Posts tagged with other

Winthrop also subscribed to the belief that the native peoples who lived in the hinterlands around the colony had been struck down by G-d, who sent disease among them because of their non-Christian beliefs:

“But for the natives in these parts, God hath so pursued them, as for 300 miles space the greatest part of them are swept away by smallpox which still continues among them. So as G-d hath thereby cleared our title to this place…”

New World Encyclopedia, via MRU

(search pictures of smallpox only if you have a strong stomach)




Lucas’ “rational expectations” revolution in macroeconomics has been tied to the ending of stagflation in the world’s largest economy, and to the reintroduction of “psychology” into finance and economics. However, I never felt like the models of “expectation” I’ve seen in economic theories seem like my own personal experience of living in ignorance. I’d like to share the sketch of an idea that feels more lifelike to me.

http://www.olivierlanglois.net/images/voro2.jpg

First, let me disambiguate: the unfortunate term-overlap with “statistical expectation” (= mean = average = total over count = ∑ᵢᴺ•/N = a map from N dimensions to 1 dimension) indicates nothing psychological whatever. It doesn’t even correspond to “What you should expect”.

If I find out someone is a white non-Hispanic Estadounidense (somehow not getting any hints of which state, which race, which accent, which social class, which career track…so it’s an artificial scenario), I shouldn’t “expect” the family to be worth $630,000. I “expect” (if indeed my expectation is not a distribution but rather just one number) them to be worth $155,000. (scroll down to green)

Nor, if I go to a casino with 99% chance of losing €10,000 and 1% chance of winning €1,000,000 (remember the break-even point is €990,000). “On average” this is a great bet. But that ignores convergence to the average, which would be slow. I’d need to play this game a lot to get the statistics working in my favour, and I mightn’t stay solvent (I’d need to get tens of millions of AUM—with lockdown conditions—to even consider this game). No, the “statistical expectation” refers to a long-run or wide-space convergence number. Not “what’s typical”.

Not only is the statistical expectation quite reductive, it doesn’t resemble what I’ve introspected about uncertainty, information, disinformation, beliefs, and expectations in my life.

File:Coloured Voronoi 3D slice.svg

A better idea, I think, comes from the definition of Riemann integration over 2+ dimensions. Imagine covering a surface with a coarse mesh. The mesh partitions the surface. A scalar is assigned to each of the interior regions inscribed by the mesh. The mesh is then refined (no lines taken away, only some more added—so some regions get smaller/more precise and no regions get larger/less precise), new scalars are computed with more precise information about the scalar field on the surface.
a scalar field

NB: The usual Expectation operator 𝔼 is little more than an integral over “possibilities” (whatever that means!).

(In the definitions of Riemann integral I’ve seen the mesh is square, but Voronoi pictures look awesomer & more suggestive of topological generality. Plus I’m not going to be talking about infinitary convergence—no one ever becomes fully knowledgeable of everything—so why do I need the convenience of squares?)

I want to make two changes to the Riemannian-integral mesh.

image
image

 

First I’d like to replace the scalars with some more general kind of fibre. Let’s say a bundle of words and associations.

(You can tell a lot about someone’s perspective fro the words they use. I’ll have to link up “Obverse Words”, which has been in my drafts folder for over a year, once I finish it—but you can imagine examples of people using words with opposite connotation to denote the same thing, indicating their attitude toward the thing.)

http://i780.photobucket.com/albums/yy90/AlexMLeo/felixsbrain.jpg

Second, I’d like to use the topology or covering maps to encode the ignorance somehow. In my example below: at a certain point I knew “Rails goes with Ruby” and “Django goes with Python” and “Git goes with Github” but didn’t really understand the lay of the land. I didn’t know about git’s competitors, that you can host your own github, that Github has competitors, the more complex relationship between ruby and python (it’s not just two disjoint sets), and so on.

