Posts tagged with Hoosiers

## Something Stupid That Stupid Magazines and Blogs Said Is True, Is Not True.

Peter Todd has been misquoted about the mathematics of dating here, here, here (here), here, here, here, herehere, here, here, and in at least five trillion issues of Cosmo. (Surprisingly, this and this did not misquote him.) It’s enough to make me want to write a strongly worded DEAR SIR to the Hearst Tower.

Here is what they say:

• Only after you’ve dated twelve people, are you ready to decide who’s “The One”!

An even wronger version of the story goes like this:

• The twelfth guy you date — he’s The One! Science says so! No pressure!!!!!!!

Not only is this wrong, but I’ve heard Peter rant in person, specifically about these misquotations. The problem he studies is known colloquially as "The Search for a Parking Space”.

1. When you arrive at the movie theatre, you circle around the car park until you see an opening. (Let’s assume it’s below freezing outside.)
2. When you see that opening, you can immediately tell how far away it is from the theatre. So you know how far you will have to walk in the cold.
3. At that moment, you have to decide whether to drive on (keep looking for somewhere closer) or accept the probably-imperfect husband — oops, I mean parking space — that you’re staring straight in the face (oops, I mean tarmac).
4. You can’t back up; you can’t see ahead; all you can do is remember the past, guess about the future, and assess the situation you’re in. That’s all you’ve got to go on. Try to solve that problem optimally.

The paper that’s being referenced (though apparently not read) in these magazines deals with an even stricter problem, known as "The Vizier Wants to Keep His Head":

1. In this version of the blind forward-search problem, the greedy, vindictive, lazy Prince has to choose a wife.
2. Being lazy, he tasks the Vizier with solving his problem. Being vindictive, if the Vizier gets it wrong, the Vizier loses his head. Being greedy, the Prince wants the Vizier to find him the wife with the richest dowry.

(I believe dowry is chosen because it’s seen as a one-dimensional, objectively valuable quantity — as opposed to beauty, which is multifaceted and arguable. If we’re talking about various land holdings, I think dowry would also be multifaceted; that things have a single price is an illusion <link> of simplistic economic thinking.

Imagine a woman whose family had holdings in modern-day Lebrija, Huelva, Palma del Condado, Aracena, and Ayamonte. Each taxable area will bring in unpredictable revenues year upon year, and the natural beauty of each estate is just as disputatious as a woman’s face. So how is that a one-dimensional value? Oh, well. The point is to assign a scalar to each woman.)

3. The debutantes enter the Prince’s chamber one at a time; as each enters, a courtier reads her name and family holdings. So the Prince and Vizier assign a scalar to that maiden. Then the Prince either proposes marriage or declines.

4.  Once an heiress has been declined, the Prince can’t call her back. In other words, even if he thinks to himself: “Crap! B_tch Number 37 had a nice rack and a fabulous estate in Milano. I should have gone with her!”, that’s just too bad. Even a handsome, powerful, jerk of a Prince can’t un-dump a ladyfriend.

5. So the Vizier is set up a similar, but more constrained, problem to the Car Park Dilemma. Except the Prince can’t circle around the way a driver could.

6. Also, this is important: exactly one-hundred dames will appear before the prince. The solution changes if an infinite progression of dames (or even just all the singles in your greater metropolitan region of choice) paraded before him.

7. If a richer girl is to be found among either the post-wife sequence of the pre-wife sequence of heiresses, off with the Vizier’s head.

Given that problem: pick the highest scalar from a forward-blind, one-by-one sequence of scalars, the Vizier maximises his probability of living past the ritual (to something like 30%) with the following strategy:

1. Observe the wealth / beauty / scalar value of the first 12 women.
2. Whatever is the highest wealth / beauty / scalar out of that group, becomes your “aspiration level” A.
3. As soon as you see an heiress with wealth/beauty/scalar ≥A, tell the Prince to marry her.

Again, that strategy doesn’t make the Vizier win (i.e., it doesn’t make you pick the perfect boyfriend every time); it merely maximises the chances of maximisation, within this narrowly specified problem.

