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Posts tagged with norms

The tropical semiring is arithmetic piped through a log with base →.

Also if you or someone you know  is first encountering a squeeze theorem or other a≤x≤A type reasoning, remark 2.1 might be a relatively painless calisthenic to warm you/them up to a≤x≤A type arguments.

by Gregory Mikhalkin










The Nielsen PRIZM groups people into 66 “demographic and geographic market segments” for the purpose of advertising to them.

Each of the segments has a nice description to go along with it. It’s the kind of story you want to hear as a marketer: it uses relatively in-depth knowledge of Americans, plus stereotypes or shallow summaries, to draw a character with enough roundness that you could pitch to him/her. That is, you could write copy or film a creative spot that you believe could speak to members of this cohesive segment.

As I read more deeply into the Nielsen-Claritas PRIZM, however, the 66 segments started to sound like perhaps they were generated by a simple formula. From their slideshow I learned that they divide the US population by:

  • affluence
  • population density
  • kids/no kids + age

Rather than use continuous on the implied cube (3 dimensions above), they lump various ranges together. They also lump the interaction terms unevenly—for example, (suburban & income) is lumped more finely and (urban & income) is lumped more coarsely. Specifically,

  • 4 totally -ordered levels of urbanity (measured by population density per zip code) urban  suburban  second city  town & rural
  • 14 levels of Affluence Groups (so they consider finer gradations of wealth & income within suburban and low-density zip codes and coarser income gradations in cities and second-cities)
     
  • Three life-stage categories, accommodating both those who do and don’t raise children at some point. {youngish && no kids, kids, oldish && no kids at home}.

    Younger folks (this is under-35’s or under-45 DINKs) are less graduated by affluence than families or older folks (over-55’s or over-45 DINKs).

    By the way, over-65’s are outside PRIZM’s marketing groups. I guess it’s assumed that they won’t buy big-ticket items or change their ways much unless the Monday lima-bean special becomes 25cents cheaper at Lida’s Diner than Bill’s Diner. Then you’ll see the entire community switch to Lida’s.

Like the MBTI, it assumes that: People fit in rectangles.

Unlike the MBTI, rather than using four sliding scales [0,1]⁴, the PRIZM uses discrete, totally ordered sets—something you could build with the letters and combn functions in R.

I started to wonder: is it really true that members of segment 26 are “urbane” and “love the nightlife” — even the empty-nesters and older homeowners of the segment? Is there really a “laid-back atmosphere” to segment 25? Or are these merely colourful papier-mâché rudely draped over a box?

Mostly, of course, I’m concerned with segment 31, the well-known Urban Achievers:

And proud we are of all of them.

HOW I SEE IT

When I look at a painting, I’m tempted to glance quickly and pass on. In order to appreciate a piece, I imagine the strokes and colour choices that make up the painting. I imagine myself painting the same thing. What would it have felt like to be inside Cy Twombly's hand while he painted Apollo 17? That gives me a better feeling of the art.

When I look at the Nielsen Prizm the same way — try to get inside the heads of its creators — I sense that they adopted the [0,1]⁸ rectangular structure simply because they’re not aware of alternatives. MBA’s do plenty of mathematics, but I’ve never seen any business mathematics cross over into CW-complexes, 3-tori, arborescences, or Lobachefskyan geometries. It could be that the people who designed the Prizm simply didn’t have anyone on their team who had heard of this stuff. All the quants were working on Wall Street rather than Madison Avenue. (Wacker Drive rather than Michigan Ave.)

The ribbon-farm guy (Venkatesh Rao) is a rocket scientist who crossed over into marketing, but so far I haven’t read enough of his stuff to say if he dove into algebraic geometry—it seems he did more functional analysis, optimisation / control theory, and differential geometry. Which is what I would expect rocket science consists of.

