel to the one you’re talking about (but could be shifted anywhere within 𝔸³)
  • The set of all successful people;
  • The set of all companies that pay enough tax;
  • The set of all borrowers who will make at least three late payments during the life of their mortgage;
  • The set of all borrowers with between 1% and 5% chance of defaulting on their mortgage;
  • The set of all Extraverted Sensing Feeling Perceivers;
  • The set of all janitors within 5 years of retirement age, who have worked in the custodial services at some point during at least 15 of the last 25 years;
  • The set of all orchids;
  • The set of all ungulates;

The boundaries of some of these (Aristotelian, not Lawverean) categories may be fuzzy or vague

  • if you cut off a cat’s leg is it still a cat?
    What if you shave it? What if you replace the heart with a fish heart?
  • Is economics a science? Is cognitive science a science? Is mathematics a science? Is  Is the particular idea you’re trying to get a grant for scientific?

and in fact membership in any of these equivalence classes could be part of a rhetorical contest. If you already have positive associations with “science”, then if I frame what I do as scientific then you will perceive it as e.g. precise, valuable, truthful, honourable, accurate, important, serious, valid, worthwhile, and so on. Scientists put Man on the Moon. Scientists cured polio. Scientists discovered Germ Theory. (But did “computer scientists” or “statisticians” or “Bayesian quantum communication” or “full professors” or “mathematical élite” or “string theorists” do those things? Yet they are classed together under the STEM label. Related: engineers, artisans, scientists, and intelligentsia in Leonardo da Vinci’s time.)

But even though it is an old thought-form, mathematicians have done such interesting things with the equivalence-class concept that it’s maybe worth connecting the mathematical type with the everyday type and see where it leads you.

Characteristic property of the quotient topology

What mathematics adds to the equivalence-class concept is the idea of “quotienting” to make a new equivalence-class. For example if you take the set of integers you can quotient it in two to get either the odd numbers or the even numbers.


  • If you take a manifold and quotient it you get an orbifold—an example of which would be Dmitri Tymoczko’s mathematical model of Bach/Mozart/Western theory of harmonious musical chords.
  • If you take the real plane ℝ² and quotient it by ℤ²
    (ℤ being the integers) you get the torus 𝕋²
  • Likewise if you take ℝ and quotient it by the integers ℤ you get a circle.

  • If you take connected orientable topological surfaces S with genus g and p punctures, and quotient by the group of orientation-preserving diffeomorphisms of it, you get Riemann’s moduli space of deformations of complex structures S. (I don’t understand that one but you can read about it in Introduction to Teichmüller theory, old and new by Athanase Papadopoulos. It’s meant to just suggest that there are many interesting things in moduli space, surgery theory, and other late-20th-century mathematics that use quotients.)
  • If you quotient the disk D² by its boundary ∂D² you get the globe S².
  • Klein bottles are quotients of the unit rectangle I²=[0,1]².


So equivalence-classing is something we engage in plenty in life and business. Whether it is

  • grouping individuals together for stereotypes (maybe based on the way they dress or talk or spell),
  • or arguing about what constitutes “science” and therefore should get the funding,
  • or about which borrowers should be classed together to create a MBS with a certain default probabilities and covariance (correlation) with other things like the S&P.

Even any time one refers to a group of distinct people under one word—like “Southerners” or “NGO’s” or “small business owners”—that’s effectively creating an (Aristotelian) category and presuming certain properties hold—or hold approximately—for each member of the set.

File:Gastner map redblue byarea bystate.png
File:Gastner map redblue byarea bycounty.png
File:Gastner map purple byarea bycounty.png
File:Red and Blue States Map (Average Margins of Presidential Victory).svg

Of course there are valid and invalid ways of doing this—but before I started using the verb “to equivalence-class” to myself, I didn’t have as good of a rhetoric for interrogating the people who want to generalise. Linking together the process of abstraction-from-experience—going from many particular observations of being cheated to a model of “untrustworthy person”—with the mathematical operations of

  • slicing off outliers,
  • quotienting along properties,
  • foliating,
  • considering subsets that are tamer than the vast freeness of generally-the-way-anything-can-be

—formed a new vocabulary that’s helpfully guided my thinking on that subject.

Ordine geometrico demonstrata!

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”:

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!