When I didn’t know about Economics or Business or Accounting or Finance, I classed them all together. But now they’re so clearly very very different. I don’t even see Historical Economists or Bayesian Econometricians or Instrumental Econometricians or Dynamical Macroeconomists or Monetary Economists or Development Economists as being very alike. (Which must imply that my perspective has narrowed relative to everyone else! Like tattoo artists and yogi masters and poppy farmers must all be quite different to the entire class of Economists—and look even from my words how much coarse generalisation I use to describe the non-econ’s versus refinement among the econ’s.
image
These meshes can have a negative curvature (with, perhaps a memory) if you like. You know when you think that property actuaries are nothing at all like health actuaries that your frame-of-reference has become very refined among actuary-distinguishment. Which might mean a coarse partitioning of all the other people! Like Bobby Fischer’s use of the term “weakies” for any non-chess player—they must all be the same! Or at least they’re the same to me.)

image

Besides the natural embedding of negatively-curved judgment grids, here are some more pluses to the “refinement regions” view of ignorance:

  1. You could derive a natural “conservation law” using some combination of e.g. ability, difficulty, how good your teachers are, and time input to learning, how many “refinements” you get to make. No one can know everything.

    (Yet somehow we all are supposed to function in a global economy together—how do we figure out how to fit ourselves together efficiently?

    And what if people use your lack of perspective to suggest you should pay them to teach you something which “evaluates to valuable” from your coarse refinement, but upon closer inspection, doesn’t integrate to valuable?)
  2. Maybe this can relate to the story of Tony—how we’re always in a state of ignorance even as we choose what to become less ignorant about. It would be nice to be able to model the fact that one can’t escape one’s biases or context or history.
  3. And we could get a fairly nice representation of “incompatible perspectives”. If the topology of your covering maps is “very hard” to match up to mine because you speak dialectics and power structures but I speak equilibria and optima, that sounds like an accurate depiction. Or when you talk to someone who’s just so noobish in something you’re so expert in, it can feel like a very blanket statement over so many refinements that you don’t want to generalise over (and from “looking up to” an expert it can also feel like they “see” much more detail of the interesting landscape.)
  4. Ignorance of one’s own ignorance is already baked into the pie! As is the beginner’s luck. If I “integrate over the regions” to get my expected value of a certain coarse region, my uninformed answer may have a lot of correctness to it. At the same time, the topological restrictions mean that my information and my perspective on it aren’t “over there” in some L2-distance sense, rather they’re far away in a more appropriately incompatible-with-others sense.

In conclusion, I’m sure everyone on Earth can agree that this is a Really Nifty and Cool Idea.

File:ApproximateVoronoiDiagram.png

 

I’ll try to give a colourful example using computers and internet stuff since that’s an area I’ve learned a lot more about over the past couple years.

A tiny portion of Doug Hofstadter’s “semantic network”.  via jewcrew728, structure of entropy

First, what does ignorance sound like?

  • (someone who has never seen or interacted with a computer—let’s say from a non-technological society or a non-computery elderly rich person. I’ve never personally seen this)
  • "Sure, programming, I know a little about that. A little HMTL, sure!”
  • "Well, of course any programming you’re going to be doing, whether it’s for mobile or desktop, is going to use HTML. The question is how.

OK, but I wasn’t that bad. In workplaces I’ve been the person to ask about computers. I even briefly worked in I.T. But the distance from “normal people” (no computer knowledge) to me seems very small now compared to the distance between me and people who really know what’s up.

A few years ago, when I started seriously thinking about trying to make some kind of internet company (sorry, I refuse to use the word “startup” because it’s perverted), I considered myself a “power user” of computers. I used keyboard shortcuts, I downloaded and played with lots of programs, I had taken a C++ course in the 90’s, I knew about C:\progra~1 and how to get to the hidden files in the App packages on a Mac.

