Math 105/Fall 2012

Math 105, SF Art Institute, Fall 2012

Systems, Networks and Strategies

This course surveys contemporary thinking about complex systems, coexistence and strategy, through a mathematical lens. We will explore system theories, cooperation, networks, and related ideas. We will use them as a framework to develop relevant math concepts, such as sets, algebra and statistics, and simultaneously explore their social context and think critically about ways to use and question them. Students will gain broadly applicable math skills and resources to develop them further, and a survey of culturally relevant discourses. The instructor will work with students to develop class projects relevant to their interests. Students will be evaluated on a mix of coursework and projects.

Lecture Wednesday 7:30PM - 10:15PM, Main Campus Building, Room 20B

• syllabus (odt)

                             August              September        
                      Su Mo Tu We Th Fr Sa  Su Mo Tu We Th Fr Sa  
                                1  2  3  4                     1  
                       5  6  7  8  9 10 11   2  3  4  5  6  7  8  
                      12 13 14 15 16 17 18   9 10 11 12 13 14 15  
                      19 20 21 22 23 24 25  16 17 18 19 20 21 22  
                      26 27 28 29 30 31     23 24 25 26 27 28 29  
                                            30                    

      October               November              December        
Su Mo Tu We Th Fr Sa  Su Mo Tu We Th Fr Sa  Su Mo Tu We Th Fr Sa  
    1  2  3  4  5  6               1  2  3                     1  
 7  8  9 10 11 12 13   4  5  6  7  8  9 10   2  3  4  5  6  7  8  
14 15 16 17 18 19 20  11 12 13 14 15 16 17   9 10 11 12 13 14 15  
21 22 23 24 25 26 27  18 19 20 21 22 23 24  16 17 18 19 20 21 22  
28 29 30 31           25 26 27 28 29 30     23 24 25 26 27 28 29  
                                            30 31                 

15 weeks.

To do

• Make outline of weeks.
• Flesh out later subjects.
• Algebra and other calculations
• Examples and activities.

• Peer instruction stuff:
  • Break into discrete little concept units
  • use video lectures for some parts
  • Concept tests for each
  • Short lectures for each
  • Reading and homework to match

Video:

• BBC, Secret Life of Chaos?
• Visionaries: In Grave Danger of Falling Food (on permaculture)
• http://topdocumentaryfilms.com/six-degrees-of-separation/
• http://topdocumentaryfilms.com/the-trap/ (Adam Curtis, on game theory and individualism)
• http://topdocumentaryfilms.com/how-music-works/ (Note to self, move this to next semester's music class notes)

Leftover parts of outline

• Have to go over some algebra, which might be review for some and not for others.
  • Where does the system stop? At a place where $f\left(x\right)=x$. Where's that?
    • Develop the specific homework questions and be sure of what's needed.
  • Maybe some simple quadratic cases? (I.e. the logistic map $x\mapsto rx\left(1-x\right)$.)
    • squares, exponents, powers.
    • If necessary, FOIL, factoring.
  • http://www.venturaes.com/curriculum/algebra_concepts/ might be useful to refer to while planning
• Attractor as function of initial conditions, as function of parameters.
• make sure to work with idea of spaces

• Something about spaces of possibilities and "lines of flight"?
• complex systems
  • ants, brains, etc
  • cellular automata incl. game of life
  • nonlinear corn starch
  • artificial life
  • Kauffman's origins of order (introduce some network ideas)
• Closed vs. open-ended creative dynamics
  • Catalytic chemistry, evolution, learning, creativity, culture
  • Jets, eggs and whatever.

Week 1: August 29

• Qigong intro.

• Welcome. What is this class.

• Go around, intro, attendance.

• Talk about the themes of the class: systems, networks, and strategy.
  • two layers
  • why learn about systems?
    • to understand what people are talking about: climate change; social change; economics; physiology; ecology; relationships, etc.
    • aesthetic reasons: a way many prefer to see the world; and a source of beauty and interest
    • useful in making sense of the world and being effective
• some examples: climate change; physiology; cosmology; economics
• something more or less experiential to motivate the coming weeks' focus on numbers
  • steady state balance of supply/demand?
  • body temperature

• Orientation toward stereotype threat:
  • Maybe mention gender and math (the 6th-grade thing), as well as race, and say it's also cool if people do feel good about the subject and like it!
  • it happens to various people
  • math ability isn't something you're stuck with, you can develop it
• this Math Anxiety Bill of Rights. See also Math Teacher's Ten Commandments.

• Do first-day activity. Instead of doing the Tragedy of the Commons, I'm thinking of doing a Story Circle activity. This has multiple uses:
  • to get to know each other
  • to have people talk about things they're proud of about themselves or their cultures - this is supposed to be a good antidote to stereotype threat, which can be important in math anxiety.
    • it's general self-affirmation, helping reduce the threat to people's self-integrity. can be about other attributes.
    • affirm something you value about yourself
    • talk about a time you learned to do something that seemed too hard
  • and (if it works) a way of introducing some concepts and language about systems, collective behaviors, and dynamics where the whole exceeds the sum of the parts.

• talk about what just happened

• Pass out syllabus and talk about the structure of the class. Establish expectations.
• Give them the math assessment homework
  • (doc, pdf)
    • But can I do it interactive on the moodle, so students can review and
    • watch videos to bone up on this stuff?
    • http://prepworks.org/2012/04/27/math-whole-numbers-on-khan-academy/
    • A: Yes. But let's not do that this week.

To do for 8/29

• print the lecture notes (pdf).
• First-day activity.
• Print the homework (math assessment)
• Videos for at home
• Upload stuff to Moodle, maybe

Week 2: Sept. 5

Sept. 10: end of add/drop period.

• Start with last week's assessment, review basic properties of real numbers and algebra as needed.
  • There are always more of them, there are no holes in between them
  • Fractions and decimals.
  • Adding and subtracting, negative, zero, commutative, associative
  • Multiplying and dividing, fractions, inverse operations, whatever, distributive, multiplying out and factoring
  • Understanding parentheses. Understanding multiplying without a times symbol as in $2x\left(1-x\right)$ (or with a little dot, as in $3\cdot 4$). Precedence.
  • The commutative property.
  • The associative property.
  • The special roles of 1 and 0.
  • The distributive property.
  • Multiply through a set of parentheses.

• I think though that a lot of this basic arithmetic has to be considered prerequisite for this class. I could spend a whole semester teaching it, and that wouldn't serve the other students.
  • I can tutor students, or give them time to bone up using the videos. They can catch up on some arithmetic skills while we go ahead with iterating maps and stuff.
  • But also, we can do review over a few weeks while moving into the dynamics stuff.
• That was one thought, but also the class was announced without prerequisites. I don't think it's fair to grade students down for not having had algebra or prealgebra in high school. I am going to try to see how far I can go without requiring the background. This means if we need fractions and decimals, say, which we do, I have to give them what they need. I guess a short lecture plus some resources to use might be okay. (But the one I did on the first full day wasn't enough.)

ideas lecture

• the idea of a "system"
  • Find Adam Smith quote about "imaginary machines". Maybe think of the whole class as being about imaginary machines.
  • A system is a thing that has parts, and it moves or changes, and how it moves or changes depends on the state of its parts. Its moving and changing is called dynamics.
  • ecosystem; human body; nervous system; solar system; political system
• history of systems thinking in society + roundup of concepts
  • holism/reductionism; parts and wholes; structure and organization; dynamics
• some video (and commentary about what's good/bad about the videos).
  • these are sort of pro-system-theory videos, laying the path to the Adam Curtis video about what's wrong with this point of view
  • The climate system: https://www.youtube.com/watch?v=lrPS2HiYVp8
    • explain "radiation" of heat
    • carbon dioxide, methane, ozone, nitrogen and oxygen
    • "to balance this inequality" between hemispheres "the atmosphere moves heat"?
  • David Attenborough convinced about climate change: https://www.youtube.com/watch?v=H-XpUacV-TE
    • involves models
    • what's on the temperature axis?
  • regulation in the body https://www.youtube.com/watch?v=_QbD92p_EVs (or maybe not, see below)
    • in weight gain example, what's a signal?
    • define homeostasis
    • pause and talk about sodium channels; glucose and insulin
  • I like this homeostasis video better https://www.youtube.com/watch?v=_QbD92p_EVs
    • explain the oscillating curve gesture
    • disagree about positive feedback being rare in biology
    • compare overheating example to climate change issues
  • https://www.youtube.com/watch?v=igmGve7zGmQ
    • Newtonian vs quantum; chaos; complexity; systems tend to self-organize; emergent behavior
    • the invisible hand; self-interest
    • biology - organisms, not machines
  • Let's do Alan Greenspan on the global financial system without video: http://www.federalreserve.gov/boarddocs/testimony/1998/199803032.htm
  • And let's not show Wallerstein, the world system https://www.youtube.com/watch?v=h43c2uC-y4A (note: "epistemological")
    • but mention: "antisystemic movements", "the system"
  • The economic system, the political system, the prison system, criminal justice system
    • how big do you need to look? What things are in feedbacks with each other?
    • example from ISSS: is getting better CEOs an answer to climate crisis?
  • https://www.youtube.com/watch?v=5dapzdEWMRA
    • around 1:41 - "System change, not climate change"
  • "A system is an imaginary machine invented to connect together in the fancy those different movements and effects which are already in reality performed. The machines that are first invented to perform any particular movement are always the most complex, and succeeding artists generally discover that, with fewer wheels, with fewer principles of motion, than had originally been employed, the same effects may be more easily produced."
• Mathematics and the sciences
  • Physics's triumphs
  • Models, equations, "physics envy"

math lecture

• OK, let's get into dynamics.
  • Iterating a discrete-time map (with as little notation as possible)
  • At first: no notation. Call them "multiply by two", etc.
  • Later: $x\to \text{something}$. A variable. The rule, the right hand side. The right hand side is a little bit of algebra.
  • No plotting, no equals sign, no trying to solve what happens, just look at what happens, and do some thinking about it in plain English.
  • just a few simple examples. geometric growth, secular growth, geometric decay maybe.

