#equivalence classes

[studyingwithjessie]

Equivalence Relations, Equivalence Classes, and some Mathematical Induction

[infiniteforloop]

I just spent 20 minutes coloring an equivalence class graph with dying highlighters and then I spilled chai on it. turning it in anyways.

[isomorphismes]

Any smooth compact manifold is diffeomorphic to the configuration space of some planar linkage.

pictures by Robert Ghrist and Kevin Walker

By the way: this is how windscreen wipers work. (last picture)

[isomorphismes]

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. Oral 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

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]

[isomorphismes]

[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 ℒ (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 ℒ and ℝ⁺, but one can abandon … units and just work with ℒ 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

[isomorphismes]

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.