I will admit that the PRIZM’s use of two “matrix” presentations with colour-coding, pictures, defined ranges, and toss-away combinations is quite clear. Probably works better than when I tell clients “Just picture a 5-dimensional manifold, I won’t say the norm because I think it’s spaced differently in the center than the edges—and let’s not get into the interaction terms yet”. But—the bones of their model are really just [0,1]³. They’ve dressed it up and they’ve done more than that (segmenting and dropping). But a cube is the underlying architecture.

Is the Prizm simple or oversimplified? I feel it’s the latter. Not that I object to mathematical models of behaviour, emotions, or any human thing—but the hypercube metaphor just doesn’t fit my presumption of the shape of the space.

  • Does consumer space have 8 corners to it?
  • What’s the best interpretation of “distance” in the consumer space?
  • Do all of the lines really cross at right angles, in a hyper-grid? Was that supposed to be implied?

WHEREUNTO

I don’t want to carp about somebody else’s work without at least offering constructive criticism. What are some potentially better ways to think about the space of all consumers—potential buyers of houses, cars, vacations, DVD’s, washers, ‘n’all that?

Mathworld’s picture of a few topological objects gives one starting point:

One thing I noticed pretty quickly: you remember playing Star Fox battle mode? Or any video game where there is a lower-right thumbnail of you on a limited square map—such that when you go leftwards off the map you appear on the right, and when you go upwards off the map you appear on the bottom? As a kid I thought I was flying on the surface of a planet, but in fact it was the surface of a torus. (Why? If you go up to the top of the North Pole you don’t come out again at the South Pole. See the picture of the sphere with B ≠ C, i.e. N ≠ S.)

In other words, a torus (donut) is the product of a_loop × a_loop. Whereas a sphere (ball) is the product of a_loop (east/west) × a_line_segment (north/south).

GEOMETRY

Following from this short lesson in topology, one alternative to multiplying only “linear” dimensions of characteristic attributes would be to multiply lines with loops. For example a_loop × a_loop × a_line_segment. I’m not sure what the name for that shape is, but you can imagine it — like a cylindrical torus. And it’s logically possible that there are two circle-like dimensions in marketing. Something like, as politics goes further and further left, it starts to resemble the far right more than the middle. But relevant to marketing.

A second alternative then might be to consider, like in the image above, the endpoints of some line segments from the 3 dimensions of Nielsen. What if some of them were identified rather than left distinct? What kind of shapes could you create with that and would that resemble the consumer space more than a rectangle?

Some other ideas of things to question:

  • How do angles meet up? (inner product)
  • How do distances work? (norms)
  • Look through an algebraic geometry book, or Solid Shape. Are there any shapes—umbilics, furrows, biflecnodes, dimples, trumpets—that have an analogue in the space of all consumers?
  • Is backwards just the opposite of forwards? Or does that wrongly assume commutativity?

I don’t know if that would result in a better model. I don’t know if thinking about things this way would reduce wasteful ad spending. I don’t have data to test these ideas on. I just wanted to share this thought.




369 Plays • Download

The Enclosures (45 min)

I had no idea that so recently people roamed about each other’s land, no fences dividing the farms and folds.


http://upload.wikimedia.org/wikipedia/commons/9/9e/Mississippian_cultures_HRoe_2010.jpg
 


File:Mott Mounds Coles Creek culture HRoe 2011.jpg

http://upload.wikimedia.org/wikipedia/commons/b/b3/Eaker_Site.jpgMoundville site
Mound and pit


Uxmal Mexico Map

File:View from Pyramide de la luna.jpg


http://ambergriscaye.com/pages/mayan/art/tikal22sm.jpg
http://www.mexicovacationtravels.com/wp-content/uploads/2010/04/Mayan-Ruins-Tulum-Mexico-pictures.jpg

http://ambergriscaye.com/pages/mayan/art/cerros22sm.jpg
http://ambergriscaye.com/pages/mayan/art/lamanai.gif


http://ambergriscaye.com/pages/mayan/art/xunantunich22sm.jpg
http://ambergriscaye.com/pages/mayan/art/altunha22sm.jpg

http://static5.businessinsider.com/image/4dc1578ccadcbbd57f130000/image.jpghttp://ambergriscaye.com/pages/mayan/art/cuello22sm.jpg

http://ambergriscaye.com/pages/mayan/art/lamanai22sm.jpg

The modern structure of towns, like so many things, is an outcome of economic structure.