So here are the reasons the magazines & blogs are wrong:

• A boyfriend is not a scalar.
• Who says that a date equals a sample? I’ve been getting to know the human race my whole life. Every day I spend single, married, or it’s complicated — I am learning more information that can be used to set my aspiration level for a partner.
• You can go back sometimes — either to rekindle a relationship that, in retrospect, was red-hot, or to revisit a crush you didn’t get far enough with to make things awkward.
• There aren’t just 100 boys to look through. Let’s face it, there might as well be an infinite number of fish in the sea.
• Um: a boyfriend is not a scalar. Love depends on you as well; if you could reduce your feelings to a scalar, you’d still want to model the relationship as a 2-equation dynamical system. Interplay; choices; reactions.

SCIENCE IS MORE EXCITING THAN MAGAZINE HYPE

The original paper is called “Satisficing in Mate Search”. (I couldn’t find it online). Here is much, much more material on both data on dating and the science of thinking smarter by Dr. Todd.

You can also read Simple Heuristics That Make Us Smart (it’s on my to-read list — and it contains “Satisficing in Mate Search”), and if you look at Amazon’s similar books for the title you’ll come across all kinds of fascinating stuff: about Bayes’ rule, thinking from the gut, less is more, why it’s good to be stupid, willpower, and even an intro to game theory. (I haven’t read that particular treatment, but I do recommend reading just-a-little-bit of game theory as an awesome way to expand your imagination.)

You can get instant gratification with a free chapter of each, so these popular treatments are just as candy-like as Wired or Cosmo.

MORAL OF THE STORY

Just like with modern physics, this modern psychological science is super interesting. Way too interesting to justify wasting time on false and farcical narratives that totally miss the point.

To gild refined gold, to paint the lily, to throw perfume on a violet, … is wasteful and ridiculous excess. —King John

## Space

The word ‘space’ has acquired several meanings, which is what you would expect of such a sexy, primitive, metaphorically rich, eminently repurposeable concept.

1. Outer space, of course, is where cosmonauts, Hubble telescopes, television satellites, and aliens reside. It’s ℝ³, or something like that.

2. Grammatical spaces keep words apart. The `space` bar got a little more exercise than the `backspace` key while I was writing this list.

3. Non-printable area (space) is also free from ink or electronic text in newspapers: ad space. Would you like to buy one?

4. Closely related is the negative area in sculpture, architecture, and other visual arts.

5. Or in music. Don’t forget to “play” the notes you don’t play, Thelonious!
6. Or the space you need to give someone in a relationship, if you want to allow them to be themselves whilst also being with you.
7. Space on my hard drive to store an exact digital replica of all my vinyl? This kind of space also applies to human memory capacity, computer RAM, and other electronic pulsings which seem rather more time-based than spatial & static.

8. Businessmen refer to competitive neighbourhoods: the online payments space; the self-help books category; the \$99-and-under motel space; and so on.

9. Space as distinct from time. Although cosmologists will tell you that spacetime is a pseudo-Riemannian manifold which looks locally like ℝ⁴, a geographer or ecologist will tell you that locally space looks like ℝ² (since we live solely on the surface of the Earth).

I believe the ℝ² view is also taken by programmers who geotag things (flickr photos, twitter tweets, 4square updates): second basement = 85th floor and canopy = rainforest floor as far as that’s concerned.

Both perspectives are valid. They’re just different ways of modelling “the world” with tuples. Is it surprising that cold, rigid, soulless mathematics allows for different, contradictory viewpoints? Time is like space in the grand scheme of things, but for life on Earth time-averages and space-averages are very different.

10. Parameter space. The first graphs one learns in school plot input x versus output ƒ(x).

But another kind of plot — like a solid liquid gas diagram

— plots input a versus input b, with the area coloured or labelled by output ƒ(x). (In the case of matter’s phases, the codomain of ƒ is the set {solid, liquid, gas, plasma} rather than the familiar .)

• When I push this lever, what happens? What about when I push that one?
• There are connections to Fourier spectrum.
11. Phase space. Paths, orbits, and trajectories taken through other spaces. Like the string of (x₍ᵤ₎,y₍ᵤ₎,z₍ᵤ₎)-coordinates that a water rocket takes across the lawn. Or the path of temperature (temp₍ᵤ₎) during a year in Bloomington.

Roger Penrose uses the example of the configuration space of a belt to explain that phases can happen on non-trivial manifolds. (A belt can take on as many configurations as a string, plus it can be twisted into a Moebius band, but if it’s twisted twice that’s the same as twisted zero times.)

