Quantcast

Posts tagged with language

If people are rational and self-interested, why do they incriminate themselves after being Mirandised?

After minute 31 an experienced Virginia Beach interrogator-cum-3L explains how he convinces criminals to confess, against their interest, even after advising them that “Anything you say may be used in court”.

Especially after minute 34, 36, 38, 39, 40, 45, 47 he explains how he has outsmarted several criminal archetypes over 28 years.

Also check the interrogator’s view (at min 45) on cultural prejudice and presumption of guilt in Virginia Beach criminal court.




  • In a foreign country the easiest people to talk to are the young and the old. They speak slowly. And they don't mind that, due to my language disability, I don't know anything about anything.
  • Children especially seem proud to explain something to a big person. And they never tire of my small vocabulary.
  • Child: You wanna come see my puppy?
  • Me: (let's see here ... conjugate the "na" ... subjugate the "py" ... wait, no, you're supposed to abjurate the "na" then proclamate the "see" ...)
  • Me: Ummm....
  • Me: What is f u p p y?
  • Child: Ha ha! PUPPY!
  • Me: Oh. What is p u p p y?
  • Child: It's an animal that you keep in your house, it's really fun, and it runs around a lot, and it chases you, and you can play with it.
  • Me: I no understand.
  • Child: Puppy. It's a cutie little baby dog!
  • Me: O! Dog. I know dog. Thank you. Much thank you.




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 Swift Luminescent Energy Drink of the Psyche, or, When Goorialla Whirls and Whorls and Roars

(por diproton)







the putative Benue–Congo linguistic subfamily of the Niger–Congo language family

the putative Benue–Congo linguistic subfamily of the Niger–Congo language family

(Source: Wikipedia)


hi-res




More than a few readers have remarked that isomorphismes is getting ever harder to navigate, and that was after being hard to navigate since the start. I apologise! I rarely make the time to work on that.

As a stopgap measure this here can be an index to the original stuff I wrote so far in 2013. My focus this year is on mathematics-as-a-language. Trying to describe familiar things using the abstractions of “pure maths”. And trying to give nontechnical descriptions of some mathematical ideas, or ways of thinking that are common in maths but different to normal thinking.




Topology gets appropriate for qualitative rather than quantitative properties, since it deals with closeness and not distance.

It is also appropriate where distances exist, but are ill-motivated.

These approaches have already been used successfully, for analyzing:

  • • physiological properties in Diabetes patients
  • • neural firing patterns in the visual cortex of Macaques
  • • dense regions in ℝ⁹ of 3×3 pixel patches from natural [black-and-white] images
  • • screening for CO₂ adsorbative materials
Michi Johanssons (@michiexile)

(Source: blog.mikael.johanssons.org)




This post should give you the feeling of bijecting between domains without knowing a lot of mathematics. Which is part of getting the intuitive feeling of mathematics with less work.

Besides automorphisms, there’s another interesting kind of bijection. I’ll try to give you the feeling of bijecting between different domains (a kind of analogy) without requiring much prior knowledge.

Like I said yesterday, a bijection is an invertible total mapping. It ≥ covers ↓ the target and ≤ injects ↑ one-to-one into the target. This is thinking of spaces as wholes—deductive thinking—rather than example-by-example thinking. (There’s a joke about an engineer and a mathematician who are friends and go to a talk about 47-dimensional geometry. The engineer after the talk tells the mathematician friend that it was hard to visualise 47 dimensions; how did you do it? The mathematician replies “Oh, it’s easy. I simply considered the problem in arbitrary N dimensions and then set N=47!” I used to be frustrated by this way of thinking but after X years, it finally makes sense and is better for some things.)

So, graphically, a bijection is surjective/covering/ ≥ / 

image

and injective/one-to-one (not one-to-many)/ ≤ /  ↑
image
.

This amounts to a mathematical way of saying two things are “the same”, when of course there are a lot of ways in which that could be meant. The equation x=x is the least interesting one so “sameness” has to be more broad than “literally the same”. The same like how? Bijection as a concept opens the door to ≥1 kinds of comparison.

 

That was a definition. Now on to the example which should give you the feeling of bijecting across domains and the feeling of payoff after you come up with an unintuitive bijection.

Let’s talk about an “ideal city” where the streets make a perfectly rectangular lattice. I’m standing at 53rd St & 140th Ave and I want to walk/bike/cab to 60th St & 147th Ave.

How many short ways can I take to get there?

The first abstraction I would do from real life to a drawing is to centre the data. A common theme in statistics and mathematically it’s like removing the origin. I can actually ignore everything except the 7×7 block between me and my destination to the northeast.

image

(By the way, by “short” paths I mean not circling around any more than necessary. Obviously I could take infinitely more and more circuitous routes to the point of circling the Earth 10 times before I get there. But I’m trying not to go out of my way here.)

image

Now the problem looks smaller. Just go from bottom left corner to top right corner.