My knowledge of internet business was a scatty array of:

  • Mark Zuckerberg
  • "venture capital"
  • programer kid internet millionaires
  • Kayak.com — very nice interface!
  • perl.
    Regular Expressions
    11th Grade
  • mIRC
  • TechCrunch
  • There seem to be way more programming going on to impress other programmers than to make the stuff I wanted!
  • I had used Windows, Mac, and Linux (!! Linux! Dang I must be good)
  • I knew that “Java and Javascript are alike the way car and carpet are alike”—but didn’t know a bit of either language.
  • I used Alpine to check my gmail. That’s a lot of confusing settings to configure! And plus I’m checking email in text mode, which is not only faster but also way more cooly nerdy sexy screeny.
  • Object-Oriented, that’s some kind of important thing. Some languages are Object-Oriented and some aren’t.
  • "Python is for science; Ruby is for web"
  • sudo apt-get install
    Sandwich
  • I had run at least a few programs from the command line.
  • I had done a PHP tutorial at W3CSchools … that counts as “knowing a little PHP”, right?

So I knew I didn’t know everything, but it was very hard to quantify how much I did know, how far I had to go.

image

A mediocre picture of some things I knew about at various levels. It’s supposed to get across a more refined knowledge of, for example, econometrics, than of programming. Programming is lumped in with Linux and rich programmer kids and “that kind of stuff” (a coarse mesh). But statistical things have a much richer set of vocabulary and, if I could draw the topology better, refined “personal categories” those words belong to.

Which is why it’s easier to “quantify” my lack of knowledge by simply listing words from the neighbourhood of my state of knowledge.

Unfortunately, knowing how long a project should take and its chances of success or potential pitfalls, is crucial to making an organised plan to complete it. “If you have no port of destination, there is no favourable wind”. (Then again, no adverse wind either. But in an entropic environment—with more ways to screw up than to succeed—turning the Rubik’s cube randomly won’t help you at all. Your “ship” might run out of supplies, or the backers murder you, etc.)

File:2Ddim-L2norm-10site.png

Here are some of the words I learned early on (and many more refinements since then):

  • Rails
  • Django
  • IronPython
  • Jython
  • JSLint
  • MVC
  • Agile
  • STL
  • pointers
  • data structures
  • frameworks
  • SDK’s
  • Apache
  • /etc/.httpd
  • Hadoop
  • regex
  • nginx
  • memcached
  • JVM
  • RVM
  • vi, emacs
  • sed, awk
  • gdb
  • screen
  • tcl/tk, cocoa, gtk, ncurses
  • GPG keys
  • ppa’s
  • lspci
  • decorators
  • virtual functions
  • ~/.bashrc, ~/.bash_profile, ~/.profile
  • echo $SHELL, echo $PATH
  • "scripting languages"
  • "automagically"
  • sprintf
  • xargs
  • strptime, strftime
  • dynamic allocation
  • parser, linker, lexer
  • /env, /usr, /dev,/sbin
  • GRUB, LILO
  • virtual consoles
  • Xorg
  • cron
  • ssh, X forwarding
  • UDP
  • CNAME, A record
  • LLVM
  • curl.haxx.se
  • the difference between jQuery and JSON (they’re not even the same kind of thing, despite the “J” actually referring to Javascript in both cases)
  • OAuth2
  • XSALT, XPath, XML

http://www.financialiceberg.com/uploads/iceberg340.jpg
http://www.emeraldinsight.com/content_images/fig/1100190504002.png


http://www.preventa.ca/images/im_risk_anatomy.jpg

This is only—as they say—“the tip of the iceberg”. I didn’t know a ton of server admin stuff. I didn’t understand that libraries and frameworks are super crucial to real-world programming. (Imagine if you “knew English” but had a vocabulary of 1,000 words. Except libraries and frameworks are even better than a large vocabulary because they actually do work for you. You don’t need to “learn all the vocabulary” to use it—just enough words to call the library’s much larger program that, say, writes to the screen, or scrapes from the web, or does machine learning, for you.)