• • Mention: the lecture notes are the textbook, so do take notes.
  • growth: lily pads, each one creates another one each day. Doubling ($x\to 2x$)
  • Qualitative behavior.
    • What direction is it going?
    • Is it speeding up or slowing down?
    • What will it do if we keep going?
    • What if we go even longer?
    • Will it ever stop?
    • Is there an upper limit?
    • How would you describe what it is doing?
  • Distinguishing the rules from the outcome
  • comparison: tripling ($x\to 3x$)
    • What's the rule there? What are some ways we could describe the outcome?
    • same models for different things - growth of a population of plants; a disease; adoption of a behavior
  • Geometric Growth might as well include interest and debt
    • Q: We're looking at things that grow... what are other things that grow?
    • at least the basic: $x\to 110\mathrm{\%}x$, say (or rather, add 10%). 100, 110, 121. Compounding.
    • Something about subprime crisis and student debt? Maybe leave that for now.
  • Q: how would we describe growth of a person?
    • small groups?
    • add one ($x\to x+1$)?
  • Decay: Divide by two ($x\to \frac{1}{2}x$)
    • Q: so what happens here?
    • (Q from homework) Can you describe the behavior of this model in words? If you don't know what initial value I have in mind, can you tell me what this model will do with it anyway?
  • Comparison: subtract one - $x\to x-1$
  • These are models for things. You can make up a model, and see whether it does what you want.
    • Make a model of the two-in-one-out game
      • How would we make this into a rule that works with numbers?
• Now revisit those with $x$.
  • A variable. The rule, the right hand side. The right hand side is a little bit of algebra.
  • ...
• How to make a model of a draining bathtub?
  • • How about $x\to x-1$?
    • Sometimes models don't do the right thing.
      • What do lily pads do in a pond, when they get big? Note "infinite growth", the "ideology of the cancer cell"
    • For bathtub, how about $x\to (x-1\text{if}x>0,0\text{if}x=0)$
• Back to the multiply-by models
  • compare $x\to 2x$, $x\to \frac{1}{2}x$. What do other numbers do? Two kinds of outcomes - what are they? What does $x\to 1x$ do?
  • what are the two kinds of models for $x\to x+\text{something}$
  • steady state vs. growth or decline. Note steady-state vs. big bang theories.
  • Oscillation example in negative-feedback video?
    • maybe steering to left/right of the center of the lane?
    • Fits a model like $x\to -\frac{1}{2}x$
  • That's 3 kinds of $x\to \text{something}x$ models, any others?
    • Sure, $x\to -2x$ of course.
    • (what would this be a model of??)
    • What are the four cases? Any special cases besides 1?
  • logistic growth, if simple growth and decay aren't enough. $x\to 2x\left(1-x/10\right)$
  • maybe a 2-variable system - predator/prey? 2-dimensional oscillation from last year?
    • $x\to x/2$; $y\to 2y$
    • $\left(x,y\right)\to \left(y,-x/2\right)$ - start with (64,32)

• Reading: "Deep Ecology - A New Paradigm" chapter of Capra, Web of Life
  • pre-lecture to fend off confusion.
  • An example of how some people think about systems. I don't agree with everything he says. Later we'll talk about problems with this kind of "system thinking".
  • 1996:
    • extinction $\leftrightarrow$ debt in Southern Hemisphere countries
    • scarcity and env. degradation, population growth $\to$ breakdown of communities, "ethnic and tribal violence that has become the main characteristic of the post-cold war era."
  • mechanistic worldview vs. holistic/ecological worldview
    • A paradigm shift
  • Cartesian

• If time to spare:
  • Joanna Macy, "The Systems View of Life" video about the Work that Reconnects:
    • https://www.youtube.com/watch?v=J7gt1YePVaE
    • https://www.youtube.com/watch?v=xgcOC8y-9_8

To do for 9/5 class

• Write lecture
  • questions, small group discussions, what
  • use questions to find out if I'm being understood
• Write homework assignment
• Reading
• Reading questions
• Contact media services about showing video

just before

• get videos ready, shut off new-email announcements etc.
• print lecture notes
• print Alan Greenspan
• print reading
• print homework

postmortem

I am so unhappy! Last year I started off by throwing too much at people all at once, assuming the functions, variables, and stuff would make sense to them. I tried to fix that this year and it didn't work! What good is it to do a 15-minute crash review of the associative law and all that? Even if it was effective, I ended up losing people by suddenly dropping the rule for simplifying fractions on them without any review. The anxiety level in the room went up and up, and I sent them home with a homework assignment that I'm afraid they're going to cry over.

Maybe I should have made two versions of the homework, one with the logistic question and one without. Also, maybe I should just insist that the class has algebra as a prerequisite.

Update: I told them not to do the math homework, I'll give them homework for the coming week.

Week 3: Sept. 12

To do for Sept. 12

• Revisit the iteration lecture, with detailed attention to introducing specific concepts.
• Write the homework and work backwards to what needs to be covered in class.
• What do they need to know when? I need to tell them so they know what to study.
• Clarify my grading policy.
  • We will do some math in this class. But I won't drop it on you without preparing you.
  • You don't have to ace it all. There are opportunities for extra credit.
• Get reading together for coming week
  • http://www.context.org/iclib/ic32/meadows/
• print + mark the reading questions, record the points

• Print lecture notes.
• Print homework questions.
• Print reading.
• Make concept questions.

Lecture notes

Math lecture.

• Look back to last week
  • What I'm trying to do in this class
    • I think I can teach some fundamental concepts about dynamics without math background.
    • Because I'm a practitioner, you get my perspective on the craft - including things that aren't in any textbook.
    • I'm trying something new with this class, and I might make some missteps. It's very important to me to be reasonable and not ask you to do things that you can't be expected to do.
    • We'll have to review some things along the way, like fractions, negative numbers, variables.
    • I'll also introduce some fundamental concepts that you probably never had, like unbounded growth.
  • I threw in a lot of things together last week, and it would have been better to take on a small number of new things at a time.
    • It might have made it hard to follow, and hard to know what's expected of you. It might have given you reason to worry about whether I'm going to put you in a difficult position by asking you to do things you aren't in a position to do, and it might have given you reason to worry about whether I can be trusted to treat you fairly. These things are important to me, and I'm committed to checking myself when I need to and being fair.

