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Posts tagged with metaphor

Doug Hofstadter is an inveterate observer of his own mind. His wonderful satirical essay on sexist language builds on a multi-year compilation of sexist words: every time he heard a word that privileges males (like “chairman”, “clergyman”, “foreman”, “manpower”) he’d write it down. After years (maybe decades?) of acutely observing language for this phenomenon, he wrote that killer essay. Only someone like that could explain the personal-ness of ideas and metaphors so well.

Around minute 33, he cuenta la cuenta of how an image/feeling from his past was called up years later, mogrified, contrasted, or intertwined with something else that he was experiencing at the time.

He also talks about what I would call “personal slang”. For example Doug says he thinks of the words “sourgrapes” as just one word. I would take this farther and say that, for me, entire stories can sometimes take up “just one mental unit”. For instance ∃ an economic argument that harsh factory labour in the Philippines is good because it’s better than picking trash out of a garbage heap. I could rattle this argument off so quickly in my head that I would be still spitting it out in person for five to ten minutes after I finished the thought to myself. What I mean is that since the argument is encoded as a single unit in my mind, it just takes me a millisecond to conceive the whole thing. So if I’m reading something on-topic that misses the point of that argument, I can quickly call up the entire speech and dismiss what I’m reading in the next millisecond.

I have many other personal metaphors (some of them are more original than repeating a popular argument). Some of them come from a mathematics textbook. And from these, a blog is born.

 

Do you ever watch a really awesome musical and wish, when you’re leaving the theatre, that people in regular life would burst out in choreographed song-and-dance? Or that no-one would look askance at you if you belted a song while you’re walking down the street, feeling intensely cheerful or melancholic? …Well, I’ve wished that before…and I’ve also wished that I could use mathematics in my speech (it’s sometimes possible with the right crowd, if there’s a chalkboard handy). And that’s what i’m trying to accomplish here.

A few months before I started this blog, I was auditing a class on the applications of differential geometry to psychology. I realised that due to X years of mathematical modelling and reading books about median voter theorems, hyperfunctions or Fourier transforms, my vision of the world had become totally abnormal.

Over the past several years I’ve felt increasingly like I need to share these shapes. You know, because it’s extremely weird to want other human beings to understand what you’re thinking / feeling on a deeper level, and stuff. And I feel like there are certain ideas that I have where some abstract idea from a maths book interpenetrate with whatever else is going on in my life, such that I think these unshareable thoughts like:

The difference between this blog and what you’d read on Wikipedia or in a paper or textbook is, I’m trying really hard to not be didactic. Sometimes I do define something or share an “aha!” moment in case it’s also the “aha!” for someone else. But when I do that it’s as preparation for a story I want to tell later. Like my wish that people would spontaneously break out in song-and-dance, I wish that I could just talk normally using these metaphors—like at a party or something, and people would know what i mean, and it wouldn’t seem either braggy or academic.

 

In Plato’s vision of an ideal government, The Republic, the “gold” people—the philosopher-kings—would train during their youth in corporalità and geometry, among other things. Geometry was supposed to teach the philosopher-kings abstract reasoning and about the true forms of the world.

Well, mathematics came a long way in the 19th & 20th centuries, such that I find the Platonic ideal much more believable now. For me, I want to understand quasimetrics, homotopy, cohomology, CW-complexes—those are the imagination-tools I want to have to see the world. If a deep understanding of surgery theory is more valuable to Gregory Perelman than a million bucks—that’s pretty convincing proof, to me, that this is personal language worth acquiring.




I’m working on a longish post about dichotomies. It’s going to be about mathematical objects that can serve as metaphors to think beyond binary opposition.

In researching the article, I found the following in the Internet Encyclopedia:

According to Jacques Derrida,[citation needed] meaning in the West is defined in terms of binary oppositions, “a violent hierarchy” where “one of the two terms governs the other.”

I don’t know if Derrida actually said that. But I can already think of a counterexample from mathematics.

 

The number √−1 is logically equivalent to −√−1. In other words i and −i are indistinguishable.

Doug Hofstadter was fond of making this point to us.

  • Complex conjugation would work the same.
  • Addition, subtraction, multiplication, and division would work the same.
  • The anticlockwise direction in the complex plane is arbitrary. If the “southern” i were the one we currently call +i, then we’d do things clockwise and everything would work out the same.
  • So integration and differentiation would work the same as well.
  • (On the other hand, −1 is not the same as +1. −1 instantiates an “alternating” pattern whereas +1 instantiates a “stay the same” pattern, under multiplication.)
  • It’s like group theory. Say we’re talking about the group P₃. Any of the atoms could be called “first”, “second”, or “third”. It wouldn’t matter.

    What matters is the structure, the relationships, the way they do things. Neither is “worse”, “better”, “before”, “after”, or “dominated by” the others—they simply relate to each other in the P₃ way.

So right there, you’ve got a binary opposition where neither term governs the other.




