isomorphismes

Counting generates from the programmer’s successor function ++ and the number one. (You might argue that to get out to infinity requires also repetition. Well every category comes with composition by default, which includes composition of ƒ∘ƒ∘ƒ∘….)

But getting to one is nontrivial. Besides the mystical implications of 1, it’s not always easy to draw a boundary around “one thing”. Looking at snow (without the advantage of modern optical science) I couldn’t find “one snow”. Even where it is cut off by a plowed street it’s still from the same snowfall.

And if you got around on skis a lot of your life you wouldn’t care about one snow-flake (a reductive way to define “one” snow), at least not for transport, because one flake amounts to zero ability to travel anywhere. Could we talk about one inch of snow? One hour of snow? One night of snow?

Speaking of the cold, how about temperature? It has no inherent units; all of our human scales pick endpoints and define a continuum in between. That’s the same as in measure theory which gave (along with martingales) at least an illusion of technical respectability to the science of chances. If you use Kolmogorov’s axioms then the difficult (impossible?) questions—what the “likelihood” of a one-shot event (like a US presidential election) actually means or how you could measure it—can be swept under the rug whilst one computes random walks on trees or Gaussian copulæ. Meanwhile the sum-total of everything that could possibly happen Ω is called 1.

With water or other liquids as well. Or gases. You can have one grain of powder or grain (granular solids can flow like a fluid) but you can’t have one gas or one water. (Well, again you can with modern science—but with even more moderner science you can’t, because you just find a QCD dynamical field balancing (see video) and anyway none of the “one” things are strictly local.)

And in my more favourite realm, the realm of ideas. I have a really hard time figuring out where I can break off one idea for a blogpost. These paragraphs were a stalactite growth off a blobular self-rant that keeps jackhammering away inside my head on the topic of mathematical modelling and equivalence classes. I’ve been trying to write something called “To equivalence class” and I’ve also been trying to write something called “Statistics for People Who Program Computers” and as I was talking this out to myself, another rant squeezed out between my fingers and I knew if I dropped the other two I could pull One off it could be sculpted into a readable microtract. Leaving “To Equivalence Class”, like so many of the harder-to-write things, in the refrigerator—to marinate or to mould, I don’t know which.

But notice that I couldn’t fully disconnect this one from other shared-or-not-shared referents. (Shared being English language and maybe a lot of unspoken assumptions we both hold. Unshared being my own personal jargon—some of which I’ve tried to share in this space—and rants that continually obsess me such as the fallaciousness of probabilistic statements and of certain economic debates.) This is why I like writing on the Web: I can plug in a picture from Wikipedia or point back to somewhere else I’ve talked on the other tangent so I don’t ride off on the connecting track and end up away from where I tried to head.

The difficulty of drawing a firm boundary of “one” to begin the process of counting may be an inverse of the “full” paradox or it may be that certain things (like liquid) don’t lend themselves to counting in an obvious way—in jargon, they don’t map nicely onto the natural numbers (the simplest kind of number). If that’s a motivation to move from discrete things to continuous when necessary, then I feel a similar motivation to move from Euclidean to Hausdorff, or from line to poset. Not that the simpler things don’t deserve as well a place at the table.

We thinkers are fairly free to look at things in different ways—to quotient and equivalence-class creatively or at varying scales. And that’s also a truth of mathematical modelling. Even if maths seems one-right-answer from the classroom, the same piece of reality can bear multiple models—some refining each other, some partially overlapping, some mutually disjoint.

One example of a total ordering ≤ is the “hotness” scale from 1–10.

Because of the widespread disagreement about the meaning of the numbers, the only thing one can infer based on a man’s rating of a woman is that she is more attractive than those who score below her.

YOUTH

The Hotness Scale derives, I think, from a need to explain one’s tastes to peers and hear them justified.

It typically surfaces in sleepover conversations like this:

• Chris (secretly likes Kelly Russell): Who do you think is hotter: Liz Jones, or Kelly Russell?
• Dave: Are you kidding?! Liz Jones is waaaay hotter.
• Chris: Oh, yeah. I mean, obviously. I was just checking. I just meant, you know, that I think Kelly Russell is like maybe a 7.
• Dave: Are you crazy?! She’s like a 2.
• Chris: Come on, 2 is like people who have skin grafts. 2 is people who were burned in fires.
• Dave: Whatever. Maybe.
  (sheepish retreat to celebrity hotness)
  But man, Jane March is the hottest woman on the planet.
• Chris: No way, Jenny McCarthy is hotter.

I’m a little embarrassed to admit (though I’ll still admit it) that when my best friend and I started using the hotness scale, we scored girls in different categories, like

1. tan
2. boobs
3. personality
4. legs
5. I forget what else
6. overall score

Yeah, we were really cool. (Also we were really twelve.)

DISPUTES

Fast forward to college. We guys were joking about using the ten-point scale, which by then was passé (although I did once use the phrase “a Bloomington 6 is a hometown 9”). We were trying to answer, what is the difference between a 6 and a 7 anyway? And is the distance between 6 and 7 greater or less than the distance between a 9 and a 10?

Everybody had taken calculus by this point so statements involving derivatives were bandied about (even though none of us meant to use real numbers … it was calculus as metaphor).

One guy proposed that each number should correspond to a decile —

• 1 to the ugliest decile,
• 10 to the hottest decile,
• and so on.

 Someone else said that one’s initial reaction put a girl either in

• the >4 (most of the time) or
• <5 — but that since no one would ever hit on someone in the latter category,
• in fact 1≈2≈3≈4.

Another said that he never assigned a 10 to anybody because that would mean he had met his wife. Um, yeah … we were still super cool.

Also contentious was whether each of us accepted the truth of whatever our own numerical ranking was. All I know is that whatever I said my score was, I secretly hoped it was 2 points higher.

There was a lot of inconsistency to the scores, which is why I’m bringing this up under the topic of ≤ rank without distance measure. Although I would wager that transitivity is violated, so perhaps this scale does not have a rational basis.

PEOPLE CAN BE ATTRACTIVE IN DIFFERENT WAYS

As I’m writing all of this I desperately want to jump ahead to partial orderings. But I haven’t defined them yet and I refuse to link to Wikipedia, so I’ll have to put that topic off.

Suffice to say that attraction is a perfect jumping-off point for one further generalization I want to make in order to get mathematics into bed with human experience.

Not everyone can be ranked side-by-side against everyone else. People can be attractive for different reasons (multiple >’s) and some people you just aren’t comparable. All of these reasons and more are great justifications for switching to posets.

POSTSCRIPTA

I’d be interested to hear other interpretations of the hotness scale, or other scales of attractiveness, and how they evolved over time. What measure, if any, do you give guys when you’re in your 30’s or 40’s?

PS When I was 24, my girlfriend told me that “24 is a very hot age”. Ha ha.

PPS Tim Ferriss claims to be able to quantify the difference between a 6 and a 9. Tell that to Jimi Hendrix.