• Revisit the doubling model, and the halving model. First, just the doubling.
  • We start with the behavior of the parts. Each individual creates one more each day. Every one individual becomes two.
  • We ask a question about the number of pads. We could ask other questions, like how the sun changes their color, but we are asking this one.
  • It seems natural to consider the time period of one day. Never mind what time of day they do it. This is a choice we make, how to look at the system, which parts to leave out.
  • The system is the population. Its behavior is driven by what the parts do. The individuals are the parts.
  • How to get from behavior of the parts to behavior of the system?
  • At least two steps. First rule for the parts behavior to rule for behavior of the system. Then from system-level rule to outcome. Last week we really just looked at getting from rule to outcome.
  • How do we get the rule?
    • Here are two ways. First: At the beginning of the day, there are so many lily pads. They are all there at the end of the day, and each of them has produced a new one. So there are so many parents, and the same number of children. The original number, and that many more. Add the number to itself.
      • Draw a rectangle: row of parents above, row of children below.
      • What is multiplication? Multiplying a number by two means adding that number to itself - adding two of that number together. So adding the number of parents to the number of children means multiplying the number of either by two.
    • Second: Here are these lily pads at the beginning of the day. There are so many of them. Then each of them becomes two. Now there are that many pairs. The new number of pairs is two times the number of parents. Two plus two plus ...
      • This is adding together all the columns of the rectangles instead of the rows.
        • This is why $2\times 4$ is the same as $4\times 2$.
  • This may seem like beating a dead horse, because you already know about adding and multiplying, but it actually does something profound for us: it turns what the parts do into what the whole does. Sometimes it's not easy to do that. The reason we're doing it with these lily pads is because we can actually do this step with them, using only addition and multiplication.
  • This gives us the rule for the system: Multiply by two.
  • Now we try to understand what that rule does to the system, and what outcomes it produces.
• Some terminology!
  • We created a system by choosing to investigate the number of lily pads. In this case they are imaginary, but they could be real. Anyway, we investigate the system by creating a model. First a mental model then a mathematical one. The system is different from the actual lily pads, and the model is different from the system. We omit various things: for instance, an industrial accident or disease could decimate the pond, but it'll never happen in our model.
    • Even more so: someone pointed out last week, they will fill the pond and stop growing. Above a certain number, we shouldn't trust the model.
    • In real situations, sometimes all we have is a model (the Attenborough climate video, for example). So we have to do two things: try to understand what the model does, and try to understand when you can and can't take the model's behavior as prediction.
  • The rule "multiply by 2" is the update rule for the system's dynamics.
  • We follow the model's dynamics through time by iterating the rule.
  • When we iterate the rule, we're looking at a dynamical system. A dynamical system has a state and an update rule. In this one, the state is the number of pads.
    • The next state is determined by the current state, according to the update rule. The state is all you need to know.
  • The subject of discreteness came up, but without the words. Discrete means separate, well-defined objects. In this model there are always discrete lily pads - we can count them. This means the state is always a counting number: 1, 2, 3, .... Time is also discrete in this model: we consider one day, then the next day, then the next day. We don't try to describe what happens each second, each fraction of a second. This is how we can talk about the "next state".
    • Not discrete is continuous. We won't do continuous time because it requires more advanced math tools. We will talk about continuous state, when the time comes.
• So the behavior of the multiply-by-two system.
  • We've boiled it down to this rule, multiply by two, so the only thing remaining to get us from rule to outcome is to ask what happens when you iterate the rule - what happens when you multiply by two repeatedly?
  • We did that some last week. 1, 2, 4, 8, 16.
  • The sizes get bigger. Actually they get a lot bigger pretty quickly.
  • The expansion speeds up. The bigger it gets, the faster it grows.
  • Does it ever stop? We had some discussion about this last week. It's worth discussing.
    • First of all, we have to distinguish between the model and the actual system. Of course the pond will stop filling up. There are probably several ways that can happen. But what does the model do?
    • This connects to a fairly profound question: is there a biggest number? And it lets us look at how we answer the question.
    • Suppose there was a biggest value for the system. Then what would happen next? We would multiply it by two! The next value would be even bigger! So that value wouldn't actually be the biggest. This is contradictory, so there can't be a biggest value. This is a proof. Proof is how mathematicians figure out what's true about things we can't investigate with our eyes and hands, like very big numbers. Note: this is probably a distraction that I should skip.
  • The state of the model grows without bound. A bound would be a "ceiling value" that it never crosses.
  • Also, it grows exponentially, as someone pointed out. Another term is that it grows geometrically. That's the term for the same thing when we use discrete time steps. Why?
    • https://en.wikipedia.org/wiki/Exponential_growth#Exponential_stories: rice on the chessboard; water lilies
    • So we can pretty much sum up the answer - what's the outcome - by saying exponential growth at the rate of doubling every day.
    • We've done some quantitative work with numbers, but a lot of the time we're really interested in qualitative outcomes - do the lily pads go extinct? Will they fill up the pond?
    • Lots of things exhibit exponential growth (at least for a while) - plants, germs, contagious behavior patterns, the chain reaction inside an atomic bomb, guitar feedback. New combinations of words like "The 99%".
  • boy, this seems like a really boring lecture. :(
• OK, now let's go back to the other example.
  • If we want to look at declining population without getting into complications immediately, we shouldn't look at lily pads, or at least not small numbers of them. Consider pond scum - algae on the pond - that's cheerful, because more lily pads and less algae makes people happy.
  • So for simplicity, let's say that they don't give birth, and each day half of them die.
  • Algae are microscopic creatures, so there are really a lot of them. It's easier to measure them in pounds, or kilograms, than to try to count them.
  • Why do half of them die? Maybe because of weed killer in the water. If you are a tiny creature floating around, every day is like flipping a coin whether you encounter a lethal dose.
  • So to really make all the steps from what each cell does to what the whole does we would have to deal with probability and statistics...
• Like the growth model, lots of things behave like this model too: a bouncing ball, the temperature of a bottle of vodka in the freezer, the amount of radiation in our milk from the Fukushima meltdown
  • Let's not do a bouncing ball, because the bounces get smaller but they also come sooner and sooner
  • Bottle of vodka: especially simple if we use Celsius temperature because 0 is freezing. So if the temperature of the freezer is 0, the temperature of the bottle will keep getting closer to zero. It'll be half as much about every 30 minutes or something.
    • To completely work this one from the beginning we would have to know some things about molecules and heat.
  • Unfortunately, the amount of radiation from Fukushima takes longer to cut in half (but fortunately there isn't very much of it here in North America).
    • To do this one we would have to know some things about atoms and radiation.
  • So, good things and bad things. Anyway, let's stick with pond scum, and we can make an approximation by saying that exactly half of them die every day. It's very reasonable.
  • So the total amount of scum, in pounds, cuts in half each day. If we start with 1000 pounds, next day we have 500 pounds. If we have 2 pounds, next day we'll have one pound. And if we have one pound, next day we'll have one half pound.
  • We leave the realm of counting numbers. We are measuring now, and fractions of pounds are completely sensible for this.
    • So one half is less than one. Where is it on the number line?
    • Q: if we have one half pound, what do we have the next day?
      • A: one quarter pound. Enough to make an awful hamburger. Where is that on the number line?
    • Stop and talk about (positive) fractions and positive numbers. Use the number line, or a yardstick for height.
    • We are dealing with amounts of things rather than counting numbers. Distances, weights, fractional parts of a whole. Slicing a pie, for example. Or dividing up a gram or an ounce of something. The amount can get smaller and smaller; zero is at the bottom. These measurements are not discrete, they're continuous, although our time measurements - day after day - are still discrete. A discrete-time, continuous-state model.
    • This is what happens when our population keeps cutting in half. It keeps getting smaller, and closer to zero. Zero is the floor. In fact, each day we go half of the remaining way to zero. So our steps also get smaller and smaller - we are slowing down.
    • So it's getting smaller, and it's going to keep doing that for a good while. Eventually, in the real-life system, we'll get down to things that don't divide, just like we would with lily pads, but it's a long way off. In the model that will never happen, we just keep splitting in half forever.
    • So some qualititative behavior: the amount of pond scum always gets smaller; it shrinks more and more slowly; it never gets to zero, and it never goes beyond zero. So zero is a lower bound.
• Is that enough? If not:
  • If we start the decay with different amounts, we get the same kind of behavior.
  • Zero is a special value: what happens if we start with zero?
    • It's a fixed point. Starting away from zero, the dynamics always approaches the fixed point.
  • What about if we start the growth model with zero?

• if there's extra time to go forward: Save for next time:
  • oscillation, fluctuation.
  • Stability vs. perturbation.
  • how to write things about these using $x$ and $t$.
• Look at a range of discrete maps and their dynamics and outcomes (unbounded growth, monotonic decay, steady state, oscillating growth, oscillating decay)
  • with tangible examples - population dynamics, bouncing ball, swaying building, spring, pendulum, social dynamics, dyadic relationship...

Systems lecture.

• More about modeling this time.
• some history of math modeling
  • Math has always been about explaining the world
    • It has a sort of dual nature. It's its own world, and it's also about our world. Mystery: why is it effective at describing actual things?
  • Geometry in antiquity, Pythagoras and music
    • Euclid was Syrian, many of the Greek founding fathers studied in Egypt and what's now Ethiopia
    • Example: Use of right triangles to make buildings.
  • Enlightenment period of Europe: Newton's triumph.
    • Using equations to explain and predict how things move.
    • Unified the heavens and the earth.
    • Laws of nature. A form of the rules -> outcomes theme. Explaining the things we see as emergent from invisible laws of nature.
  • Inspired many attempts to math-ify various sciences. Still going on.
    • Laws of social movement - sociology, repeated attempts to analyze with math.
      • Some things are really hard, others you can do: example: demography and the baby boom
    • Statistics and gambling. Deciding when something's a good bet. Pascal's wager. Becomes all the tools we use to understand polls, trends, observations in general.
  • Condorcet and investigation of democracy and government
  • Economics: What drives inequality, social change, starvation... what to do... Marxism
  • Chemistry, Astronomy, Particle physics, various physical sciences - what's the shape of the universe, its history, origins, future
  • Biology and Ecology: conservation strategies, sustainable fishing, how does diversity work...
    • We've been starting with some biology models.
  • Climate models as in the video we saw. They are central to establishing what the climate is doing and why.

• 20th century, the cybernetics community
  • • it doesnt' really start with Wiener, goes back to the Chinese, the Greeks and the steam engine. Also Ampere, Cybernetique, science of governance
    • Condorcet on voting and progress?
    • WWII and emergence of idea of feedback along with operations research
      • negative feedback: guided missile.
        • Cybernetics = steersman? Governance.
      • a feedback comes with a causal circuit, which gives us a system.
      • the need to study the system, in distinction to the components
        • examples: maybe Meadows' shower example
        • lynx and hare
        • governance... managing interest rates... people organize or riot when things get bad... Bateson + Haley's debate about power, political consent
    • negative vs. positive feedback
      • video: Jimi Hendrix and/or Sonic Youth
    • People study systems, including complicated ones
      • Idea of "general systems theory"
      • graphical examples of system diagrams and models
    • things cybernetics people did
      • Cybernetics and Norbert Wiener's life
      • Is it military in nature? It seems to go to other places - study of governance, freedom, love.
      • Ashby's machines + learning (kitten learning from mouse)
      • Pask's color machine, growing computers
      • McCullough's neural networks
      • Margaret Mead, how do different practices manifest in different cultures
      • Stafford Beer, Project Cybersyn in Chile, corporate downsizing
      • Grey Walter and his dancing turtles
      • Gaia, ecosystems
    • Cybernetics in the 60s
      • became popular as way of rethinking the world, new paradigm, holistic... whole earth... connectedness...
      • Gregory Bateson, Stewart Brand and Jerry Brown
      • holistic vs linear thinking.
      • the structure of the system vs. the nature of its parts.
      • idea of a system seems to contradict idea of leadership.
    • Cybernetics and computers/machines/corporations/military/domination
      • Deleuze, society of control, globalization

• Emergence and dynamics
• Rules of games and outcomes
  • scientific explanation = rules? ("laws")
  • Rube Goldberg machines
  • Grey Walter, artificial life
• Two models that changed the world: Darwin's "invisible hand" and Smith's

• talk about games and rule-based art?