The path should go something like: at first knowing programming languages ⊃ ruby. Then knowing programming languages ⊃ ruby ⊃ rubinius, groovy, JRuby. At some point uncovering topological connections (neighbourhood relationships) to other things (a comparison to node.js; a comparison to perl; a lack of comparability to machine learning; etc.)

I could make some analogies to maths as well. I think there are some identifiable points across some broad range of individuals’ progress in mathematics, such as:

  • when you learn about distributions and realise this is so much better than single numbers!

    a rug plot or carpet plot is like a barcode on the bottom of your plot to show the marginal (one-dimension only) distribution of data

    who is faster, men or women?
  • when you learn about Gaussians and see them everywhere
    Central Limit Theorem  A nice illustration of the Central Limit Theorem by convolution.in R:  Heaviside <- function(x) {      ifelse(x>0,1,0) }HH <- convolve( Heaviside(x), rev(Heaviside(x)),        type = "open"   )HHHH <- convolve(HH, rev(HH),   type = "open"   )HHHHHHHH <- convolve(HHHH, rev(HHHH),   type = "open"   )etc.  What I really like about this dimostrazione is that it’s not a proof, rather an experiment carried out on a computer.  This empiricism is especially cool since the Bell Curve, 80/20 Rule, etc, have become such a religion.NERD NOTE:  Which weapon is better, a 1d10 longsword, or a 2d4 oaken staff? Sometimes the damage is written as 1-10 longsword and 2-8 quarterstaff. However, these ranges disregard the greater likelihood of the quarterstaff scoring 4,5,6 damage than 1,2,7,8. The longsword’s distribution 1d10 ~Uniform[1,10], while 2d4 looks like a Λ.  (To see this another way, think of the combinatorics.)
  • when you learn that Gaussians are not actually everywhere
    kernel density plot of Oxford boys' heights.

    histogram of Oxford boys' heights, drawn with ggplot.A (bimodal) probability distribution with distinct mean, median, and mode.
  • in talking about probability and randomness, you get stuck on discussions of “what is true randomness?” “Does randomness come from quantum mechanics?” and such whilst ignorant of stochastic processes and probability distributions in general.
  • (not saying the more refined understanding is the better place to be!)
  • A brilliant fellow (who now works for Google) was describing his past ignorance to us one time. He remembered the moment he realised “Space could be discrete! Wait, what if spacetime is discrete?!?!?! I am a genius and the first person who has ever thought of this!!!!” Humility often comes with the refinement.
  • when you start understanding symbols like ∫ , ‖•‖, {x | p} — there might be a point at which chalkboards full of multiple integrals look like the pinnacle of mathematical smartness—
    http://www.niemanlab.org/images/math-formula-chalkboard.jpg
  • but then, notice how real mathematicians’ chalkboards in their offices never contain a restatement of Physics 103!
    Kirby topology 2012
    http://whatsonmyblackboard.files.wordpress.com/2011/06/21june2011.jpg
    A parsimonious statement like “a local ring is regular iff its  global dimension is finite” is so, so much higher on the maths ladder than a tortuous sequence of u-substitutions.
  • and so on … I’m sure I’ve tipped my hand well enough all over isomorphismes.tumblr.com that those who have a more refined knowledge can place me on the path. (eg it’s clear that I don’t understand sheaves or topoi but I expect they hold some awesome perspectives.) And it’s no judgment because everyone has to go through some “lower” levels to get to “higher” levels. It’s not a race and no one’s born with the infinite knowledge.
 

I think you’ll agree with me here: the more one learns, the more one finds out how little one knows. One can’t leave one’s context or have knowledge one doesn’t have. And all choices are embedded in this framework.




jkottke:

This video is 13 minutes of traffic accidents in Russia and totally amazing.