I drew one shortest path in red and two others in black. To me it would be boring to go north, north, north, north, north, north, north, east, east, east, east, east, east, east. But if I want to count all the ways of making snakey red-like paths then I should bracket the possibilities by those two black ones.

When I try to draw or mentally imagine all the snakey paths, I lose track—looking for patterns (like permute, then anti-inner-permute, but also pro-inner-anti-inner-inner-permute…these are words I make up to myself) that I probably could see if I understood the fundamental theorem of combinatorics, but I’ve never been able to fully see the additive pattern.

But, I know a shortcut. This is where the bijection comes in.

image

Every one of these paths is isomorphic to a rearrangement of the letters NNNNNNNEEEEEEE.

Eureka!

 

Every time I “flip” one of the corners in the picture—which is how I was creating new snakes in between the black brackets—that’s just like interchanging an N and an E.

Of course! It’s so obvious in hindsight.

And now here’s the payoff. Rearrangements of strings of letters like AAABBBCCCCD are already a solved problem.

I explained how to count combinatorial rearrangements of letters here. It’s 1026 words long.

The way to get the following formula is to [1] derive a trick for over-counting, [2] over-count and then [3] quotient using the same trick.

image

Now,

  • since the rearrangements of AAAAAAABBBBBBB are isomorphic to the rearrangements of NNNNNNNEEEEEEE,
  • and since the rearrangements of NNNNNNNEEEEEEE are isomorphic to the short paths I could take through the city to my destination,

the correct answer to my original question—how many short ways to go 7 blocks east and 7 blocks north—is 14!/7!/7!.

I asked the Berkeley Calculator the answer to that one and it told me 3432. Kind of glad I didn’t count those out by hand.

 

So, the payoff came from (1) knowing some other solved problem and (2) bijecting my problem onto the one with the known solution method.

But does it work in New York? Even though NYC is kind of like a square lattice, there may be a huge building making some of the blocks not accessible.

image

Or maybe ∃ a “Central Park” where you can cut a diagonal path.
image

And things like “Broadway” that cut diagonally across the city.

image

And some dead ends in certain ranges of the ciudad. And places called The Flat Iron Building where roads meet in a sharp V.

image

So my clever discovery doesn’t quite work in a non-square world.

 

However now I maybe also gave you a microcosm of mathematical modelling. The barriers and the shortcuts could be added to a computer program that counts the paths. We could keep adjusting things and adding more bits of reality and make the computer calculate the difference. But the “basic insight”, I feel, is lacking there. After all I could have written a computer program to permute the letters NNNNNNNEEEEEEE or even just literally model the paths in the first place. (At least with such a small problem.) But then there would be no Eureka moment. I think it’s in this sort of way that mathematicians mean their world is more beautiful than the real one.

As mathematical modellers we inherit deep basic insights—like the Poisson process and the Gaussian as two limits of a binary branching process—and try to construct a convoluted sculpture using those profound insights as the basis. For example maybe I could stitch together a bunch of square lattice pieces together. Maybe for instance two square lattices representing different boroughs and connected only by a single congested bridge. Since I solved the square lattice analytically, the computational extensions will be less mysterious to me if I use the understood pieces. Unless I can be smart enough to figure out how to count triangles & multi-block industrial buildings & shortcuts & construction roadblocks and find an equally excellent insight into how the various discrepancies change the number at the end of my computation (rather than just reading it off and having an answer but no wisdom), I’m left using the excellent insight as a starting point and doing some dirty computations from there—no wisdom at all, no map, just scrapping in the wilderness—a lot of firepower and no idea how to use it. I might as well be spraying a tree with a shotgun instead of cutting the V with an axe and letting its weight do the work.




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
image
image
image
image
image
image
image
a covering, drawn by Robert Ghrist www.math.upenn.edu/~rghrist
image
image
imageimageimageimageimageimageimageimageimageimageimageimage

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 ≥ 

image
image
and one-to-one-ness 
≤  ↑
image
successor function and square function
image
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.

imageimage

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;
    image
  • the space of possible protein configurations;
    image

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)
    image
    image
  • how big of a suitcase you need to fit the thing in (L-∞ norm)
    image
    image

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).

image
image

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

image
image
image
image
image
image
image

image
image

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.

image
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
    image
    imageimage
  • 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)
    image
  • 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?
    image
    What if you shave it? What if you replace the heart with a fish heart?
    image
  • 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.

image

  • 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.
    image
    image
  • If you take the real plane ℝ² and quotient it by ℤ²
    image
    (ℤ being the integers) you get the torus 𝕋²
    image
  • Likewise if you take ℝ and quotient it by the integers ℤ you get a circle.
    image

  • 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.

image

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
image
image
image
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!