Posts tagged with holism

Both direct sum and tensor product are standard ways of putting together little Hilbert spaces to form big ones. They are used for different purposes. Suppose we have two physical systems…. Roughly speaking, if … a physical system’s … states are either of A OR of B, its Hilbert space will be [a] direct sum…. If we have a system whose states are states of A AND states of B, its Hilbert space will be [a] tensor product….
```MEASURE SPACE   disjoint union  Cartesian product
HILBERT SPACE   direct sum      tensor product
```

John Baez

(Source: math.ucr.edu)

## Dummyisation

Statisticians are crystal clear on human variation. They know that not everyone is the same. When they speak about groups in general terms, they know that they are reducing N-dimensional reality to a 1-dimensional single parameter.

Nevertheless, statisticians permit, in their regression models, variables that only take on one value, such as `{0,1}` for `male/female` or `{a,b,c,d}` for `married/never-married/divorced/widowed`.

No one doing this believes that all such people are the same. And anyone who’s done the least bit of data cleaning knows that there will be `NA`'s, wrongly coded cases, mistaken observations, ill-defined measures, and aberrances of other kinds. It can still be convenient to use binary or n-ary dummies to speak simply. Maybe the marriages of some people coded as `currently married` are on the rocks, and therefore they are more like `divorced`—or like a new category of people in the midst of watching their lives fall apart. Yes, we know. But what are you going to do—ask respondents to rate their marriage on a scale of one to ten? That would introduce false precision and model error, and might put respondents in such a strange mood that they answer other questions strangely. Better to just live with being wrong. Any statistician who uses the `cut` function in R knows that the variable didn’t become basketed←continuous in reality. But a `facet_wrap` plot is easier to interpret than a 3D wireframe or cloud-points plot.

To the precise mind, there’s a world of difference between saying

• "the mean height of men > the mean height of women", and saying
• "men are taller than women".

Of course one can interpret the second statement to be just a vaguer, simpler inflection of the first. But some people understand  statements like the second to mean “each man is taller than each woman”. Or, perniciously, they take “Blacks have lower IQ than Whites” to mean “every Black is mentally inferior to every White.”

I want to live somewhere between pedantry and ignorance. We can give each other a break on the precision as long as the precise idea behind the words is mutually understood.

` `

Dummyisation is different to stereotyping because:

• stereotypes deny variability in the group being discussed
• dummyisation acknowledges that it’s incorrect, before even starting
• stereotyping relies on familiar categories or groupings like skin colour
• dummyisation can be applied to any partitioning of a set, like based on height or even grouped at random

It’s the world of difference between taking on a hypotheticals for the purpose of reaching a valid conclusion, and bludgeoning someone who doesn’t accept your version of the facts.

So this is a word I want to coin (unless a better one already exists—does it?):

• dummyisation is assigning one value to a group or region
• for convenience of the present discussion,
• recognising fully that other groupings are possible
• and that, in reality, not everyone from the group is alike.
• Instead, we apply some ∞→1 function or operator on the truly variable, unknown, and variform distribution or manifold of reality, and talk about the results of that function.
• We do this knowing it’s technically wrong, as a (hopefully productive) way of mulling over the facts from different viewpoints.

In other words, dummyisation is purposely doing something wrong for the sake of discussion.

roots of `x²⁶•y + x•z⁶ + y¹³•z + x⁹•y¹³ + z²⁶     =   0`

$\dpi{200} \bg_white \large x^{27} \cdot y+x \cdot z^6+y^{13} \cdot z+x^9 \cdot y^{13}+z^{26}$

(Source: imaginary.org)

The purpose of a family is the enhancement of the individual pursuits of happiness … in the overall … preservation of the family as a whole.

—from Family Wealth

Is this a logical sentence, or not?

hi-res

## Regression on Complexes III: Modcloth

My father used to tell me that when people complimented him on his tie, it was never because of the tie—it was because of the suit. If he wore his expensive suit, people would say “Nice tie!” But they were just mis-identifying what it was that they thought was nice. Similarly if you’re interviewing candidates and accidentally doing your part to perpetuate the beauty premium to salaries, you aren’t going to think “She was really beautiful, therefore she must be more competent”. You might just notate that she was a more effective communicator, got her point across better, seemed like more of a team player, something like that.

` `

Achen (2002) proposes that regression in the social sciences should stick to at most three independent variables. Schrodt (2009) uses the phrase “nibbled to death by dummies”.

