Quantcast

Posts tagged with language

English predominantly talks about time as if it were horizontal, while Mandarin … commonly describes time as vertical.

Lera BoroditskyDoes Language Shape Thought?: Mandarin and English Speakers’ Conceptions of Time

 

see also

  • Jenn-Yeu ChenDo Chinese and English speakers think about time differently? Failure of replicating Boroditsky (2001)
    By estimating the frequency of usage, we found that Chinese speakers actually use the horizontal spatial metaphors more often than the vertical metaphors.




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.