  • When shepherds no longer roamed freely through the hills
  • and it became efficient for homes to be built in a rotary array around some kind of centre,
  • then pubs (public houses = free houses) became the meeting place

This is one of the most influential things I’ve heard, period. Think about how much longer you have to walk and how much lonelier life became once you don’t cut across another person’s land.

  

My pessimistic image of the culture that I live in is

  • city people all in their separate flats, with their separate computers, or separate televisions, on separate couches, alone in the space they’ve paid for with the career they fought to dominate
  • going out to a restaurant, pub, or coffee shop to experience the unexpected bumpings into people
  • so everything costs money. It costs money to have friends, costs money to hang out, costs money to flirt, costs money to meet people, costs a lot of money to meet rich people, costs money to put yourself in a place where people will happen to encounter you—unless you do it over the internet—and then people wonder why nobody makes friends after college
  • suburban people the same, except also having their own pools instead of sharing a community pool
  • having their own medium-sized lawns — big enough to keep the neighbours from peeping in the window, or seeing you on the porch and say hello — instead of sharing a large park cutting all the medium lawns down to small lawns (not that they individually choose this — the decision is made by real estate developers)
  • country people even more isolated because land tracts are so huge
  • and nobody, but nobody, knows their neighbours.

William GedneyYoung girl standing on a porch leaning against a support beamKentucky, 1964[via the Duke University Libraries]
rocksteady
http://www.everyculture.com/images/ctc_02_img0346.jpg

Coney Island, 1984



More old New York


brooklyn

Danny Lyon/ NARA
Apartment house across from Fort Green Park in Brooklyn

On Bond Street in Brooklyn

Boys in Brooklyn

Fourth Annual Veuve Clicquot Polo Classic
Fourth Annual Veuve Clicquot Polo Classic


Basketball playground in Brooklyn

http://www.tfaoi.com/am/7am/7am322.jpg









Crooklyn told you what to dream

This is no 1970’s Brooklyn or 1950’s Appalachia, with people sitting on their porches and knowing each other and generally being outside and around each other at the same time.

  • When I think about more "primitive" cultures, I imagine if I’d been part of them then my identity would be so tied up in my relationships to other people—what I was born into, tight & maintained family relations, never redefining myself or my history, never escaping childhood, but with a sense of self and belonging and continuity from youth to parenthood to old age. Imagine if you slept this close together:

    (and of course, relatives and friends would hear or even—gasp—see you having sex—but which is more normal, to hide it or display it?)

    ^ Apparently the Holocene (little ice age) is the reason NW European culture with its individualism and small family norms—which propagated the world over thanks (I guess) to guns, germs, and steel—changed that NW European culture from a practice of public sex (in the manor—like a barn, sleeping to private.
  • Would people be kinkier? Or maybe it would depend on the initial conditions (if sex-copying is like an Ising spin then perhaps the first mover (wink wink) decides whether the culture becomes kinky or not)….
  • Can you imagine flirting, teasing, when you’re young, and then seeing the one you wanted to be with have sex with the one you competed with? I can’t fathom what would happen next. Would it be easier to move on? Harder? Would things just be so different that I can’t even conceive it? (yuk yuk)
  • …And I won’t even go into the sexual norms of Babylon or Sparta
  • I can’t say which culture I would prefer to live in, because my preferences are a function of the way I was raised. Economists usually leave aside where utility hypersurfaces come from and just treat them as good (or at least, unimpeachable—or, at the very least, immalleable).
  • But from a deep-past, anthropological perspective like this, it’s easy to see, “Yeah, maybe I just think monogamy is good because, duh, I live in a monogamous culture”. More broadly, I live in a culture of monogamy, where crushes and attraction are repressed, where physical attraction should not be confused with “real love”, where you probably have never met your spouse yet when you’re 16, where what the family thinks of him/her is less important than what you think, where equality among the sexes is valued, where young people don’t date anymore, they just party and eventually have sex with their friends (except for certain religions where that subculture exerts a dominating influence or sometimes the subculture itself has been magnetised toward the prevailing culture), where ambition is good, where people want to be footballers, where monetary compensation is negotiated in secret and kept secret, where compensation is based on measurable individual achievement rather than arguable perceptions of morality, where shame and guilt are not uncommonly attached to sex, where people opine about who should have sex with whom and why and when and where and how, where people break up because they finish school and get a job in a different city, where classically sexual relationships are supposed to happen with one person over a lifetime but serial monogamy seems to be taking over, where people puff themselves up to impress strangers or newly-met friends-of-friends, especially when they’re afraid or ashamed of themselves on the inside.
  • I mean of course there are various arguments you could make (at least I’ve heard some) as to why monogamy is good, or why love as it’s conceived by us is the right way to conceive it, or why everybody having sex in public would be weird, or gross, and people in my culture argue back and forth both directions about these things—but at least for me, I really can’t extricate myself enough from the expectations and the learned behaviours and the way things have always worked for me and my expectations of others and … so on.
  • Things I take for granted. Anyway, back to private property