I understand the gripes. These two men are talking about political analysis, where the “macro” variables are shaky to begin with. What does it mean that the Heritage Foundation rated two countries `7` versus `9` points apart on corruption or freedom? Acts of corruption are individual and localised to a geography. Even “ethnofract”, which seems like a valid integral, still maps `∼10⁷` individual variation down to `10⁰`. But this is statistics with fraught macro measures trying to answer questions that are hard to quantify in the first place—like the Kantian peace or center–periphery theories of global political structure.

What about regressions on complexes in more modest settings with more definitive data measurements? Let’s say my client is a grocery store. I want to answer for them how changing the first thing you see in the store will affect the amount purchased of the other items. (In general trying to answer how store layout affects purchases of all items … this being a “first bite”.) Imagine for my benefit also that I’m assisted or directed by someone with domain knowledge: someone who understands the mechanisms that make X cause Y—whether it’s walking, smelling, typical thought patterns or reaction paths, typical goals when entering the store, whatever it is.

I swear by my very strong personal intuition that complexes are everywhere. By complexes I mean highly interdependent cause & effect entanglements. Intrafamily violence, development of sexual preference, popularity of a given song, career choice, are explained not by one variable but by a network of causes.  You can’t just possess an engineering degree to make a lot of money in oil & gas. You also need to move to certain locations, give your best effort, network, not make obvious faux pax on your CV, not seduce your boss’ son, and on and on. In a broad macro picture we pick up that wealth goes up with higher degrees in the USA. Going from G.E.D. to Bachelor is associated with `tripling ± 1` wealth.

I think this statistical path is worth exploring for application in any retail store. Or e-store or vending machine (both of which have a 2-D arrangement). Here as the prep are some photos of 3-D stores:

And for the 2-D case (vending machine or e-store) here are some screen shots from Modcloth, marked up with potential “interaction arrows” that I speculated.

Again, I don’t have a great understanding of how item placement or characteristics really work so I am just making up some possible connections with these arrows here. Think of them as question marks.

• purse, shoes, dress. Do you lead the (potential) customer up the path to a particular combination that looks so perfect? (As in a fashion ad—showing several pieces in combination, in context, rather than a “wide array” of the shirts she could be wearing in this scene.)
• colours. Is it better to put matching colours next to each other? Or does that push customers in one direction when we’d prefer them to spread out over the products?

• variety versus contrastability. Is it better to show “We have a marmalade orange and a Kelly green and a sky blue party dress—so much variety!” or to put three versions of the “little black dress” so the consumer can tightly specify her preferences on it?

And if you are going to put a purse or shoes along with it (now in 3-ary relations) again the same question arises. Is it better to put gold shoes and black shoes next to the “cocktail dress” to show its versatility? Or to keep it simple—just a standard shoe so you can think “Yes” or “No” and insert your own creativity independently, for example “In contrast to the black shoes they are showing me, I can visualise how my gold sparkly shoes would look in their place”? More and more issues of independence, contrast, context, and interdependence the more I think about the design challenge here.

• "random" or "space" or "comparison". You put the flowers next to the shelves to make the shelves look less industrial, more rather part of a “beautiful home”. Strew “interesting books” that display some kind of character and give the shopper the good feelings of intellect or sophistication or depth.

Or, what if you just leave a blank space in the e-store array? Does it waste more time by making the shopper scroll down more? or does it create “breathing room” the way an expensive clothing store stocks few items?
• price comparisons. You stock the really really expensive pantsuit next to the expensive pantsuit not to sell the really-really-expensive one, but to justify the price or lend even more glamour to the expensive one.

• more obvious, direct complements like put carrots and pitas next to hummous so both the hummous looks better and you will enjoy it more. Nothing sneaky in that case.

Did you ever have the experience that you buy something in the store and it read so differently in the store and when you were caught up in the magic of the lifestyle they were trying to present to you, but now it’s hanging up with your stuff it reads so different and doesn’t actually say what you thought it said at the time?

For me if I’m clothes shopping I’m thinking back on what else I own, what outfits I could make with this, how this is going to look on me, how its message fits in with my own personal style. And at the same time, the store is fighting me to define the context.

` `

In the Modcloth example I’m talking mostly about 2- or 3-way interactions between objects. In analogy to simplicial complexes these would be the 1-faces or 2-faces of a skeleton.

