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In 20th-century abstract mathematics, one builds up ideas and properties—not assuming anything except what one is told. You think 2+3=5? Well in my space that I just made up, e₂⊕e₃ = e₁, and 5 doesn’t even exist!

Concepts are added in incrementally, like

  • ‖A‖ means the “size” of A. size exists
  • ‖A − B‖ means the “distance” between A and B. plus exists & negative exists; or, comparison exists
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  • (If zero exists, we could say the size of A = the distance between A and 0: ‖ A − 0 ‖ = ‖A‖.)
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  • ⟨ A | B ⟩ means A “times” B. times exists
  • arccos ⟨A|B⟩ ‖A‖⁻¹ ‖B‖⁻¹ inverses exist. times exists. so angle exists
  • topology adds in neighbourhood relationships—not necessarily in a way that you can infer size or distance (∵¬□∃ metric), but so that you could talk about paths or connectedness
  • order or ranking — is it a total order? a transitive order? a partial order? a lattice? Order is subordinate to size, to distance, and to linearity.
  • dimensionality — a set containing { ‘a’, ‘b’, the moon, 12, the vector (0 1 1 0 1)∈ℝ⁵, my cat’s hairball } doesn’t inherently have dimensions to it — so structured sets like ℝ² are supposed to explain how their universe breaks down
  • linearitypossibly the scariest word in mathematics class? I’ve tried and will continue to try to explain it elsewhere, but “linear” is an extremely-restrictive-but-not-that-restrictive-because-so-many-things-are-linear-once-you-allow-calculus-and-maps-across-domains-for-example-fourier-transforms property. Linearity presumes monotonicity (order preservation), size, and a kind of “constancy” that tells you if 2 went to 4, then 13 is going to go to 26. Or “the 26 of the present land”.

Someone GPL’ed this nice (but not comprehensive) chart of two paths through the theory space—starting with a pair (thing, operation) [“magma”—sweet name, right?] and gradually adding more and more axioms until you get to a group.

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Mathematical words obtain everyday meaning—sometimes unexpected meaning—in applications. For example

  • "angle" might mean "correlation" — the angle between two pulse-trains would be their correlation; and in recommendation engines the matrix “cosine distance” is a basic measure of similarity
  • "multiplication" — well what if you want to multiply two functions together? You could convolve them. Convolution doesn’t seem very much at all the same action as 3×8 = three groups of eight. Neither do Photoshop blends seem like multiplication, but some of them are.
  • "size" — well maybe I mean "how well the business did" on a slew of different metrics — in which case, are there 20 different conceptions of "size"? I guess so.

Could you multiply two trees together? Could you define the angle between two natural numbers? The angle between two business models? Sure. If you know what you’re doing and why, you might even come up with a conclusion that makes sense. It all depends on (a) your ingenuity, (b) domain knowledge of the real-life situation, and (c) mathematical vocabulary.

Sometimes there is more than one interpretation that works with a given set. For example, {0,1} × {0,1} → {0,1} might be joined to operations that define “logical AND" and "logical OR”, or it might be interpreted just as on/off. Or it might be interpreted as the story of unrequited love.

 

All of that preface is meant to dislodge any notions you might have that ℝ² is somehow a “default” or “standard” paradigm. Sometimes number×number is an appropriate metaphor and sometimes not.

For example in the movie Rogue Trader, Nick Leeson’s boss is portrayed talking about “synergy” and “the information curve”. “Nick has positioned himself right there on the information curve!” It’s a parody and nobody seems to know quite what “the information curve” is (what’s on the axes? why is it curved?) but because Nick appears to be earning 70% of Barings’ profits, nobody questions the information curve.

Your typical crappy airport “business advice” books—Thomas Friedman kind of crap—will throw around 2-D charts that make no sense as well. Please leave some pics in the comments if you know what I’m talking about and examples come to mind. Here are a few dubious 2-D metaphors:

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The “political compass” labels reduce the complexity of the world in particular ways that suit the rhetorical aims of these libertarian authors. For example projecting totalitarianism and populism into the same neighbourhood when one could just as well project them onto opposite ends of some other spectrum.

Here are some dubious scales—where either order, linearity, or 1-dimensionality is suspect.