Week 4: Sept. 19

Start with discussing the homework

Math lecture

graphs and coordinates

• Plotting things, cartesian coordinates
  • A powerful way of using our visual abilities to make sense of things that we wouldn't ordinarily picture visually.
    • Interested in students' perceptions of this business, because I'm a verbal learner, and there may be more visual learners in the class, and using graphs and plots requires using both modes together.
    • Much of math is using the verbal mode. The verbal mode is privileged in our society, so there can be a lot of value judgements attached. I think of them as just different kinds of intelligence.
• Does plotting mathematical objects with coordinates and axes start with the story of Descartes and the fly?
  • Actually, no, it starts with maps, and longitude and latitude.
  • Project some maps?
  • Latitude and longitude are very different from each other
    • Latitude can be measured directly
    • Longitude is difficult - long-term competition
• Imagine we record latitude-longitude pairs during an ocean voyage across the Atlantic (or Pacific?)
  • Use the Pacific view here, and go from maybe Somalia to Thailand, so that we can have longitude and latitude numbers increase from left to right (even start from latitude zero)
  • We could put pins in the map, or make marks, and see the route of the ship.
  • Raise the screen and let the map project on the chalkboard, so I can add dots.
• Using the same system to understand things that aren't maps... took humans a while to come up with
  • though with geometry sometimes it practically is maps - laying out a building for instance
  • Other times it's not geographical, but we use the map as a metaphor to help us understand our system.
    • Talk about metaphors for a minute.
• Consider our growth system.
  • We have a sequence of pairs of numbers. Day, population size.
  • Not traveling from west to east, but traveling forward in time. We treat time as if it were space. A strange thing to do. Allows us to use our visual intuitions.
• So from last time, here's some pairs from the doubling system.
  • Act as if time and size were longitude and latitude
  • A map? If so, a map of a strange place. Wonder if it has continents....
  • So here is our trail across the map. Where does it lead?
  • It will always keep going to the right.
  • It will always keep going up, faster and faster.
  • It leads to parts of the map that are farther into the future and larger in population size.
  • What does going up faster and faster look like?
  • It has no upper bound. What would an upper bound look like?
• use the simple growth applet I made? No doubt.
  • Compare "multiply by 2" to "multiply by 3".
• Look at the decay model (multiply by 1/2).
  • We keep going half the distance down to zero.
  • compare this curve, stretching out to the right, with the measurements on a yardstick that I drew last time.
    • Again, we're using spatial location as a metaphor for other quantities.
  • We can see that it gets smaller, and gets slower and slower.
  • We can see that it has a lower bound at zero.
  • Also, check it out - if we went from right to left, the one model would turn into the other one! (This is probably a distraction)
• Some terms: axes; coordinates;
  • Go over the (3,2) question from the survey.
• independent quantity; dependent quantity; time series
  • Mention the climate-change time series in the video we saw.
• If time permits: maybe an in-class exercise:
  • Introduce the "add one" rule.
  • Hand out graph paper, ask people to try plotting it.
• You can put other things on your axes.
  • example size-lifetime plot from Tree of Knowledge? It's pretty and I'll have it scanned already...
  • somewhat advanced, maybe mention and leave the use of them to another time: supply-demand curves
  • preview of a 2-D state space?
• You can use things besides location
  • Color - heatmaps; size of points; thickness of line.
  • See Tufte's book.
  • show some nice examples. Here are two famous ones: http://blog.wolfire.com/2009/02/design-principles-from-tufte/
  • graphical summaries for medical patients http://www.edwardtufte.com/bboard/q-and-a-fetch-msg?msg_id=0003mm
  • movie box office charts http://www.xach.com/moviecharts/
• If more time:
  • Experiment with plotting one thing against another:
    • lily pads vs. pond scum
    • better: lily pads now vs. tomorrow. And pond scum now vs. tomorrow.
    • Amount of growth in lily pads vs. time
      • better: amount of growth in "add one" model vs. time

systems lecture

• We'll start on projects a little later. But let's talk a little in advance.
• Rules and systems in the art world
  • Cage, Rauschenberg, Xenakis
  • Sol LeWitt
  • Allan Kaprow
  • Herbert Brun
    • contradictions, perturbations, composing, anticommunication, false statements
  • Eno
    • One of Eno's favorite quotes, from the managerial-cybernetics theorist Stafford Beer, would become a fundamental guiding principle for his work: Instead of trying to specify it in full detail," Beer wrote in his book The Brain of the Firm, "you specify it only somewhat. You then ride on the dynamics of the system in the direction you want to go." Eno also derived inspiration from Stafford Beer's related definition of a “heuristic.” “To use Beer's example: If you wish to tell someone how to reach the top of a mountain that is shrouded in mist, the heuristic ‘keep going up’ will get him there,” Eno wrote. Eno connected Beer's concept of a “heuristic” to music.
  • 12-tone composers? Reich, Riley, Oliveros, ...
  • Kesey's Acid Tests, Brand's Trips Festival, "be-in"
    • the Well and creating environments for online communities
• Rules and games
  • Brand and New Games
  • some new games, play one or two
  • 3-puzzles game
  • Nomic
  • I don't know if these everlasting games are relevant but they're fun
  • the singing game where you switch to someone else's note
  • modifying an existing game is like evolution

• video about the last Meadows reading: https://www.youtube.com/watch?v=kIQvBYOtgMg

Base it on the Meadows reading: http://www.sustainabilityinstitute.org/pubs/Leverage_Points.pdf Things to preteach for Meadows:

• leverage points (The silver bullet, the trimtab, the miracle cure, the secret passage, the magic password, the single hero or villain who turns the tide of history)
• Jay Forrester; limits to growth (Forrester in Curtis video?)
• parameters: for example, what to multiply by.
  • size of bathtub, max rate in, rate out
• stocks
  • buffer = big stock
• flows (inflows, outflows)
  • like in 2-in, 1-out game: consider the add-one model
    • spend a few minutes on that one.
    • Now compare the 2-in, 1-out game. It's the difference that matters. Similar with banking account.
  • speaking of stocks and flows, it makes a difference whether you have savings, doesn't it?
• delays
  • shower on 4th floor
  • note also, difference you can make by changing pay-out schedule in checking account
• feedback
  • negative feedback:
    • The safety drain in a bathtub.
    • a bucket with a V-shaped gap??
    • controlling faucet + drain in bathtub; density dependence
  • positive feedback: population explosion
    • tragedy of commons = missing negative feedback?
• modeling a system in a computer
• changing the structure of the system
• self-organization = system changing its own structure?
• paradigms

• • • Circular causation and circuits?

hand out the reading and homework, and go home.

• graph paper to print http://havefunteaching.com/worksheets/math-worksheets/graphing-worksheets/

• System Operations exhibition to open at Eli Ridgway Gallery, 172 Minna St., SF. Includes SFAI's Anne Colvin.

To do before class

• Write this week's homework
• Print this week's homework
• Grade last week's homework
• Get last week's homework ready to hand back
• Homework homework
• Figure out something about projects
• Preload the Dennis Meadows video
• Print this week's reading (Donella Meadows)

Week 5: Sept. 26

Math lecture

I think this week we don't necessarily get to logistic population growth yet. I think the sequence is

• plots with different independent axes; variables; functions and cobwebbing; then changing the linear growth function to a density-dependent one.
• I should also set them up with equilibrium example. Make logistic-growth applet, show it but wait to develop the model?
• Also I wanted to give them an overview. I am teaching them a streamlined introduction to dynamical systems. This will include things like state space, parameters, equilibria, attractors, basins of attraction, bifurcation, chaos. Later in the semester we'll also talk about self-organization, complex systems, evolution, networks, swarms.
• We'll get to these things in time.
• I am going to use the terms I gave you last week in this week's lecture, however.

• I want to plot
  • Outcomes vs. parameters: bifurcation diagram
  • Next state vs. current state: update function, a.k.a. rule. Cobweb diagram
  • back to map example: snow line vs. longitude?

• • An example from our class? Multiply by 2, multiply by 3, etc.? Start with parameters and then go to names?
  • Use the growth applet (with slider, this time).
    • Look what we can do now that we have graphs. We can compare lots of these things at a glance.
    • Sometimes up, sometimes down. What do you think makes the difference?
      • A. Above or below 1!
      • Also, what does "Multiply by 1" do?
    • Two classes of outcomes and a borderline case.
    • 1 is a special parameter value. We'll see more of this kind of thing when we do bifurcations.
    • We can do that now: plot ultimate value vs. growth rate.
      • Terminology: plot something against something else.
        • We can get nimble, plot all kinds of things against other things, ask various questions.
      • let's do this one starting at, say, 1
      • starting at a whole lot of different initial values.
• We could also plot ultimate value vs. initial value.
  • In this system, it wouldn't be much of a plot except when growth rate is 1.
  • No - this is silly to do here.
    • oops, I should have taken it out of the notes. I did it in class.