Leonardo da Vinci’s ability to embrace uncertainty, ambiguity, and paradox was a critical characteristic of his genius. —J Michael Gelb Say you want to use a mathematical metaphor, but you don’t want to be really precise. Here are some ways to do that: Tack a +ε onto the end of an equation. Use bounds (“I expect to make less than a trillion dollars over my lifetime and more than $0.”) Speak about a general class without specifying which member of the class you’re talking about. (The members all share some property like, being feminists, without necessarily having other properties like, being women or being angry.) Use fuzzy logic (the ∈ membership relation gets a percent attached to it: “I 30%-belong-to the class of feminists | vegetarians | successful people.”). Use a specific probability distribution like Gaussian, Cauchy, Weibull. Use a tempered distribution a.k.a. a Schwartz function. Tempered distributions are my favourite way of thinking mathematically imprecisely. Tempered distributions have exact upper and lower bounds but an inexact mean and variance. T.D.’s also shoot down very fast (like exp{−x²} the gaussian) which makes them tractable. For example I can talk about the temperature in the room (there is not just one temperature since there are several moles of air molecules in the room), the position of a quantum particle, my fuzzy inclusion in the set of vegetarians, my confidence level in a business forecast, ….. with a definite, imprecise meaning. Classroom mathematics usually involves precise formulas but the level of generality achieved by 20th century mathematicians allows us to talk about a cobordism between two things without knowing everything precisely about them. It’s funny; the more advanced and general the mathematics, the more casual it can become. Like stingy stickler things that build up to a chummy, whatever-it’s-all-good.   Our knowledge of the world is not only piecemeal, but also vague and imprecise. To link mathematics to our conceptions of the real world, therefore, requires imprecision. I want the option of thinking about my life, commerce, the natural world, art, and ideas using manifolds, metrics, functors, topological connections, lattices, orthogonality, linear spans, categories, geometry, and any other metaphor, if I wish.

Leonardo da Vinci’s ability to embrace uncertainty, ambiguity, and paradox was a critical characteristic of his genius. —J Michael Gelb

Say you want to use a mathematical metaphor, but you don’t want to be really precise. Here are some ways to do that:

  • Tack a +ε onto the end of an equation.
  • Use bounds (“I expect to make less than a trillion dollars over my lifetime and more than $0.”)
  • Speak about a general class without specifying which member of the class you’re talking about. (The members all share some property like, being feminists, without necessarily having other properties like, being women or being angry.)
  • Use fuzzy logic (the ∈ membership relation gets a percent attached to it: “I 30%-belong-to the class of feminists | vegetarians | successful people.”).
  • Use a specific probability distribution like Gaussian, Cauchy, Weibull.
  • Use a tempered distribution a.k.a. a Schwartz function.

Tempered distributions are my favourite way of thinking mathematically imprecisely.

Tempered distributions have exact upper and lower bounds but an inexact mean and variance. T.D.’s also shoot down very fast (like exp{−x²} the gaussian) which makes them tractable.

For example I can talk about the temperature in the room (there is not just one temperature since there are several moles of air molecules in the room), the position of a quantum particle, my fuzzy inclusion in the set of vegetarians, my confidence level in a business forecast, ….. with a definite, imprecise meaning.

Classroom mathematics usually involves precise formulas but the level of generality achieved by 20th century mathematicians allows us to talk about a cobordism between two things without knowing everything precisely about them.

It’s funny; the more advanced and general the mathematics, the more casual it can become. Like stingy stickler things that build up to a chummy, whatever-it’s-all-good.

 

Our knowledge of the world is not only piecemeal, but also vague and imprecise. To link mathematics to our conceptions of the real world, therefore, requires imprecision.

I want the option of thinking about my life, commerce, the natural world, art, and ideas using manifolds, metrics, functors, topological connections, lattices, orthogonality, linear spans, categories, geometry, and any other metaphor, if I wish.




Vector fields

Vector fields pervade. I think about them every time I throw a frisbee in wind.

In a social context, I think about vectors of intent attached to people talking at a party — vectors of flirtation, vectors of eye movement and attention, and more abstract vectors representing jokes, topics of discussion, dance moves, or songs that are playing.

Also when I’m thinking about international trade or just the local flows of money in my community, it’s natural to use the vector-field metaphor to “see” the flows.

Electric field of 3 point charges

I also think of history (at different scales) using vector fields. Wars are like nation-states or soldiers aiming weapon vectors at each other. Commerce has many more dimensions since goods and money are both multi-dimensional. Ideas and culture also transmit in a vector-field-like way. Epidemics — well, there’s a reason mosquitoes are referred to as disease vectors.

Information flows, thoughts, internet bits — anything that can be characterised as a vector, you can expand that thought into a more complicated vector-field thought. Turbulent versus laminar flows of ideas and culture? Maybe it wouldn’t deserve a research grant but it’s fun to think about.

There are pretty obvious physical examples of vector fields — rivers, wind, geological eroding forces, magnetism, gravity, flying machines, bridge engineering, parachute design, weather patterns, your entire body as it does martial arts or dances. Being measurable, these are the source of most of the neat vector-field pictures you can find online.

(Or you find programmatically simple theoretical vector fields like the above: a vector facing [−y,x] is attached to every point (x,y). So for instance the point (3,4) has a pointer going out −4 south and 3 east, which equals a total force of 5.)

The same metaphors and visualisations, though, are open to interpretation as social or economic variables too. For example a profitable business is more of a “sink” or attractor for 1-D money flows, while a benefactor is a “source”. Likewise a blog that receives lots of links and traffic is a 2-D attractor on the graph of the web — and Google recognises that as PageRank.

I know of at least one paper that tries to best economists’ utility theory models by imagining a person on a 1-D vector field, trying to avoid minus signs and find a path to plus signs in the space.

Lotka-Volterra-Goodwin Predator-Prey Model

There is also a game theory connection. Basins of attraction can draw you into a locally optimal place that is not globally optimal. You can imagine examples in the evolution of animals, in company policies or business practices, or in whole economic systems.

On the one hand it may seem frivolous or crackpottical to generalise these concrete physical concepts to the social or psychological. On the other hand — that’s the power of the generality of mathematics!

Vector fields are surfaces or spaces with a vector at each point. That’s the mathematical definition.