This chart additionally uses way too many significant figures. How is it you gauge "total novelty in the universe" again?
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(Remember: {"heroic", ”pragmatic”, ”circumspect”, ”brazen”} also comprises or belongs to a scale—in the ggplot sense of the word as well as other senses.)

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Wow! You mean that losses are bad and earnings are good? That is some insightful business insight.

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Crappy reductions needn’t be 2-D. The MBTI is a crappy reduction of personality in 4-D. And here are some in 1-D and other-D:

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I like how step 5 leads to step 2. This should be a list rather than a flow.

Bloom's taxonomy is unjustified, both the projections and the order

Order, 1-dimensionality questionable.
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Again, a list. This one has a heading. Apparently headings deserve 4 connecting wires whereas list items only deserve 3?
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This is just a list of things. There is no “center” or “flow” or “order” or “cycle” relationship. Maybe “give them” and “get them” could have used a two-way arrow between them.

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8-D and I just do not understand what these axis labels mean.
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I actually spent hours finding the worst graphics evar. Not gonna tell you my google keywords though.

 

And, not to be critical all the time, here’s a 2-D metaphor that does work:

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Stagepiece one: undermine the conceit that ℝ² is a default. Stagepiece two: cruddy graphics from various domains that force a metaphor that doesn’t really work. And now, the main act.

Today, I want to take aim at a highly suspect 2-D chart from the world of psychology:  the affect × intensity description of feelings.

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Right away when I look at this, it seems like an overly limiting and not internally valid picture of emotional range. Like so many taxonomies, it gets deeply under my skin in a way that I can’t explain, except to shout: Bad theory! Bad theory!  I mean — how does it make sense to say

  1. that each of these states is a point, as opposed to a spray or splotch or something else
  2. that this precise “point” is the same for all individuals
  3. "delighted" is slightly to the left of "happy" but happy is directly above "pleased"
  4. that “sleepy” is to the right of “tired” instead of the other way around
  5. that tired and sleepy are the same distance from each other as “pleased” and “glad”
  6. WTF is “droopy”? It sounds like a word to be applied to a plant, not a person. I also don’t think it qualifies as an emotion. "Droopy" sounds like a word Good Housekeeping would use to shame a 1950’s American married woman for not being perky! happy! sexy! listening! rubbing his feet! when her husband returns home from work.
  7. Are “sleepy” and “tense” actually moods or emotions? They sound like physical states.
  8. All of these emotions are near the perimeter, but some are closer to the origin than others
  9. sad minus gloomy = satisfied minus calm
??? because all of those are implicit in the drawings.

Remember what I was outlining at first. In abstract mathematics and in deciding the shape of a theory, we shouldn’t assume anything that doesn’t have to be assumed to explain the results.

I could attack the valence-intensity model in at least two ways.

  1. First would be to exclaim “But you didn’t justify any of that stuff! Linearity? Dimensionality? Order? You skipped it all! Where’s the justification?”
  2. Second, perhaps a little stronger than merely asking for backup, would be to point out flaws. For example if I could find a counterexample showing that emotional states don’t have magnitude, can’t be added, don’t break down on dimensions, or aren’t linear across dimensions.
The easiest critique of type [2] I could think of is to question the existence of a “zero-point” emotion. It might be possible to have low-or-zero activation of an emotion on the intensity axis, but on the valence axis? Could I have high intensity of zero valence? What about high intensity in the negative direction at zero valence? It doesn’t make sense.

I came up with a list—several years ago—of different feelings which all could contend for “emotional zero”.

  • neither happy nor sad
  • neutral
  • feel blank
  • both happy and sad (bittersweet)
  • not sure
  • ambivalent
  • "I feel nothing"
  • kinda sort
  • middling

That’s just feelings we have the words for. There are lots of nameless emotions (or emotional superpositions) that could contend for the neutral canvas — the origin from which all other emotions are measured.

The fact that so many clearly distinct feelings all contend for the “origin” made me think there is, in fact, no origin. But making the space affine (removing zero) doesn’t fix the problems I had begun to notice with the circumplex view of the emotional spectrum. I think we just have to think of the range of emotions as a totally different kind of space. I don’t know its topology; I do believe there should be some “activation level” (like a scalar) at least sometimes; I do believe that superpositions are possible.

http://isomorphismes.tumblr.com/post/6559201759/graph-tradeoffs-design-pattern

http://isomorphismes.tumblr.com/post/4840897988/logic-emotion

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