Posts tagged with partial order

‘The Descent of the Modernists’, cartoon by E. J. Pace, first published in Seven Questions in Dispute by William Jennings Bryan (1924).
with some nice autobiographical details, via cmnotes




Railing against “grey areas” has become a favourite rant topic. People think that they’ve covered their bases and are being really open-minded when they switch from {0,1} to [0,1]—but no false dichotomies are avoided in this transition from discrete to continuous.

Let’s take the example of sex & gender. Most of the tick-boxes and bathrooms we face in life are labelled “M” or “F”, which covers most of us but not all.

http://i.huffpost.com/gen/57607/thumbs/s-TRANSGENDER-BATHROOM-RULE-large.jpg
http://www.understandthetimes.org/images/nir152i.gif
http://i.telegraph.co.uk/multimedia/archive/01001/460-toilet_1001290c.jpg

http://farm6.staticflickr.com/5054/5424192256_e8aaaf7256.jpg
http://burstupdates.files.wordpress.com/2012/04/burstupdates-women-urinal-e1338061475122.jpg
http://images.sodahead.com/profiles/0/0/2/1/7/2/1/0/9/hmm-33753092570.jpeg
http://i1.stay.com/images/venue/147/77/e90c14ec/science-fiction-museum-and-hall-of-fame-sfm.jpg
http://www.funnypictures.net.au/images/men-women-alien-bathroom-signs1.JPG

http://www.scatmania.org/wp-content/uploads/2008/09/29082008165.jpg

(And I want to apply a kernel weighted to extra-count the forgotten individuals, since as minorities they’re more vulnerable. This can be seen in data such as e.g. higher suicide rates and higher murder rates.) The University of Hawai’i’s guidelines for dealing with individuals possessing ambiguous genitalia (Archives of Pediatrics and Adolescent Medicine) use words like

  • chromosomes—XX, XY, or other
  • micropenis, labia-scrotum fusion, gonadal dysgenesis
  • androgen insensitivity syndrome, hypospadias, kiinefelter syndrome, congenital adrenal hyperplasia, Turner’s syndrome
  • true hermaphroditia

which raises the question of where the “grey area” between [M,F] ~= [0,1] could come from. Chromosomes either come in whole units — for example people with Klinefelter’s syndrome have 47 chromosomes “XXY” — or have a much more complicated structure if you want to dig into the DNA string. Other aneuploidies include XYY, monosomy or partial monosomy, trisomy 21 (which I don’t think affects genitals or sex assignment), distal 18q−, mosaicism, the list goes on. How are we going to assign a total order there in order to define a continuous variable? I don’t see any way to—just more possibilities to add to the domain of a categorical variable (and making it much more confusing than the usual gender dummy!).

The paper above, to give another example of non-orderability, notes that various chemicals usually squirt at you in fœtal development but they vary in their squirtular timing. So androgen, progesterone, and so on aren’t mutually fungible (as the different “coloured edges” in Ramsey theory), and además we’re dealing with time series like Ed Küpfer’s pictures of sports scores:

image

image
image

Those kinds of pitures, but with different coloured spiketrains representing the incommensurability of androgen vs testosterone and so on.

So how do you get total orderability (necessary for a “grey area”) from a time series of incommensurable chemtrains? I don’t see it. The geometry is more interesting than just a line segment.

Further reading: transgender mathematician (Leigh Noble), transgender computer programmer (Tim Chevalier @eassumption), transgender economist (Deirdre McCloskey @deirdremcclosk), transgender electrical engineer (Lynn Conway). Jeff Eugenides’ Middlesex.

(Source: twitter.com)




image

  • I like Indonesian food better than Japanese food i ⪰ j, and
  • I like Japanese food better than English food j ⪰ e.
  • I also like French food better than English food f ⪰ e, but
  • I see French food as so different from the “exotic Eastern” foods that I can’t really say whether I prefer French food to Indonesian f≹i or Japanese f≹j.

    I would just be in a different mood if I wanted French food than if I wanted “exotic Eastern” food.

So my restaurant preferences are shaped like a poset. In a poset some things are comparable  and some things ain’t . Popularity is shaped as a poset and so is sexiness. Taste in movies is a poset too. The blood types have the same mathematical form as a poset but only if you reinterpret the relation  as “can donate to” rather than “is better than”. So not really the same as ethnic food.

image

Partial rankings | orders are transitive, so

  • (indonesian ⪰ japanese and japaneseenglish) implies indonesian ⪰ english.

That means I can use the “I prefer " symbol to codify what I said at the outset:

  • Indonesian  Japanese English
  • French English
  • neither⪰j nor⪰ f … nor⪰ f nor⪰ j (no comparison possible )


Posets correspond nicely to graphs since posets are multitrees.

image

image

Total orders — where any two things can be ranked  — also correspond to graphs, but the edges always line up the nodes into a one-dimensional path. So their graphs look less interesting and display less weird dimensional behaviour. Multitrees (posets) can have fractional numbers of dimensions, like 1.3 dimensions. That’s not really surprising since there are so many kinds of food / movies / attractiveness, and you probably haven’t spent the mental effort to precisely figure out what you think about how you rate all of them.

Rankings | orders are a nice way to say something mathematical without having to use traditional numbers.

I don’t need to score Indonesian at 95 and score Japanese at 85. Scores generated that way don’t mean as much as Zagat and US News & World Report would like you to think, anyway — certainly they don’t have all the properties that the numbers 85 and 95 have.

It’s more honest to just say Indonesian ⪰ Japanese lexicographically, and quantify no more.




At my secondary school, the high-scoring wide receiver was more popular than the fat lineman. And the fat lineman was more popular than the team statistician. But you couldn’t really compare the wide receiver’s popularity to that of the actor who got most of the lead roles. They were admired in different circles, to different degrees, by different people. With so little overlap, a hierarchy must treat them as separate rather than comparable.

So popularity = a partial order (and, possibly, an inverted arborescence or join-semilattice). Sometimes there is a binary relation  between two people such that one is-more-popular than the other. Sometimes you just can’t say. And no such relation exists. (neither geoff ≻ ian nor ian ≻ geoff)

Transitivity did hold at my school, so if you were more popular than geoff, you were by extension more popular than anyone than whom geoff was more popular. ( ari, shem, zvi: arishem and shemzvi implied arizvi)

And, by definition, even I was more popular than the nullset. (thanks, mathematics)