But in general in a branded store, the overall effect is closer to let’s say the N-cells or N−1-cells. Maybe it’s not as precise as the painting in http://isomorphismes.tumblr.com/post/16039994007/thoroughly-enmeshed-composition-perturbation or a perfectly crafted poem or TV advertisement, where one change would spoil the perfection.

But clothing stores are definitely holistic to a degree. By which I mean that the whole is more than the sum of the parts. It’s about how everything works together rather than any one thing. And a good brand develops its own je ne sais quoi which, more than the elements individually, evokes some ideal lifestyle.

More on this topic after I finish my reading on Markov basis.

## Holistic

Some say science & mathematics are reductive.

• Galileo showed us how to break apart space into three pieces and that `ẋ`, `ẏ`, `ż` function independently. (The speed the soccer ball falls downward off the cliff is unrelated to how much forward momentum you kicked it.)
• Experimental science tests just one thing and isolates it as perfectly as possible.
• Some say that the entire progress of empiricist science has been the systematic isolation and testing of small parts of reality, combined with “rigourous technical analysis” (by which they mean, theories in the language of mathematics).
• (I’ve seen this view in the CP Snow ish arguments where “literary types” or “critical theory types” want to attack science or scientists en masse, or where “science types” want to attack philosophers/postmodernists/cultural theorists/Marxists/Deleuze/liberal academia en masse.)
• Some philosophers argue that the world is really atomic in some way, not just the particles but the causes and forces in the world are reducible to separable elements.

The reductionistic approach to science has to pair with qualifiers and caveats that “The lab is not the real world” and “We’re just trying to model one phenomenon and understand one thing”: hopefully combining `A` with `B` doesn’t introduce complexity in the sense that `A+B` is more than the sum of its parts — in statistical modelling language, that the interaction terms don’t overwhelm the separable, monic terms.

But is it really true that mathematics is reductionistic? I can think of both separable mathematical objects and not-separable ones. You could argue, for example, that a manifold can be decomposed into flat planes—but then again, if it has a nontrivial genus, or if the planes warp and twist in some interesting way, wouldn’t you be nullifying what’s interesting, notable, and unique about the manifold by splitting it up into “just a bunch of planes”?

Or with set theory: you could certainly say that sets are composed of atomic urelemente, but then again you could have a topological space which is non-decomposable such as an annulus or network or 1-skeleton of cells, or a non-wellfounded set (cyclic graph) which at some point contains a thing that contains it.

How about a sort-of famous mathematical object in the theory of links & knots: the Borromean rings.

The Borromean rings are famous for the fact that they cannot be decomposed into a simpler atom, whilst retaining their Borromean nature. In other words the smallest atom you can find is the 3 rings themselves. If any one or more rings were removed then they would not be linked together.

So not only is it an interesting case for causality (is `ring 1` binding up `ring 3`? No. Is `ring 2` binding up `ring 3`? No. In a way none of the rings is locking up any other ring, and yet they are locked by each other.

It takes the interaction term `[ring 1 ∩ ring 2]`only together do they bind up `ring 3`, but together they do bind it completely. (Did something like this come up in Lord of the Rings or some other fantasy or myth? Like the weakest link or that square battle arrangement with spears, or having a ton of archers in Warcraft 2, but not like Captain Planet, if any member of a group breaks then its entire strength is lost (super convexity) but together they’re nearly undefeatable.)

Also, circularly, as well as `ring 3` being bound-by `[ring 1 ∩ ring 2]`, it also binds — or again, with-its-mate-binds — the other rings.

I don’t know if historically the Borromean rings were a symbol of holism, although one would think so given this picture from the Public Encyclopedia:

Even if it hasn’t been, we certainly could use the Borromean rings now as a symbol of holism, complexity, integratedness, un-separability, irreducibility, and convolution.

## (Thoroughly) Enmeshed Composition

Imagine you were a wealthy writer — so wealthy that you could pay servants to look stuff up for you. Instead of drudging through tomes (or internet searches) to fact-check yourself, find original references, and so on. You just do the fun part: pontificate on paper.

Now let’s say after you have finished an essay, your servant / employee / virtual personal assistant comes back from his footnote research and tells you that statement #13 should be revised based on the best-known research on the topic. In fact, statement #13 is almost the reverse of the truth.

I can imagine things going one of two ways from here.

1. In the less interesting case, statement #13 is an offhand remark upon which little else in the essay depends. You correct yourself, modify some text directly before and after statement #13, and move on. The only neighbours of the concepts in statement #13 are the transition sentences directly before & after it on a 1-dimensional topological line.