  • "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

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 \texttt{ \{turkey, vulture, dove \} } is quite different to the set \{ \forall \texttt{ cones,\ the\ plane,\ a\ sheaf\ of\ rings} \} 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 ( \texttt{ things, distances\ between\ the\ things } ). (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 mapsInvertible, 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

John Baez and James Dolan, From Finite Sets to Feynman Diagrams

  • an explosion of ideas
  • equality x=x is boring
  • why is 6÷2=3 ?


The last time I was reading the Zhoangdze, I got sidetracked by footnotes about the following 白馬論:

  • "A white horse is not a horse."

This is apparently attributable to Gongsun Long 公孙龙, a Chinese philosopher of the 3rd century B.C. = Era of Warring States 战国时代. How could such a wack philosopher be worth dignifying with a mention in the  庄子?

Juang Tzu

I got my answer from page 40 of James Gleick’s The Information. Like most ancient debates, this all relates back to Bill Clinton and Monica Lewinsky.

  • Big Bill Clinton, Rhodes Scholar: “That depends what the meaning of the word is, is.”

Too true, my man.

As Bill Thurston pointed out, students of mathematics regularly use the = symbol when it’s not appropriate (maybe an or or or set comprehension is what’s needed) just because it’s the only “connecting word” they know. But the meaning of “is” is too multifarious to always translate to the = symbol. For more tools of how to think about what exactly we’re saying with “is”, check out two papers I’ve linked on this site: Barry Mazur's When is a thing equal to some other thing? and John Baez + James Dolan’s From Finite Sets to Feynman Diagrams.

You might have thought I would stoop to a seductive pic of Ms Lewinsky, but I'm too good for that. Here's an ancient map of China, Warring States period.

Back to the ancient Chinese stuff. What 公孫龍 was trying to say, was that

  • "white horse" “horse”

in the sense of the = sign. The = sign means you can freely substitute one thing for another—to the point of ridiculousness if you wish—without distorting the truth value. But using Gleick’s example,

  • "Lana doesn’t like white horses” does not mean “Lana doesn’t like horses”

Really "white horse" ⊂ "horse", a white horse is a kind of horse, but that means in an object-oriented programming sense we’re talking about class inheritance, not ===.

Now afore you go runnin afeart that the English | Chinese language is being shoehorned into mathematical symbology, read a couple sentences of Quine. In this case a bit of set theory and technical statements like “The set of all referents satisfying the criteria X also satisfy the criteria Y”, and thinking about alternatives to = like  and , actually makes our English-language thinking clearer.


UPDATE: Jeremy Tran explains that:

[As] I understand it. For “is ≡”, we use the phrase 一樣, while for “is ⊂”, we have 一種. These two concepts are relatively distinct from each other. The ‘proper’ translations for the two are “the same” and “a type of”, respectively. But often, those two are translated into English simply as “is”, which can lead to issues.

My Chinese isn’t exactly the best, so hopefully I haven’t made any big mistakes. But this is the gist of it. :D

When I was a maths teacher some curious students (Fez and Andrew) asked, “Does i, √−1, exist? Does infinity ∞ exist?” I told this story.

You explain to me what 4 is by pointing to four rocks on the ground, or dropping them in succession — Peano map, Peano map, Peano map, Peano map. Sure. But that’s an example of the number 4, not the number 4 itself.

So is it even possible to say what a number is? No, let’s ask something easier. What a counting number is. No rationals, reals, complexes, or other logically coherent corpuses of numbers.

Willard van Orman Quine had an interesting answer. He said that the number seventeen “is” the equivalence class of all sets of with 17 elements.

Accept that or not, it’s at least a good try. Whether or not numbers actually exist, we can use math to figure things out. The concepts of √−1 and serve a practical purpose just like the concept of (you know, the obvious moral cap on income tax). For instance

  • if power on the power line is traveling in the direction +1 then the wire is efficient; if it travels in the direction √−1 then the wire heats up but does no useful work. (Er, I guess alternating current alternates between −1 and −1.)
  • allows for limits and therefore derivatives and calculus. Just one example apiece.

Do 6-dimensional spheres exist? Do matrices exist? Do power series exist? Do vector fields exist? Do eigenfunctions exist? Do 400-dimensional spaces exist? Do dynamical systems exist? Yes and no, in the same way.