Posts tagged with equivalence classes

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,… https://en.wikipedia.org/wiki/Clifford_algebra#Basis_and_dimension 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.

(Source: arxiv.org)



















Walter Ong turns to the fieldwork of the Russian psychologist Aleksandr Romanovich Luria among illiterate peoples [of] Uzbekistan and Kyrgyzstan … in the 1930’s.

Luria found striking differences between illiterate and even slightly literate subjects, not in what they knew, but in how they thought.

Logic implicates symbolism directly: things are members of classes; they possess qualities, which are abstracted and generalised.

Les Mots & Les Images by René MagritteOral people lacked the categories that become second nature even to illiterate individuals [living] in literate cultures…. They would not accept logical syllogisms.

A typical question:

—In the Far North, where there is snow, all the bears are white.

—Novaya Zembla is in the Far North and there is always snow there.

—What colour are the bears?

—I don’t know. I’ve seen a black bear. I’ve never seen any others…. Each locality has its own animals.

….

"Try to explain to me what a tree is," Luria says, and a peasant replies: "Why should I? Everyone knows what a tree is, they don’t need me telling them."

James Gleick, The Information, citing Walter J. Ong and Aleksandr Romanovich Luria

image

from http://en.wikipedia.org/wiki/William_Blake#Royal_Academy:

Over time, Blake came to detest Joshua Reynolds attitude towards art, especially his pursuit of “general truth” and “general beauty”. Reynolds wrote in his Discourses that the “disposition to abstractions, to generalising and classification, is the great glory of the human mind”; Blake responded, in marginalia to his personal copy, that "To Generalize is to be an Idiot; To Particularize is the Alone Distinction of Merit".[20]

image




The category of categories as a model for the Platonic World of Forms by David A Edwards & Marilyn L Edwards

  • Thales (7th cent. BC) made the first universal statement (proof w/o regard to the gods or mythology, just from pure reason)
  • pre-Greek mathematics was essentially engineering maths.
  • I owe ya a post on the illiterates in chapter 2 of James Gleick’s The Information. He tells the story of some illiterates in outer Soviet Union. According to the tale, they basically do not abstract at all. No abstract reasoning, no properties ascribed to members of a class, and so on.

    It sounds kind of idyllic in the way of NYT tales of the Pirahã or Jill Bolte Taylor’s story of losing the logical half of her brain. I’m not sure if Thales set us on the path to Hell or Heaven.
  • Plato set for himself the [goal] of extending geometry [beyond] triangles and circles and such, to all of human thought. He failed, but his vision has come to pass.
  • Why did Lawvere succeed where Plato and Whitehead failed?
  • He had Descartes’ already-abstract notion of a function, along with
  • Eilenberg & Mac Lane’s notions of category and functor.
  • The definition of function for infinite sets is already implicit in the choice of “which set theory”.
  • Category theory, unlike earlier formalisations (think Peano arithmetic and Goedel’s proof), is stable to the “meta” step: you do 2-categories, you do n-categories … the abstraction is ultimately a k → k+1 kind of deal rather than a “And this is the ultimate finality!” kind of deal.




[G]eometry and number[s]…are unified by the concept of a coordinate system, which allows one to convert geometric objects to numeric ones or vice versa. …

[O]ne can view the length ❘AB❘ of a line segment AB not as a number (which requires one to select a unit of length), but more abstractly as the equivalence class of all line segments that are congruent to AB.

With this perspective, ❘AB❘ no longer lies in the standard semigroup ℝ⁺, but in a more abstract semigroup (the space of line segments quotiented by congruence), with addition now defined geometrically (by concatenation of intervals) rather than numerically.

A unit of length can now be viewed as just one of many different isomorphisms Φ: ℒ → ℝ⁺ between and ℝ⁺, but one can abandon … units and just work with directly. Many statements in Euclidean geometry … can be phrased in this manner.

(Indeed, this is basically how the ancient Greeks…viewed geometry, though of course without the assistance of such modern terminology as “semigroup” or “bilinear”.)
Terence Tao

(Source: terrytao.wordpress.com)




[T]he point of introducing L^p spaces in the first place is … to exploit … Banach space. For instance, if one has |ƒ − g| = 0, one would like to conclude that ƒ = g. But because of the equivalence class in the way, one can only conclude that ƒ is equal to g almost everywhere.

The Lebesgue philosophy is analogous to the “noise-tolerant” philosophy in modern signal progressing. If one is receiving a signal (e.g. a television signal) from a noisy source (e.g. a television station in the presence of electrical interference), then any individual component of that signal (e.g. a pixel of the television image) may be corrupted. But as long as the total number of corrupted data points is negligible, one can still get a good enough idea of the image to do things like distinguish foreground from background, compute the area of an object, or the mean intensity, etc.