2. In the more interesting case, what you said before statement #13 was meant to lead up to the exact statement you made. Perhaps #13 was a key point, or the thesis of the essay. And let’s further imagine that the text following statement #13 depended critically on the exact value of statement #13 being as you wrote it. When #13 is altered, the preceding text is no longer necessary and the succeeding text no longer works.

In an especially dire scenario, your PA’s research might overturn the worldview that led you to write the essay in the first place.

Like Holger Lippmann’s “Flower Circles 13”, changing one element renders the entire whole needful of alteration. Everything is so thoroughly enmeshed (see “complete graph” below for the neighbourhood relations) that no element of the text speaks in isolation.

That’s in distinction to the calculus, where smooth functions can be approximated by a differential.

In physicists’ language, due to tightly, globally connected topology, perturbations cannot be localised. Rather, the opposite: local perturbations cause global changes in the object.

OK, someone dared me. I’ll say it: Gestalt.

## My sister isn’t “irrational”, her utility function just has large interaction terms.

What happens if, instead of doing a linear regression with sums of monomial terms, you do the complete opposite? Instead of regressing the phenomenon against $\large \inline \dpi{200} \bg_white x_t + y_t + z_t + \epsilon_t$ , you regressed the phenomenon against an explanation like $\dpi{200} \bg_white \sqrt[ \sum \text{powers} ]{{x_t}^{66} \; {y_t}^{13} \; {z_t}^{282} + {x_t}^3 \, {y_t}^9 \, {z_t}^7 + \ldots + {x_t}^{17} \, {z_t}^{1377} }$ ?

I first thought of this question several years ago whilst living with my sister. She’s a complex person. If I asked her how her day went, and wanted to predict her answer with an equation, I definitely couldn’t use linearly separable terms. That would mean that, if one aspect of her day went well and the other aspect went poorly, the two would even out. Not the case for her. One or two things could totally swing her day all-the-way-to-good or all-the-way-to-bad.

The pattern of her moods and emotional affect has nothing to do with irrationality or moodiness. She’s just an intricate person with a complex utility function.

If you don’t know my sister, you can pick up the point from this well-known stereotype about the difference between men and women:

"Men are simple, women are complex.” Think about a stereotypical teenage girl describing what made her upset. "It’s not any one thing, it’s everything.”

I.e., nonseparable interaction terms.

I wonder if there’s a mapping that sensibly inverts strongly-interdependent polynomials with monomials — interchanging interdependent equations with separable ones. If so, that could invert our notions of a parsimonious model.

Who says that a model that’s short to write in one particular space or parameterisation is the best one? or the simplest? Some things are better understood when you consider everything at once.

"The whole is more than the sum of the parts"

$\LARGE \inline \dpi{200} \bg_white f( \ \bigcup\limits_i \, x_i \: ) \quad \geq \quad \sum\limits_i \, f( \, x_i \: )$

Who says scientists are reductionistic? Any superadditive system—due to complexity, interaction terms, valuation by an Lₚ norm with 0<p<1, or some other reason—adds up to more in total than the pieces individually do.

$\LARGE \inline \dpi{200} \bg_white \rho( \ \bigcup\limits_i \, x_i \: ) \quad \geq \quad \sum\limits_i \, \rho( \, x_i \: )$

(Such Lₚ norms are semimetrics but not seminorms.)

Some people think of “geometric” art as being math-y, in the same sense that the band Maps & Atlases is math-y.

But I don’t think lines, circles, squares, tessellations, grids, and polygons are more mathematical than globsleaves, aleatorics, coloursnets, or scribbles. In fact, I can link to a math post about each: lines, circles, hypersquarespolytopes, aleatoricstessellations, blobs, gridsleaves, nets, scribbles, colours.

The mathematical thought that occurs to me when looking at this painting is how, in composition, every spot on the canvas influences every other spot. Holger Lippmann couldn’t have swapped a few of these circles because it would have ruined the effect.

Similarly in painting like this, if you added a splotch of yellow in the bottom right, that would affect the look of several other parts of the canvas.

Algebraically, the pieces of the composition are like a highly connected graph (in “how good it looks” space).

If you regressed compositional outcome against the content of each point in the painting (or just against the style of each circle), the relevant explanatory variables would be highly interactive terms. All the monomial, binomial, trinomial, … terms would be irrelevant.

` `

The image is: 29417FlowerCircles_13_grid3 by holger lippmann, via wowgreat

hi-res