• an especially useful one is next value vs. current value.
  • Let's do that one for growth rate of 2.
    • Start with 1, get 2.
      • review coordinates, introduce the ordered pair.
    • Given 2, we get 4. Given 4, we get 8.
  • An odd connection between the values on the two axis. We're doing a mental transportation from one axis to the other.
    • We would not do that with latitude and longitude!
    • But on this plane, there is this connection between the two different directions.
    • The next value becomes the current value.
  • But we don't have to follow the sequence. We can just draw the whole rule.
    • Given 3, we get 6. etc. Given 1/2, we get 1; given 0, we get 0.
    • Given 3 1/2, can we tell what we get by looking, without calculating?
  • When ready: let's do it for "Multiply by 1/2".
    • OK, it looks similar but different.
    • Any thoughts?
  • Now, what about for "Multiply by 1"?
    • This is a special model in which nothing changes!
    • Also, a special place on the plane where the values are equal.
      • Known as the diagonal.
    • The diagonal wouldn't be meaningful in longitude and latitude. Places where the numbers match aren't special places the way the North Pole is.
    • Likewise, if we're plotting a time series, the diagonal wouldn't be meaningful, because who cares if the number of lily pads equals the number of days - we could be measuring in hours and then it wouldn't be equal, but everything that matters would be the same.
    • But here it is meaningful. Well, so far it's just the location of this especially useless model, which is about the same as not having any dynamics at all. But look what it's useful for...
  • See, we start at 1 and go up to find that the next value is 2.
  • So we had a question and an answer. The question is "Starting at current size 1, what do we get next?" That's a location on the horizontal axis, or a vertical line. The answer is this point - or really it's the vertical location, or a horizontal line. We asked a question by providing one coordinate, and the answer is the other one.
  • Next thing, now that we're at 2, is to ask what comes next. That means we put 2 on the bottom axis. We take it off the left axis and put it on the bottom axis. We had this horizontal line, and now we use this vertical line. Look where they meet!
  • You can use the diagonal to shift your "now" from tomorrow to today. I.e. to make "next value" into "current value".
  • So what? It's a trick, but you can do that just by looking at one axis and then the other.
  • But look: we can make it into a shortcut. Here we are at (1,2). That is a question-answer pair: 1 today, 2 tomorrow. To ask the next question - 2 today, what tomorrow - we slide horizontally to (2,2) on the diagonal, and up from there. Skip the axes completely.
  • This is the cobweb diagram.
  • This is how you can read the outcome off the picture of the rule.
  • Do cobweb diagram for "Multiply by 2" with various starting points.
  • Do cobweb diagram for "Multiply by 1/2".
  • And for "Multiply by 1".
• The diagonal is where the next value is the current value - no motion.
  • What does it look like when the value is getting larger?
  • What does it look like when the value is getting smaller?

Maybe this week, maybe not:

• A little preview:
• Could I draw this loopy squiggle on the plane and use it to do a cobweb diagram?
  • Well, if I am here, where do I go next? There are three or five places on that vertical line.
  • In order to know how to do it, we need a particular shape: only one place above each location on the horizontal axis.
• When you have this question-answer relationship, it's a function. Something that given one thing, gives you another thing.
  • There are lots of functions in life, though we may not call them that.
  • Every person has a height and weight. Given a person, a quantity.
  • Names have length.
  • And something like "Add 1" or "Divide in half" is a function: given a number, it gives you the next number.
  • So is the population size time series: for any day, a size.
  • When we have an independent and dependent axis, we have a function, because we're talking about the same thing: we use the horizontal location as a question and get the vertical location as an answer.
• Another preview: the granovetter threshold model
  • There's a few people demonstrating in Tahrir Square. Maybe there are a lot of people who would join if the crowd was big enough - but how big? What if we could ask them?
  • So for a given size, there are so many people who would consider that big enough.
  • If we have 100 people, how many people say 100 is enough to get them to participate? More than 100, we hope.
  • So if there are so many people that will do it even if nobody else does - that might be how we got started - and so many people that will do it if 10 people do, which is at least as many as for zero - and so on up, how do we find out what happens? Cobweb.
  • And maybe the tipping example: this curve is just below the diagonal most of the way; but then if just a few people change from naysayers to enthusiasts, they raise the curve over there on the left at say, 50. But if those people are satisfied if there's 50, they're also satisfied if there's 100, and 200, ... They raise the curve all the way to the right (to the point where they came from).
    • And look, that's enough to set off all these other people, and make the whole thing catch fire!
    • That can happen, I think. (But how often are we right on the edge like that, we don't know...)
• A third preview, sometime soon we're going to consider the logistic map, which looks like this on the cobweb diagram.
  • What?
  • Well, not going into the details right now, but the relationship between current size and next size of the population goes like this.
  • Over here, it's above the diagonal, which means it goes up; over here, it's below the diagonal, which means it goes down. Where it crosses, it doesn't go anywhere. So maybe that's where it ends up?
  • Fire up the logistic population dynamics applet.
  • Small $r$: sure enough, it goes partway up and stays.
  • If $r$ is larger, it still goes to carrying capacity, but with overshoot and bouncing back and forth.
  • But look if $r$ is too large for that: the oscillations grow instead of shrinking, and never go away.
  • Beyond that: more complex oscillations, and then chaos.

systems lecture

• just take questions about the Meadows reading,
• show part of the Adam Curtis video

Reading:

• Barry Commoner's presentation on whole systems and ecological catastrophe, from Our Own Metaphor
• http://leeworden.net/lw/thresholds-1

• preteach for reading: average, standard deviation
• preteach for Commoner reading?

Week 6: Oct. 3

Overview of the class session:

• Collect the homework and pass back the old ones, and take attendance at the same time.
• Questions about the homework.
• Questions about the reading.
• Talk about the midterm and hand out the sample midterm.
  • Questions about the sample.
• Math lecture:
  • review
  • fractions
  • decimals
  • on to the logistic map (if time permits)
• break
• projects!
  • hand out the projects sheet
• second half of Curtis movie
• hand out reading
  • homework is to come next time.

review (of the math stuff)

• exponential growth and decline models.
  • growth:
    • monotonic
    • unbounded above
    • accelerates
    • multiply by 2 is an example, but so is 3, 1.2, etc.
  • decline:
    • monotonic
    • bounded below
    • decelerates
  • Both are dynamics that show up (for a while) in many different kinds of systems.
    • populations; debts; infection; explosions; new ideas; new technologies; guitar feedback
    • declining population; cooling; bouncing ball
  • They are the qualitative behavior of positive and negative feedback processes in general.
  • Neither is a good model forever. After a long decline, you arrive at the end, and after an exponential growth phase you tend to hit a limit and get a different kind of system behavior.
  • Note that it's not trivial that there are the general behaviors of self-amplifying and self-damping things - recall the "add 1" and "subtract 1" models, which are different. They do arise, but they're more exceptional.
    • • (note to self: I wonder if I'll get to present the reason why, which is the linearization of a system. I guess not, but I might want to look for a way to sneak it in from some sideways angle.)
    • We're getting to models with "limits to growth", but need to lay groundwork first.
• We looked at these two models in terms of rules and outcomes - how to start with something that follows certain rules, and figure out what the outcome of that process is.
  • We identified two different kinds of rules: the rule that the parts follow (produce 1 new offspring each day) and the rule that the whole follows (multiply the population size by 2)
    • and how to get from the parts rule to the whole rule (drawing a rectangle, which stands for how we get from addition to multiplication).
    • The update rule is powerful because you can predict the future of the state just from knowing the current state, without needing to know a bunch of other stuff.
    • and then how to get from the whole rule to the whole outcome, by iterating.
• Plotting and graphing
  • (I'm using the two words interchangeably)
  • Similar to locating things on a map. We do the same thing with more abstract things: by identifying events with pairs of numbers, we can locate them on a plane and look at patterns of events in a form that lets our visual abilities work on them.
  • There's room for a lot of creativity using the two axes, and other ways of representing things on a page/blackboard/screen, as the tools we have to work with.
  • And some standard ways.
  • Using the horizontal axis as the independent quantity.
    • The question-answer relationship; the vertical-line test for functions.
    • Time series, very useful for our systems with numerical state and update rules.
  • We also looked briefly at the growth rate as a parameter, plotting "ultimate outcome" vs. growth rate.
    • There's a general pattern here, though we haven't done enough examples yet to see that it's general: qualitatively different outcomes for different ranges of parameters, with borderline cases separating them. Like different regions on a world map.
    • Threshold values. Knowing where they are can mean knowing how a system could be radically different, and maybe even knowing what to do about it.
    • We could have also plotted ultimate outcome vs. starting pop size.
  • But instead we got into plotting the rule rather than outcome: we visualize the rule by plotting next state vs. current state.
    • In this form, the doubling system looks like a straight line, and the halving system looks like a different straight line.
    • It's the relationship with the diagonal that makes the difference between the qualitatively different outcomes, and explains the borderline case in between.
    • And we used cobwebbing to read off long-terms outcomes from the diagram of the rule, without needing to know the symbolic ("multiply by two") version of the rule.

• A preview of things to come (maybe even today):
  • Using symbols to describe rules and outcomes, and learn things about them.
  • Things that happen in a population model that includes limits to growth.