One of my least favourite aspects of modern capitalist life is the segregated non-interaction of private persons with each other and each other’s property.

  • Everyone lives in their own place—or cramp in with flatmates—they can at least be a network of friends, since the formula for friendship requires proximity and random encounters. That is, if you don’t work opposite hours….
  • Everyone rides the tube to work while not speaking to each other. Or goes running alone, listening to their own iPod’s, dreaming of a career success or thinness/sexiness and people liking them-which idea was implanted by yet another commercial interest….
  • Or outside major cities, everyone drives in their own car and listens to the radio by themselves. At least drivers who talk on their mobile phones are enjoying some person-to-person interaction.
  • Rich suburban people all have their own pool. (And if a neighbour drowns in it, it’s your fault—so better put up a fence.)

Things are too entangled, too complex, for me to state a preference. Although, I guess by staying where I am, I’m tacitly putting up with and agreeing to the norms I was raised by. Maybe I am being too pessimistic, or maybe someday I will seek out something new … or try to get together with people who want to make something new….

I would link this up to some other thoughts I’ve had about charity and need. In a clear sense, somebody who accepts charity (say thanking you for giving something that’s actually rather crappy but they won’t tell you that since they want to be polite) does something back for the giver; if we had an fMRI we could measure the utility upticks in the donor and if we knew all of the neurochemistry we could say which dopamines are flowing where.

That’s undeniably true but the first time someone pointed it out to me, it clashed with my simple and straightforward view that the rich giver is the one doing the good act and the poor receiver should be grateful. But human interactions are more complicated than that, clearly. And something similar could maybe be said of the ever-escalating wealth and comfort of our age—or at least how my culture chooses to make use of that wealth.









http://www.the-latest.com/files/rex-dance.jpg

girls with dirty dresses in kentucky 1960's

When people are poor they lean on each other, and maybe in reaching the goal of standing fine alone something else is therefore symplectically lost. We should call up some of the boomerang millennials and ask how their failure to fly out of the nest really turned out—if being a loser didn’t have its upsides in terms of strengthening familial relationships. And then I’ll ask myself what it is I’m aiming for.




commodification

  • same hopes & desires
  • same choice of stores
  • the same houses and the same decorations in those houses
  • EFFICIENCY
  • fruit from the other side of the world

part II

  • Do our sexual norms derive from the invention of chimneys in the 14th century?
  • the invention of table manners
  • furniture, music, buttons, wainscoting, and intellectual pursuits — all due to the Little Ice Age?
  • lower and upper classes slept in the same hall, with the animals, around a fire, in the manor-house days. And had sex right in front of each other! omg! Economics begetting morals (I mean seeing thru to the 19th century)
  • So the Little Ice Age was the beginning of privacy. Speaking of not having sex in front of each other, maybe it contains as well the roots of abortion as well as the roots of Victorianism. Privacy norms were made law in 1979 in the United States, to the chagrin of anti-abortionists. Since then and before, appeals to privacy as a fundamental human right have been made to justify any victimless crime (homosexuality, libertinism, drug use … some of which are no longer criminal). What if our conception of this “natural human right” is just a function of the history of global temperature?