  1. Show this to your teenagers before they take the wheel. If it doesn’t scare the p*ss out of them—or even worse, if it excites them—no more Grand Theft Auto and hide the car keys.
  2. Next time you complain about public services, boring orderliness and “safety first”, the desireability of risk, Panglossian everything-optimal economics, or forget how relatively safe you are on your German freeways, …. watch this.

    As someone else remarked (can’t remember the source), the difference between Somalia and the USA is the stuff everybody in the US completely forgets is even possible.
  3. Notice how many of the accidents are caused by people trying to zoom ahead of everyone else—off the side of the road, cutting down a tree without noticing it will land on somebody else, trying to pass on the left or on the right or across the lane. Is your time really that important relative to everyone else’s, people?
  4. Assumptions. You think you can make assumptions, like that someone won’t fell a tree on your head, or a military jet won’t fly over your head, that someone won’t spill military equipment near you, or that people from the other lane (or off the road) won’t drive completely orthogonal and attack your car. Sometimes those assumptions are wrong.
  5. How many of these people do you think actually accepted the blame on themselves for their reckless actions?

via @Alea_, @felixsalmon




Paul Bloom disproves the idea that sexual pleasure se logra by merely the proper stimulation of various genitalia with the following Gedankenexperiment:
Imagine you find out that the person you had sex with last night is not who you thought they were.
Maybe you learn that the charming gentleman is the author of white-supremacist hate literature.Maybe you find out that the beautiful woman was your long-lost sister. The feeling of wanting to crawl out of your own skin and leave the ugly husk of your body behind wouldn’t be out of place.
That such tropes appear in literature we’ve found from millennia ago suggests people have long felt this way: sexual pleasure must be tied in with not only the body of your partner, but with their spirit and inherent nature as well.
  
Pleasure is complicated. Economists know this but usually choose to forget the fact. The study of where individual demand curves come from would be a new discipline, although ink has been spilled on the topic.
However, the questions of pleasure and satisfaction are relevant to the engineering of society. If the objective function is set to: maximise output, but people derive pleasure from achieving increasingly difficult goals and receiving even artificial rewards, then the world of work is not optimised for happiness but the world of school is.
Getting more practical than grand critiques of “society”, anyone who manages more employees than herself would benefit from knowing which free-or-cheap buttons she can push to motivate and reward the people “under” her. Even more pedestrian: I know that sitting down feels better after a physical labour or constitutional, but I haven’t a quantitative knowledge of how to engineer my habits and routines to take fullest advantage of that fact.
Sound the trumpet again for a department of happiness studies.

Paul Bloom disproves the idea that sexual pleasure se logra by merely the proper stimulation of various genitalia with the following Gedankenexperiment:

  • Imagine you find out that the person you had sex with last night is not who you thought they were.

Maybe you learn that the charming gentleman is the author of white-supremacist hate literature.
Dave Chappelle playing a (blind) black white supremacist
Maybe you find out that the beautiful woman was your long-lost sister. The feeling of wanting to crawl out of your own skin and leave the ugly husk of your body behind wouldn’t be out of place.

That such tropes appear in literature we’ve found from millennia ago suggests people have long felt this way: sexual pleasure must be tied in with not only the body of your partner, but with their spirit and inherent nature as well.

  

Pleasure is complicated. Economists know this but usually choose to forget the fact. The study of where individual demand curves come from would be a new discipline, although ink has been spilled on the topic.

However, the questions of pleasure and satisfaction are relevant to the engineering of society. If the objective function is set to: maximise output, but people derive pleasure from achieving increasingly difficult goals and receiving even artificial rewards, then the world of work is not optimised for happiness but the world of school is.

Getting more practical than grand critiques of “society”, anyone who manages more employees than herself would benefit from knowing which free-or-cheap buttons she can push to motivate and reward the people “under” her. Even more pedestrian: I know that sitting down feels better after a physical labour or constitutional, but I haven’t a quantitative knowledge of how to engineer my habits and routines to take fullest advantage of that fact.