fractions

• I promised to talk about how decimals and fractions relate (why 1/2 is sometimes called 0.5) but then I didn't. So I need to do that.
  • Needs to present them as proportions, in the way I'll use for the logistic map.
• Mini-lecture on fractions and decimals!
  • Fractions and decimals are two ways of talking about the same things.
  • A fraction describes a portion of something.
    • A simple example: part of a pie.
    • I might cut the pie into quarters. Quarters is Latin for fourths, which we also call them.
    • I might eat one fourth and leave the other three. We have developed special notation for this: we write $\frac{1}{4}$. This amount of pie is one fourth of a pie.
    • So $\frac{1}{4}$ means of four equal parts, one of them. The two parts are the denominator on the bottom: of how many equal parts in total, and the numerator on the top: how many of those.
    • This is a generalization of the idea of a portion: it could be a fourth of a pie, or of a bottle of water, or of the world's ocean water, or of all the dentists in a survey...
  • Sometimes there's more than one way to write a fraction: if I took $\frac{2}{4}$ of the pie, it's the same amount as if I just cut the pie in two pieces and took one of them: $\frac{1}{2}$.
  • That's also the same as if I cut it into six and took three of them. One way to look at this relationship is: here's a pie cut in half. Here's the half I'll eat.
    • If I slice up the part I eat and the part I don't eat, I'm still eating the same amount: now there are 3 times as many parts here and 3 times as many parts there, and 3 times as many parts all together: now we would call that $\frac{3}{6}$, but it's the same amount of the pie as before.
    • We had $\frac{1}{2}$ of the pie - one slice of two - and we tripled the whole number of slices : $\frac{}{6}$ - and we tripled the number I'm eating - $\frac{3}{6}$ - and it's the same amount.
    • This is why we can multiply the numerator and denominator by the same amount and the fraction we get is equal to the one before.
    • $\frac{1}{2}\Rightarrow \frac{1\cdot 3}{2\cdot 3}\Rightarrow \frac{3}{6}$
  • We can also do that in reverse. If the pizza is cut into 8 slices and I eat two of them, it's simple enough that we can visualize: we could have just cut it into 4 pieces - we can consider pairs of little slices - and I would have eaten one of them.
    • Here's the numerical version of that: $\frac{2}{8}\Rightarrow \frac{1\cdot 2}{4\cdot 2}\Rightarrow \frac{1}{4}$.
  • To do here: what is $1-\frac{1}{4}$, how to figure it out.
  • Adding and subtracting
    • In simple cases, adding and subtracting fractions is very simple.
    • If I slice the pie into 8 pieces and eat 2, and then I eat 1 more, clearly I've eaten 3 out of 8, that's right, $\frac{2}{8}+\frac{1}{8}=\frac{3}{8}$.
    • I just introduced the equal sign.
    • If I eat 3 of those slices, and then I eat the other 5, I've eaten all 8, and it still works: $\frac{3}{8}+\frac{5}{8}=\frac{8}{8}$ and $\frac{8}{8}$ is the same as one whole pie. That's right, $\frac{8}{8}=1$.
  • Where there's adding, there's subtracting: if I eat 3 of the slices, how many are left: clearly 5. That's because we can take $\frac{3}{8}$ away from 1 just like what we just did, only backwards: 1 (the whole pie) is 8 slices, $\frac{8}{8}$, and taking away 3 of those 8 leaves 5: $1-\frac{3}{8}=\frac{8}{8}-\frac{3}{8}=\frac{5}{8}$.
    • It's fine to chain equal signs. All those things are equal to each other.
• And by the way, when we were first looking at the "divide by 2" system where half of the algae cells die off each day, and we said that it always gets smaller but never reaches zero, maybe it wasn't fully clear why that is. I think it's pretty clear when we use this pie analogy for situation. Here's the whole pie, half the pie, then when we divide again we cut it into twice as many pieces and have one of them... the piece is always half as big as the one before (which is not a surprise), but we can see that it is always smaller and there's always still some amount of pie, not zero.

decimals

• There have been times and places where people used only fractions to handle parts of things. for instance, Egyptian notation: instead of $\frac{3}{8}$, one had to write $\frac{1}{4}+\frac{1}{8}$.
• But the decimal system makes us a lot more powerful. We can handle arbitrary parts of things and easily tell at a glance how big they are, and which are bigger than others, round them up and down to various amounts of precision, and it's much simpler to multiply and divide them.
• We use it for whole numbers already. We reuse the same numbers to represent larger or smaller amounts, depending on position.
  • Example: 656 - 6 100's plus 5 10's plus 6 ones.
  • each position is "worth" 10 times as much as the one to its right.
• If we return to the convention of drawing numbers as spatial locations, that is, use a "number line",
  • look how a decimal number like that is a really efficient way of finding a precise location.
  • It's like transportation: first you fly to Boston, then you take a train to Providence, then take a local bus from the train station, then walk from the bus stop. Get into the general area, and then navigate the finer and finer details.
  • The same: here are these big regions, 100 numbers at a time, we choose one of these, and then it's divided into these regions, 10 at a time, like city blocks, choose one of these, and then get off the bus and walk to the exact number.
• In both cases, we might need even finer detail. Even after you get in the house, you might want to go into the living room, and then go and sit on the couch.
• With the numbers, we can keep going by cutting these intervals into tenths. Take this one, it has ten subdivisions, take the right one of those.
  • The way we write it is, we just keep going to the right. We didn't need a decimal point before, we just understood that the right edge of the number is the ones. Now we add the decimal point so we can tell which position is which, because now we have some numbers to the left of the ones and some numbers to the right as well.
• so I wanted to address the question, "why is $\frac{1}{2}$ also written as 0.5", because that is what came up in class a couple weeks ago, when I said I would talk about decimals and fractions.
  • So this picture is a start towards answering that.
  • every time we home in on the parts of a region, we have it cut into ten equal slots.
  • Five is right in the middle. We know that half of ten is 5, and half of 100 is 50, and so forth.
  • Half means half the distance to zero. Do you see why?
    • let's go back to the pie for that. If we compare a certain amount of the pie to a minute hand that can go around the pie, if it's at half-past we get half the pie and if it's right on the hour, no pie. However much we have, if we shift to having half as much we're going half of the way down to zero.
  • In the same way, dividing in half on the number line is going halfway between there and zero.
  • I think these shifts of reference can be confusing. Slices of things, amounts of rotation, distances on a meter stick, fractions with one number over another, decimal numbers, horizontal and vertical distances... take time with them and don't lose people.
• Once we buy that, we can see that dividing one in half takes us to 0.5 because it's the same division into ten slices with five in the middle.
• In each case, going from 10 to 5, with the 5 one position to the right of where the 1 was.

• Is it time to show "powers of ten"? https://www.youtube.com/watch?v=0fKBhvDjuy0
  • Or this remake, with Morgan Freeman (originally for IMAX): https://www.youtube.com/watch?v=qxXf7AJZ73A

This week's model

not this week. later.

random notes

• • Other things to work toward: a system of two variables. Lotka-Volterra cases, maybe, predation, competition, mutualism. Conveniently, density-dependent growth is a special one-variable case of this.
  • It would make sense to do a couple simple examples and then go to exploring it in an applet.

• So the starting point for this lecture may be what do I need for the logistic map, and work backward.

• recurrence relation; difference equation; map
  • Some recurrence relations can be solved: the density-independent growth model $x\to rx$, for instance, has the solution $x\left[n\right]=x\left[0\right]r^n$.

• note: Target Practice: cobweb-diagram game, at Robt. Devaney's website.

systems lecture

• talk about the readings.
  • I forgot to pre-teach mean and standard deviation! I have to write better notes and use them better!
• show the rest of the video

also in class

• Hand out new homework (due in 2 weeks)
• Talk about next week's midterm
• Hand out sample test as a study guide and a promise about what to expect
• Reading for 2 weeks away.

notes

Stuff I mostly haven't gotten to yet, so I'm carrying it along from week to week until I do:

• Spaces
  • the state space
  • parameter space
  • number of dimensions, maybe have fun with

• Especially: Negative feedback and self-regulation.
  • "The system goes to an equilibrium."
    • Like a thermostat. Or homestasis of the human body. Or Gaia.
    • It's going to do something according to its nature. Should we "go with the flow"? Trust it? Try to anticipate it? Or just find out?
  • Like Adam Smith's invisible hand.

• Joanna Macy video, maybe.
  • In preparation for going on to the Curtis video.

• Variables and functions
  • I introduced the "function", while drawing plots, and mentioned the vertical-line test. I haven't said anything about using letters for variables or anything that uses that since the first day (except to say we'll get to it later). I think it's time.

• notes that I might use sometime, but not right now:
  • Multiple kinds of plots: trajectory of state variable, time series of state variable vs. time, map of next state vs. current state
• Simplifying expressions.
  • Applying operations precisely vs. using intuition.
  • Add to/Subtract from both sides.
  • Divide from both sides.
    • Dividing by a negative? Multiplying by a negative?
    • The issue of dividing by zero. Cancelling something out when it's not zero.
    • Fractions and decimals. Fractions that include variables, parentheses, etc.
  • Factor something out of a set of parentheses.

• re guitar feedback, see "barkhausen criterion" and composer Robert Ashley's Wolfman (1964). http://answers.yahoo.com/question/index?qid=20071115084054AA5g49H

• What problem do variables solve?
  • Naming things concisely.
  • Maybe postpone introducing variables until I need them...

• But it's a good time to note that we're starting to have a lot of quantities, and it's time to fix up some names.
  • "When a twelfth-century youth fell in love he did not take three paces backward, gaze into her eyes, and tell her she was too beautiful to live. He said he would step outside and see about it. And if, when he got out, he met a man and broke his head - the other man's head, I mean - then that proved that his - the first fellow's - girl was a pretty girl. But if the other fellow broke his head - not his own, you know, but the other fellow's - the other fellow to the second fellow, that is, because of course the other fellow would only be the other fellow to him not the first fellow who - well if he broke his head, then his girl - not the other fellow's but the fellow who was the - Look here, if A broke B's head, then A's girl was a pretty girl, but if B broke A's head, then A's girl wasn't a pretty girl but B's girl was." Jerome K. Jerome, Idle Thoughts of an Idle Fellow, 1890 (D.M. Campbell, The Whole Craft of Number, 140)

• Notation, variables
  • State variables; time
  • parameters
• Parameters
  • "degrees of freedom"
• Functions, multiple kinds of
  • state as fn of time; next state as fn of current state
  • maybe outcome as fn of parameters
  • functions in general, have some fun with
  • the rules of the game
    • it'd be great to have a computer thing for exploring cobweb diagrams
    • population dynamics examples, others
    • consider using Granovetter model as an example

Week 7: Oct. 10

Midterm thing of some kind.