Terence Tao

If you’re thinking about points in Euclidean space, then yes — if the distance between them is nil, they are in the exact same spot and therefore the same point.

But abstract mathematics opens up more possibilities.

  • Like TV signals. Like 2-D images or 2-D × time video clips.
  • Like crime patterns, dinosaur paw prints, neuronal spike-trains, forged signatures, songs (1-D × time), trajectories, landscapes.
  • Like, any completenormedvector space. (= it’s thick + distance exists + addition exists + everything’s included = it’s a Banach space)

(Source: terrytao.wordpress.com)




Leonardo da Vinci’s ability to embrace uncertainty, ambiguity, and paradox was a critical characteristic of his genius. —J Michael Gelb
Say you want to use a mathematical metaphor, but you don’t want to be really precise. Here are some ways to do that:
Tack a +ε onto the end of an equation.
Use bounds (“I expect to make less than a trillion dollars over my lifetime and more than $0.”)
Speak about a general class without specifying which member of the class you’re talking about. (The members all share some property like, being feminists, without necessarily having other properties like, being women or being angry.)
Use fuzzy logic (the ∈ membership relation gets a percent attached to it: “I 30%-belong-to the class of feminists | vegetarians | successful people.”).
Use a specific probability distribution like Gaussian, Cauchy, Weibull.
Use a tempered distribution a.k.a. a Schwartz function.

Tempered distributions are my current favourite way of thinking mathematically imprecisely, thanks to this book: Theory of Distributions, a non-technical introduction.
Tempered distributions have exact upper and lower bounds but an inexact mean and variance. T.D.’s also shoot down very fast (like exp −x², the gaussian) which makes them tractable.
For example I can talk about the temperature in the room (there is not just one temperature since there are several moles of air molecules in the room), the position of a quantum particle, my fuzzy inclusion in the set of vegetarians, my confidence level in a business forecast, ….. with a definite, imprecise meaning.
Classroom mathematics usually involves precise formulas but the level of generality achieved by 20th century mathematicians allows us to talk about a cobordism between two things without knowing precisely everything about them.
It’s funny, the more “advanced” and general the mathematics, the more casual it can become. Even post calc 1, I can speak about “a concave function" without saying whether it’s log, sqrt, or some non-famous power series.
Our knowledge of the world is not only piecemeal, but also vague and imprecise. To link mathematics to our conceptions of the real world, therefore, requires imprecision.
I want the option of thinking about my life, commerce, the natural world, art, social networks, and ideas using manifolds, metrics, groups, functors, topological connections, lattices, orthogonality, linear spans, categories, geometry, and any other metaphor, if I wish.

Leonardo da Vinci’s ability to embrace uncertainty, ambiguity, and paradox was a critical characteristic of his genius. —J Michael Gelb

Say you want to use a mathematical metaphor, but you don’t want to be really precise. Here are some ways to do that:

  • Tack a onto the end of an equation.
  • Use bounds (“I expect to make less than a trillion dollars over my lifetime and more than $0.”)
  • Speak about a general class without specifying which member of the class you’re talking about. (The members all share some property like, being feminists, without necessarily having other properties like, being women or being angry.)
  • Use fuzzy logic (the  membership relation gets a percent attached to it: “I 30%-belong-to the class of feminists | vegetarians | successful people.”).
  • Use a specific probability distribution like Gaussian, Cauchy, Weibull.
  • Use a tempered distribution a.k.a. a Schwartz function.

Tempered distributions are my current favourite way of thinking mathematically imprecisely, thanks to this book: Theory of Distributions, a non-technical introduction.

Tempered distributions have exact upper and lower bounds but an inexact mean and variance. T.D.’s also shoot down very fast (like exp −x², the gaussian) which makes them tractable.

For example I can talk about the temperature in the room (there is not just one temperature since there are several moles of air molecules in the room), the position of a quantum particle, my fuzzy inclusion in the set of vegetarians, my confidence level in a business forecast, ….. with a definite, imprecise meaning.

Classroom mathematics usually involves precise formulas but the level of generality achieved by 20th century mathematicians allows us to talk about a cobordism between two things without knowing precisely everything about them.

It’s funny, the more “advanced” and general the mathematics, the more casual it can become. Even post calc 1, I can speak about “a concave function" without saying whether it’s log, sqrt, or some non-famous power series.

Our knowledge of the world is not only piecemeal, but also vague and imprecise. To link mathematics to our conceptions of the real world, therefore, requires imprecision.

I want the option of thinking about my life, commerce, the natural world, art, social networks, and ideas using manifolds, metrics, groups, functors, topological connections, lattices, orthogonality, linear spans, categories, geometry, and any other metaphor, if I wish.




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.