Sound the trumpet again for a department of happiness studies.


hi-res




Swimming to Cambodia




Interesting how Austrian economists see themselves:

  • Ludwig von Mises was a genius (obviously)
  • a Man of the Mind — and that’s a good thing
  • "dignified ruthlessness"
  • seriousness … also a good thing … Serious about Reality
  • quoting Ayn Rand is OK (erm, quoting her straightforwardly, not mockingly)
  • "We’re not dogmatic … we’re consistent
  • People who call us dogmatic believe in relative truth, not absolute truth

^ The last one is the only really confusing one to me.




Even the beneficiaries of hypertrophy have found it difficult to cope with extreme cultural change … they are sociobiologically equipped only for an earlier, simpler existence. Where the hunter-gatherer fills … one or two … roles out of … several available, his literate counterpart … must choose ten or more out of thousands, and replace one … with another….

Furthermore, each occupation—the physician, the judge, the teacher, the waitress—is played just so, regardless of the true workings of the mind behind the persona. [D]eviations … are interpreted … as a sign of mental incapacity…. Daily life is a compromised blend of posturing … and of varying degrees of self-revelation. Under these stressful conditions even the “true” self cannot be precisely defined….:

"…Self, then, is not … half-concealed behind events, but a changeable formula for managing … during them. Just as the current situation prescribes the official guise…so it provides where & how we will show through, the culture … prescribing what … we must believe ourselves to be….”

Little wonder that the identity crisis is a major source of modern neuroticism, and that the urban middle class aches for a return to a simpler existence.

E. O. Wilson (also quoting Erving Goffman), On Human Nature

Particularly the phrase “changeable formula” stands out to me. I think this means that our self-concept, seen as a function ƒ, takes the_environment as an input. (And that input has a nonzero derivative, i.e. it’s not a trivial input.)

Not only that; “the environment” isn’t limited to what_happened_in_our_early_years. We might feed that early_environment variable in as well, but in addition immediate conditions can change our self-concept. In equation form:

  • Self = ƒ (   ∫ early life,    present situation,   ...other stuff...  )




  • If Charles Dickens were alive today, he would rather receive £20 from a waiter friend than £100 from a banker friend, because it means more coming from the poor man. £20 is a larger fraction of the poor man’s income, so it means more than a paltry £100 to the rich man.
  • Someone in my Christmas circle confided that she would rather receive either
    • money, or
    • something personal (like a letter, a drawing, or a small gift that shows that the person knows her well and put thought or time into the gift)
    "If someone gets me shower gel or bath salts, it shows that they don’t know anything about me at all. I would rather just get money, because then at least I can get myself something I want."

Christmas obligations account for $100 billion of spending in the USA, or roughly ½% of its yearly output. If you are buying the wrong gifts, though, perhaps 20% of that is wasted—i.e., $20 billion of waste in the world’s largest economy.

This has led to the gift card, probably the best business idea ever. (Businesses get cash regardless of whether they deliver a service or product, in advance of delivering the service or product, in return for expensively allaying the culturally-induced guilt of rich people.)

But in both of the gift metrics above—the one from my Christmas circle and the one from 120 years ago—time and care are what’s really wanted in a present. Or an Xbox 360.

PS: Joel Waldfogel suggests a charity gift card: the card recipient gets to act rich, and the cash goes to a high-marginal-value end purpose.




Branes, D-branes, M-theory, K-theory … news articles about theoretical physics often mention “manifolds”.  Manifolds are also good tools for theoretical psychology and economics. Thinking about manifolds is guaranteed to make you sexy and interesting.

Fortunately, these fancy surfaces are already familiar to anyone who has played the original Star Fox—Super NES version.

In Star Fox, all of the interactive shapes are built up from polygons.  Manifolds are built up the same way!  You don’t have to use polygons per se, just stick flats together and you build up any surface you want, in the mathematical limit.