Week 8: Oct. 17

• Homework due Oct. 17: doc, pdf

Plan for this class:

• Go over the midterm and its grades
• Overview of where we're going with the population models: the logistic model
• Writing dynamics symbolically, using variables
• Review what we've seen cobweb diagrams do: equilibria, attractors
• Preview the simplest logistic model
• Talk about projects
  • More on rules/outcomes in games, art

• reading: maybe chapter from Gleick, Chaos?
• Or something else about chaos. Sci. Am. 1986 article? Can I get a copy from the library?

math lecture

• do an overview of things to come
  • dynamical systems can do a lot more than the multiply and divide models we've been looking at. In order to see more interesting behaviors we need a bit more complexity in our models.
    • In order to do that, we need better tools for working with them.
  • I'm going to develop the logistic population growth model.
    • In symbolic form ($x\to rx\left(1-x\right)$), with variables, parameters, and equations.
      • We'll work up to doing some things with variables and these symbolic forms. It's like learning a language.
    • In graphical form via the cobweb diagram
      • It looks like this, and we'll do some building up to this too.
    • Via simulation, looking at its time series.
      • It looks like this, on the computer.
      • direct damping, overshoot, chaos
  • The logistic map in historical context.
    • (fill in story of May, chaos, Laplace etc.)

• We're going to need notation.
  • First the use of a letter.
    • We can introduce a letter to some things we've been doing and it doesn't make much difference.
      • Redraw population vs. time, with letter labels on the axes. Same as using English-language labels, except that you need to be told what the letters mean.
      • Rewrite our update rules. "Multiply by 2" $\Rightarrow$ "$x\to 2x$". It's more compact, but basically the same as before. Here's what we had, and then here's what we had multiplied by 2.
• But let's pause and make sure we understand it.
  • A variable name is like a pronoun. Sometimes it can refer to a particular person and sometimes not. For instance, if I say, "whoever has the attendance sheet, could they please pass it to me?" - I don't actually know who I'm referring to. I'm just saying whoever it is, this is what I have to say to you.
  • And I might say that every week, and it might be different people at different times, even though I say the same words. I could write it on my notes and plan to say it every week. It's the same no matter who the people are.
• This is like that. This is quantitative, so instead of "whoever", it's "however much" or "however many". "However many plants there are, there will be two times that many tomorrow".
  • We can use it every day, even though there will be different amounts: $x$ will mean different things at different times.
  • So it's very similar to things we do in regular speech, but it's a different language that might take some practice.
• So that's "$x$" -- we should also talk about "$\to$": this is notation for an update rule, and it means "whatever $x$ is, it gets updated to this other thing."
  • So $x\to 2x$ is "replace $x$ by $2x$", as we all know, but it could be other things.
    • $x\to x+1$, or $x\to x-1$.
    • What does $x\to x$ do? What about $x\to 1$?
• Maybe the biggest advantage is that we can say more things easily.
  • We can talk about $x\to 3x$, $x\to 4x$, $x\to \frac{1}{2}x$, which we could already, but also $x\to 2x+1$, $x\to \left(x-1\right)x$, lots of other things. Are they meaningful? Sometimes. Sometimes it's useful to look at these things even when they're not directly meaningful.
• We have several different ways to work with the dynamics of a system now. Ideally you'll become able to bounce between them and carry several in your mind at the same time. You can use them to check each other. You might need some practice to get there.
  • Using words and thinking about the actual system. This is when we say "the population size doubles every day." If it's not too unwieldy to put into words, this might be all we need to understand the long-term outcome.
  • Using a math operation on the numerical that represents the system's state. This is "multiply by 2". It's similar to using English, but we could do it without knowing what the number actually stands for, and we could do it mechanically without having an intuition for whether the answer we get is reasonable or not. Also, we could program a calculator or computer to do it.
  • Doing it graphically on a cobweb diagram. This is a way you can get a qualitative picture of what's going on without needing to know exact numbers. If you work well with visuals, this might work well for you. But it's a peculiar, abstract visual that can take some getting used to, and it's best when you combine it with thinking about the update rule another way as well.
  • Doing it symbolically. This is similar to using a math operation verbally, but a lot more powerful.
  • You get best results when you use more than one mode at once. The last question on the midterm was like that, to let you choose which way works best for you. I drew a curve for cobwebbing, and I also wrote a description of what the model does, so you could use the description to ask whether you have it doing something that makes sense or not, or even just use it to get the answer and skip the cobweb.
• Anyway, we'll build on that. I don't expect you to become a hotshot with variables and symbolic math expressions just like that. Here's an example of the kind of question you've got on the homework this week:
  • If the update rule for our system is $x\to x-10$, and the current state of the system is 30, what is the next state?
  • Things you'll need for the HW: Fractions and division.
    • we talked about parts of things like $\frac{3}{5}$, but we didn't talk about things like $\frac{x}{5}$ or $\frac{20}{5}$.
    • but if you wanted to, you could maybe work it out from what we did talk about. But I'm not going to do that to you.
    • $\frac{3}{5}$ means you are counting fifths of a pie, and you count three of them. It's that much of a thing. $\frac{3}{5}$ of the people in the room, or the population size is $\frac{3}{5}$ of what it used to be, or whatever.
    • So what would it mean if I wrote $\frac{20}{5}$?
      • Anyone?
      • We are counting fifths, and we count 20 of them. It takes more than one pie to get to that many slices. (draw sliced pies) It takes 4 pies.
      • Why is this? Because in each pie, you get 5 of these slices, so with 4 pies you get $4\times 5=20$ slices.
      • This is the reason that $\frac{20}{5}$ is 4. But does this seem familiar? This is division. We did the same thing as if we asked, what is 20 divided by 5.
    • In fact, fractions and division are the same thing. If you take 20 divided by 5, you get a regular whole number 4, and if you take 1 divided by 5 you get this other thing, $\frac{1}{5}$, which is not a whole number, it's a fraction.
    • This notation, with an upper and lower number and a bar, means division.
      • So if you see $\frac{10}{2}$, you don't need to do anything very special, you just say "what's 10 divided by 2" and that's what it is, 5.
      • It's useful notation. I use it in the homework. I write $\frac{x-50}{2}$.
        • What does this mean? It means take $x-50$ and divide it by 2.
      • Can we do this? If $x$ is 60, what is $\frac{x-50}{2}$? In steps.
      • And if $x$ is 60, what is $50+\frac{x-50}{2}$?

combined math/systems lecture

I think this might get pushed to next week too

• Round up some of what I've been throwing at you with cobweb diagrams.
  • linear and nonlinear systems, more or less as an aside, for context.
  • The "Multiply by" systems are linear. Linear systems can only go toward zero forever, or go away from zero forever (or go basically nowhere, in boundary cases).
  • Nonlinear systems can do lots of other things, like, for instance, everything that anything in the universe does. (At least, if you are okay with the materialist faith underpinning the sciences.)
  • FYI it's called linear because in the "next value vs. current value" plot, it's a straight line.
• We don't need to get caught up in the difference between linear and nonlinear systems. But since I started giving you cobweb diagrams to work with, we've seen some of the other things systems can do.
  • remember the threshold model?
    • Should I review the threshold model?
  • When I introduced the threshold model, I drew a diagram that stops at 700 people demonstrating. That 700 is the place where the curve for the update rule meets the diagonal. Since the curve draws the next value against the current value, and the diagonal is the place where the two values are the same, the place where the curve meets the diagonal is the place where the next value is the same as the current value.
    • That's a fixed point. If your system is there, it stays there.
    • It's also an attractor, because the system is attracted to it.
  • In the homework, I gave you another threshold model, that looks more like this.
    • from here, you go up to the upper crossing, and from here, you go down to the lower crossing.
    • Here in the middle is a repellor - the system moves away from it. It's like the continental divide.
    • The number line is divided into two regions - the values that end up here, and the values that end up there. These are basins of attraction of the attractors.
• These fixed points are the simplest kind of attractor. There are other possibilities, for instance, it can oscillate up and down forever.

other notes, things that might not go anywhere

• Computer models
  • some examples. maybe bring them in earlier.
  • Agent-based models and other simulations.
  • some examples of complex models that produce the same behavior as simple models?
  • some that don't?

Week 9: Oct. 24

(class cancelled)

Week 10: Oct. 31

project presentations.

Week 11: Nov. 7

pop dyn chaos, bifurcations

• class outline:
  • scheduling - presentations last week, final the week before?
  • mistake on the hw
  • logistic map lecture
  • complex systems lecture

• Attractors.
  • cycles, maybe in cobweb diagram, maybe in computer examples.
  • initial conditions, basin of attraction.
• Deterministic vs. random.
  • mention random noise, stochastic dynamics

• Bifurcation, bifurcation diagrams.
  • a bit about more complex attractors, chaos

logistic map lecture

• Now we can talk about the logistic model
  • We were ready for it in a way, in the first couple weeks, when we were looking at the unbounded growth model, and
  • Probably the best way to motivate the logistic map: birth requires both a parent and an empty space.
  • Do it with proportions from the start. How much of the lawn has dandelions (a common weed) growing on it? half? Somewhere from none (0) to all (1). Draw.
    • • By the way, why am I doing this? Because we've been talking about exponential growth and limits to growth, and I want to give you an inside perspective on sustainability and overshoot? Yes. But also because this system will seem boring for a certain amount of time and then it will suddenly erupt into chaos. Watch for it.
    • How many seeds does a lawnful of dandelion produce? A lot. We don't actually care how many, so much as we care how many new plants it'll lead to. Say each plant's seeds is expected to produce 1.2 new plants, if all the seeds land on congenial soil.
    • So far so good, looks like a "multiply by 1.2" system, except with this extra clause.
    • Congenial soil meaning unoccupied. How much of the lawn is UNoccupied? The remainder. If $\frac{3}{4}$ of the lawn is occupied, $\frac{1}{4}$ is unoccupied, which is $1-\frac{3}{4}$.
    • "Multiply by 1.2 and then multiply by one minus the population size"
    • It's time to improve our language. It's time to call the population size $x$. This might be unfamiliar, or uncomfortable, or seem mysterious, but stay with me and see what it lets us do.
    • So if the population size is $x$, what did we have so far? It produces a lot of seeds, enough to make $1.2x$ on the next week. If that was it, we would have "multiply by 1.2" but we could also call it "replace $x$ by $1.2x$." But what about the land-use issue?
    • Because $x$ is a proportion, how much of the lawn is occupied, the inverse quantity, how much is unoccupied, is also a proportion, the rest of the lawn. $1-x$.
    • So the lawn produces enough fluff to seed $1.2x$ of the lawn. But only $\left(1-x\right)$ proportion of that is actually able to set seed.
    • I should draw pictures. Maybe a cloud of fluff, and the places it lands, and a fraction of it that is viable.
    • Yes, so the actual next amount of dandelion is $1.2x\left(1-x\right)$.
• Now that we're using variable names, let's run with it, since we want to consider various rates of reproduction, not just 1.2: call that $r$ for reproductive rate, and our rule for this model is, instead of "Multiply by 2", "replace $x$ by $rx\left(1-x\right)$".