part III

  • Jamestown, VA versus charcoal
  • Economics before capitalism. Sounds like the ruler had the economy’s interest at heart — a growing economy means more to tax.

part IV

  • before trains, each village was more-or-less a genetic island. Not that no-one swam the waters to marry someone from the next town over, but genetic interchange among geographically dispersed humans was slow. As transport became cheaper and faster, procreation between Poles and Germans, Lyonnaise and Bretagnes, Spanish and Portuguese became more common.




[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)




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




I wrote earlier about the many different ways to measure distance. One way I didn’t include is unmeasurable distance.

Sometimes A is

  • tastier,
  • sexier,
  • cooler,
  • more interesting,
  • or otherwise better endowed

than B … but it’s impossible to quantify by how much. No problem; just say that A≻B but that |A−B| is undefined.

It’s still the case that if A is sexier than B and B is sexier than C, it must follow that A is sexier than C.

Symbolically: A≻B & B≻C A≻C.

This concept opens up many parts of human experience to the mathematical imagination.

I will also express my view on moral rates of income tax using orderings ≻.

Oh, and if you’re into this kind of thing: using orders instead of measurable quantities kind of saved the economic concept of “utility”. Kind of saved it. At least instead of talking about 174.27819 hedons, nowadays you can just say X is lexicographically preferred to Y. Ordinal utility instead of cardinal utility.




A circle is made up of points equidistant from the center. But what does “equidistant” mean? Measuring distance implies a value judgment — for example, that moving to the left is just the same as moving to the right, moving forward is just as hard as moving back.

But what if you’re on a hill? Then the amount of force to go uphill is different than the amount to go downhill. If you drew a picture of all the points you could reach with a fixed amount of work (equiforce or equiwork or equi-effort curve) then it would look different — slanted, tilted, bowed — but still be “even” in the same sense that a circle is.

Here’re some brain-wrinkling pictures of “circles”, under different L_p metrics:

astroid p=⅔
p = ⅔

The subadditive “triangle inequality” A→B→C > A→C no longer holds when p<1.

p = 4p = 4 

 p = 1/2
p
= ½
. (Think about a Poincaré disk to see how these pointy astroids can be “circles”.)
 p = 3/2 p = 3/2 

 workin on my ♘ ♞ movesThe moves available to a knight ♘ ♞ in chess are a circle under L1 metric over a discrete 2-D space.




Everybody knows that

(referring to the sides of a right triangle). That formula tells you the straight-line distance on a flat plane. Say I walked three blocks east in a grid-style city and then four blocks north.  Then I&#8217;ve traveled

blocks i.e. 5 blocks, as the crow flies.  I could also go one block down to the sub-sub-&#8230;-sub-basement and then the same rule would apply in 3-D, as the mole burrows



blocks north-east-down.

You see the same formula&#8212;weird, right?&#8212;in statistical calculations of standard deviation:


Each of the |dev|'s is a data point's deviation from the center (middle, barycenter, mean) of all such observations.

So standard deviation is a Pythagorean concept &#8212; in a way, situating your n data points on the corner of an n-dimensional box and then calculating a hypotenuse.  If that doesn&#8217;t make you wary of standard deviation as the only, absolute measure of variability &#8230; well, it should.