[Sorry, I don’t have a Unicode character for subscript t, so I used u to represent the time-indexing of path variables. Maybe that’s better anyway, because time isn’t the only possible index.]

1. Personal space. I forgot personal space. Excuse me; pardon me.

2. All of the spaces above are like an existing nothing. The space between your arm and your chest, the space where I draw—all of these are conceptually “empty” but impinge on and interact with the rest of reality.

All of those senses of the word are completely nothing alike to how mathematicians use the word. Mathematicians mean “stuff plus structure to the stuff” which is not at all like the other spaces.

Abstract spaces.
These are best understood as ordered tuples, i.e. “Things plus the relationships and desired interpretation of those things.” The space—more like “the entire logical universe I’m going to be talking about here”—is supposed to contain EVERYTHING you need, in order to work with any of the parts. So for example to use a division sign ÷, the space must include numbers like and . (Or you could just do without the ÷ sign. You can make a ring that’s not a division ring; look it up.)

• A Banach space is made up of vectors (things that can be added together), is complete (there are enough things that infinite limit sequences make sense), with a notion of distance (norm), but not necessarily angle. Also two things can be 0 distance away from each other without being the same thing. (That’s unlike points in Euclidean space: (2,5,2) is the only thing 0 away from (2,5,2)).
• A group is complete in the sense that everything you need to do the operation is included. (But not complete in the way that Banach space is complete with respect to sequences converging. Geez, this terminology is overloaded with meanings!)

• A vector space is complete in the same way that a group is. In the abstract sense. Again, a vector is “anything that can be added together”. The vectors’ space completely brings together all the possible sums of any combination of summands.

For example, in a 2-space, if you had (1,0) and (0,1) in the space, you would need (1,1) so that the vector space could be complete. (You would also need other stuff.)

And if the vector space had a and b, it would need to contain a+b — whatever that is taken to mean — as well as a+b+b+(a+b)+a and so on. In jargon, “closed under addition”.
• A topological space (confusingly, sometimes called “a topology”) is made up of things, bundled together with the necessary overlap, intersection, union, superset, subset concepts so that “connectedness” makes sense.

• A Hilbert space has everything a Banach space does, plus the notion of "angle". (Defining an inner product is as good as defining an angle, because you can infer angle from inner multiplication.) ℂ⁷ is a hilbert space, but the pair ({0, 1, 2}, + mod 2) is not.

• Euclidean space is a flat, rigid, stick-straight, all-joins-square Hilbert space.
• To recap that: vector space  Banach space  Hilbert space, where the  symbol means “is less structured than”.

Topological spaces can be even more unstructured than a vector space. Wikipedia explains all of the T0 T1 ⊰ T2 ⊰ T2.5  T3  T3.5  T4 ⊰ T5 ⊰ T6 progression which was thoroughly explored during the 20th century. (Those spaces differ in how separated “neighbours” are taken to be.)

I don’t mean to imply that these spaces can only be thought of as tuples: ({things}, operations). There are categorical ways to understand them which may be better. But don’t look at me; ask the ncatlab!

1. Lastly, sometimes ‘space’ just means a collection of related things, without necessarily specifying, like above, the tools and viewpoints that we take to their relationships.
• The space of all possible faces.
• The space of all possible boyfriends.
• The space of all possible songs.
• The space of all possible sentences.
• Qualia space, if you’re a theorist of consciousness.
• The space of all possible romantic relationships.
• The space of all possible computer programs of length 17239 bytes.
• Whatever space politics occupies. (And we could debate about that.)
• (consumption, leisure, utility) space
• The space of all possible strategy pairs.
• The space of all possible wealth distributions that sum to W.
• The space of all bounded functions.
• The space of all 8×8 matrices over the field ℤ₁₁.
• The space of all polynomials.
• The space of all continuous functions from [0,1] → [0,1].
• The space of all square integrable functions.
• The space of all bounded linear operators.
• The space of all possible models of ______.
• The space of all legal configurations of the Rubik’s cube.

(Some of these may be assumed to come packaged with a particular set of interpretations as in the previous ol:li.)

Before you and I can have a productive discussion, we need to agree on the meanings of the words that we use.
Grandpa

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