The point of doing it this way, is that you can use all the power of linear algebra and calculus on each of those flats, or “charts”.  Then as long as you’re clear on how to transition from chart to chart (from polygon to polygon), you know the whole surface—to precise mathematical detail.

Regarding curvature: the charts don’t need the Euclidean metric.  As long as distance is measured in a consistent way, the manifold is all good.  So you could use hyperbolic, elliptical, or quasimetric distance. Just a few options.

 

Manifolds are relevant because according to general relativity, spacetime itself is curved.  For example, a black hole or star or planet bends the “rigid rods" that Newton & Descartes supposed make up the fabric of space.

bent spacetime

black hole photo

In fact, the same “curved-space” idea describes racism. Psychological experiments demonstrate that people are able to distinguish fine detail among their own ethnic group, whereas those outside the group are quickly & coarsely categorized as “other”.

This means a hyperbolic or other “negatively curved" metric, where the distance from 0 to 1 is less than the distance from 100 to 101.  Imagine longitude & latitude lines tightly packed together around "0", one’s own perspective — and spread out where the “others” stand.  (I forget if this paradigm changes when kids are raised in multiracial environments.)

Experiments verify that people see “other races” like this. I think it applies also to any “othering” or “alienation” — in the postmodern / continental sense of those words.

 

The manifold concept extends rectilinear reasoning familiar from grade-school math into the more exciting, less restrictive world of the squibbulous, the bubbulous, and the flipflopflegabbulous.

ga zair bison and monkey

calabi-yau manifold

cat detective




Branes, D-branes, M-theory, K-theory … news articles about theoretical physics often mention “manifolds”.  Manifolds are also good tools for theoretical psychology and economics. Thinking about manifolds is guaranteed to make you sexy and interesting.

Fortunately, these fancy surfaces are already familiar to anyone who has played the original Star Fox—Super NES version.

imageimageimageimageimage

In Star Fox, all of the interactive shapes are built up from polygons.  Manifolds are built up the same way!  You don’t have to use polygons per se, just stick flats together and you build up any surface you want, in the mathematical limit.

The point of doing it this way, is that you can use all the power of linear algebra and calculus on each of those flats, or “charts”.  Then as long as you’re clear on how to transition from chart to chart (from polygon to polygon), you know the whole surface—to precise mathematical detail.

image

image
image

Regarding curvature: the charts don’t need the Euclidean metric.  As long as distance is measured in a consistent way, the manifold is all good.  So you could use hyperbolic, elliptical, or quasimetric distance. Just a few options.

Manifolds are relevant because according to general relativity, spacetime itself is curved.  For example, a black hole or star or planet bends the “rigid rods" that Newton & Descartes supposed make up the fabric of space.

bent spacetime

black hole photo

In fact, the same “curved-space” idea describes racism. Psychological experiments demonstrate that people are able to distinguish fine detail among their own ethnic group, whereas those outside the group are quickly & coarsely categorized as “other”.

This means a hyperbolic or other “negatively curved" metric, where the distance from 0 to 1 is less than the distance from 100 to 101.  Imagine longitude & latitude lines tightly packed together around "0", one’s own perspective — and spread out where the "others" stand.  (I forget if this paradigm changes when kids are raised in multiracial environments.)

image

If you stitch together such non-Euclidean flats, you’ve again constructed a manifold.

Think about this: the pixel concept re-presents brush-stroke or natural images by a wall of sequential colored squares.  You could extend it to 3-D, for example representing humans by little blocks—white for the bone, burgundy for the blood, pink for the fingernails, etc.

In a similar fashion, the manifold concept extends rectilinear reasoning familiar from grade-school math into the more exciting, less restrictive world of the squibbulous, the bubbulous, and the flipflopflegabbulous.

ga zair bison and monkey

calabi-yau manifold

cat detective