But so, what&#8217;s up with all the 2's?  Higher mathematics doesn&#8217;t use numbers! Replace each of the ^2's with a ^p for any power, and you&#8217;ve got ∞ new, valid, measures of distance, called L_p norms. This symbolic change gives rise to some mind-expanding imagination weapons, including the much-hyped non-Euclidean geometry (p ≠ 2, Cthulhu fans).

p=1 corresponds to city-block distance &#8212; (not in Boston, Prague, or Edinburgh because the streets don&#8217;t connect squarely – in some imaginary flat, square Lattice City).

p=2 corresponds to Navy distance on a flat lake. If you sail or fly across the Atlantic, the curvature of the Earth starts to make a difference and p ≠ 2.
p=4 could measure financial volatility in a way that penalizes kurtosis. (It&#8217;s still a two-way measure, though. Better would be p=5 or another high, odd number, to get a quasimetric.)
These new conceptions of &#8220;distance&#8221; make more sense of physical reality:
General relativity (which gave us GPS&#8217;s and satellites) requires Riemannian geometry, since spacetime is curved.
Euclidean geometry (p=2) fails if you&#8217;re Magellan, since the Earth&#8217;s surface is curved.
And we wouldn&#8217;t have the atom bomb without the even more brain-wrinkling geometry of nuclear physics.
If you want to tolerate noise or slight differences among data, Lp norms let you treat similar things as the same while still maintaining différence among quite different things.
If you want to use mathematics on things besides physics – like text mining, psychology, chess, racism, self-versus-other, financial time series, cluster analysis, terms of trade, marketing data, voting, bargaining, morality, functional spaces, utility theory, strategic arms races – you&#8217;ve got to be aware of the distance measure in your relevant space.

Everybody knows that

a² + b² = c²

(referring to the sides of a right triangle). That formula tells you the straight-line distance on a flat plane. Say I walked three blocks east in a grid-style city and then four blocks north. Then I’ve traveled

sqrt( 3² + 4²)

blocks i.e. 5 blocks, as the crow flies. I could also go one block down to the sub-sub-…-sub-basement and then the same rule would apply in 3-D, as the mole burrows

sqrt( 3² + 4² + 1²) = sqrt(26)

blocks north-east-down.

You see the same formula—weird, right?—in statistical calculations of standard deviation:

sqrt( dev&sub1; ²  +  dev&sub2; ²  +  dev&sub3; ² + ...) = ST DEV

Each of the |dev|'s is a data point's deviation from the center (middle, barycenter, mean) of all such observations.

So standard deviation is a Pythagorean conceptin a way, situating your n data points on the corner of an n-dimensional box and then calculating a hypotenuse. If that doesn’t make you wary of standard deviation as the only, absolute measure of variability … well, it should.

But so, what’s up with all the 2's? Higher mathematics doesn’t use numbers! Replace each of the ^2's with a ^p for any power, and you’ve got ∞ new, valid, measures of distance, called L_p norms. This symbolic change gives rise to some mind-expanding imagination weapons, including the much-hyped non-Euclidean geometry (p ≠ 2, Cthulhu fans).

p=1 corresponds to city-block distance — (not in Boston, Prague, or Edinburgh because the streets don’t connect squarely – in some imaginary flat, square Lattice City).

p=2 corresponds to Navy distance on a flat lake. If you sail or fly across the Atlantic, the curvature of the Earth starts to make a difference and p ≠ 2.

p=4 could measure financial volatility in a way that penalizes kurtosis. (It’s still a two-way measure, though. Better would be p=5 or another high, odd number, to get a quasimetric.)

These new conceptions of “distance” make more sense of physical reality:

  • General relativity (which gave us GPS’s and satellites) requires Riemannian geometry, since spacetime is curved.
  • Euclidean geometry (p=2) fails if you’re Magellan, since the Earth’s surface is curved.
  • And we wouldn’t have the atom bomb without the even more brain-wrinkling geometry of nuclear physics.
  • If you want to tolerate noise or slight differences among data, Lp norms let you treat similar things as the same while still maintaining différence among quite different things.

If you want to use mathematics on things besides physics – like text mining, psychology, chess, racism, self-versus-other, financial time series, cluster analysis, terms of trade, marketing data, voting, bargaining, morality, functional spaces, utility theory, strategic arms races – you’ve got to be aware of the distance measure in your relevant space.


hi-res