• So we've gotten from describing how the parts update (they put out seeds and the seeds might succeed, depending on free space) to how the whole updates (according to this symbolic expression).
  • How to get from here to an outcome?

• Where do we begin with trying to understand this rule?

• well, before we even do that, maybe we should go back to the old model(s) in the new language of $r$ and $x$.
• We considered "Multiply by 2", "Multiply by 1/2", even "Multiply by 1", we were varying that number up and down (that parameter, to be precise), but we haven't considered this other object, "Multiply by $r$", so before we ask about a more complicated rule with $r$ and $x$ maybe we should do that.
  • So there's some number, called $r$, and we don't know what number it is, because it can be various numbers. It's a variable (it's also a parameter, but let's wait on that).
  • Again, a variable name is like a pronoun. Sometimes it can refer to a particular person and sometimes not. For instance, if I say, "whoever has the attendance sheet, could they please bring it up here?" - I don't actually know who I'm referring to. I'm just saying whoever it is, this is what I have to say to you. And I might say that every week, and it might be different people at different times, even though I say the same words.
• The update rule is like that. Without knowing what $x$ is, we say "$x\to 2x$", and we can keep using it even though $x$ is different things at different times.
• This is like that. We might spend some time on this "multiply by $r$", and then say, well, let's say $r$ is 2 and ask certain questions, and then later say, now let's say $r$ is $\frac{1}{2}$.
  • And we can say some things about it regardless of what $r$ is, just like I can say no matter who the person is, this is my message to them.
• Like what?
  • Well, for one thing, we know that multiplying by one leaves the number you multiply unchanged. So when the population size is 1, "multiply by $r$" gives you $r$.
  • For another thing, anything times zero is zero. So when the population size is zero, "multiply by $r$" always gives you some more zero. (A reasonable biology model will almost always give you that, because life doesn't spontaneously appear.)
• What about $x$? As long as we have to deal with it, we might as well get some use out of it, no? Instead of saying "when the population size is 1, 'multiply by $r$' gives you $r$", I could just be saying "when $x$ is 1, $rx$ is $r$". That might take some getting used to, but you have to admit it's more to the point.
  • So the familiar old "Multiply by 2" could also be called "replace $x$ by $2x$", again, and the general-purpose "multiply by $r$" would be "replace $x$ by $rx$". Or "$x\to rx$". It gets shorter and shorter.
  • Brevity is useful, by the way. The less it takes to do something, the more of it you can do...
• What else do we know about $x\to rx$? We have $0\to 0$ and $1\to r$. Not much else, except I guess that when $x$ is really large, then $rx$ is as well. This is like saying that when we drew "Multiply by 2" and the others as curves, on the "current size vs. next size" plot, they all slope upward as we go to the right.
• So we know some things about it. But as we know from last time, in some of these multiply models the population grows and in some of them it shrinks. So without knowing what $r$ is, we don't know enough to say what kind of outcome we get.
• This is true with the more complicated system too.

• The dandelion system, or logistic population growth model, has the form:
  • $x\to rx\left(1-x\right)$.
  • Even without knowing the value of $r$, we can figure out some things about this.
    • When $x$ is 0, it updates to 0: $0\to r\cdot 0\cdot \left(1-0\right)$.
    • When $x$ is 1, it updates to 0: $1\to r\cdot 1\cdot \left(1-1\right)$.
    • When $x$ is $\frac{1}{2}$, it updates to $\frac{1}{4}r$: $\frac{1}{2}\to r\cdot \frac{1}{2}\cdot \left(1-\frac{1}{2}\right)$.
  • This suggests the shape of the update rule's graph, and that is what it's shaped like. (DRAW PARABOLIC CURVE).
  • Since we know how to do cobweb diagrams, we can learn about the behavior of this model from the shape of this curve.
  • When $r$ is small, it's all below the diagonal, and the plants are doomed to fail.
  • When it's a bit bigger, it's above for a stretch, and then crosses downward: there's a fixed point.
  • When it's bigger than that, we start to get situations like you saw in the last homework, with different angles of crossing.
    • Let's go over that for a second.
      • When the curve is rising while it crosses the diagonal from above. A simple fixed point attractor.
      • When the curve is falling but not too steeply, a fixed point attractor with oscillation.
      • When it's steeper than that, the attractor becomes a repellor.
  • As we consider larger values of $r$, we pass through these different cases.
  • To the applets!
    • logistic_cobweb_0.html
    • logistic_3.html
• Develop the bifurcation diagram
  • bifurcation diagram
• Animation of Lorenz attractor: http://www.youtube.com/watch?v=_wKBqJEwO3c
• And I still like the flappy scarf video.

Week 12: Nov. 14

Running out of weeks! This week, I talk a lot about complex systems and networks! Swarms and crowds will go on Dec. 5, I think.

complex systems

• Intro to themes of complex systems
  • Chaos theory considers complex behavior in simple systems
    • Complexity theory is about relatively simple behaviors in complex systems.
• Complex system: one with a lot of parts. In complex system models, typically the parts are all pretty much alike.
  • emergence of properties of the whole from the interaction of the parts
    • not chaos but order
    • brain, economy
  • self-organization
  • systems where the simple systems approach doesn't work - too many variables, or no equilibrium (economic bubbles) or indeterminate outcome (evolution)
  • show some cool videos
    • corn starch! http://www.youtube.com/watch?v=vCHPo3EA7oE ... make some?
  • creation of Santa Fe Institute and its interdisciplinary approach
    • More is Different
  • biology
    • Kauffman's Boolean networks
      • tangle of light bulbs + wires $\iff$ genetic network
      • what happens? simple attractor! there are a small number of them!
      • Cell types! Origins of Order! The missing half of evolution!
      • Kauffman network applet: http://www-users.cs.york.ac.uk/susan/cyc/n/nk.htm
• Cellular Automaton
  • wolfram pictures: http://www.stephenwolfram.com/interviews/99-dailytelegraph/cell.html
    • 1-d video
      • CA and the "edge of chaos"
      • game of life
        • if alive, 2 or 3 live neighbors to stay alive
        • if not alive, 3 live neighbors to come to life
        • video - some configurations are open-ended, some aren't
• The two big founding models of complex system: Darwin's natural selection and Smith's invisible hand
  • economics
    • Brian Arthur, increasing returns
      • alternative to supply/demand equilibrium - economic system in flux, chaotic, changing
    • modeling stock markets, bubbles
  • Simulating evolution
  • Artificial Life
    • • Tom Ray version
      • Chris Langton version / von Neumann
      • back to Chris Langton
      • aside: Grey Walter version
        • 'At the end of an essay on cybernetics that Grey wrote in Colin Ward's Anarchy #25, 1963, (one shilling and sixpence), and which lies here open in front of me, he concluded: "we find no boss in the brain, no oligarchic ganglion, or glandular Big Brother...If we must identify biological and political systems our own brains would seem to illustrate the capacity and limitations of an anarcho-syndicalist community."'
      • contemporary version (Venter, synthetic biology)
    • genetic algorithm
      • Darwin = patron saint of SFI?
      • evolving CA
      • classifier system
    • Kauffman and Fontana's autocatalytic networks
      • Open-ended dynamics: a whole different kind of animal.
      • origin of life; economics, ecosystems, industry, computing, culture/science
      • Rosetta Stone for Connectionism
        • neural nets, classifier systems, immune networks, autocatalytic reaction networks (and others)
  • ants + swarm intelligence
    • Ants vs. neural networks
  • sandpile
    • has been used to describe economics, puctuated equilibrium, cancer, political/social change, scientific paradigm shifts, climate change
  • Immune system information processing
    • designing an immune system for computers
  • mention Scott Page's nice result
  • SFI researchers published an important result in HIV research at one point - what was it?
  • Agent-based models in ecology and social sciences
  • mercury beating heart
  • Iain Couzin flocks
  • Mandelbrot

• Ecology
  • and permaculture, maybe

Networks and graphs

This week's lecture: Networks and graphs

• Networks have been the symbol of complex systems since the beginning of system theory in the 1940s or before. Now that we have easy access to computers we can study them effectively.
  • As I have mentioned before, a curious thing about mathematical models is that when a model become popularly known, the main take-away message is often not the conclusions of the model but its assumptions. (And this has clear implications in understanding how math can be used in service of propaganda.) For instance, the most influential message of classical economics is not that supply and demand balance each other, but that people are self-interested; the most influential message of game theoretical models of cooperation is not that defection is the prediction unless there is reciprocity, kin selection or various other things, but that cooperation is hard because of temptation to defect - that's an assumption of the models; and with networks it's not that degree distribution matters, or that there are special positions in a network, but that it's possible to be organized in a way that doesn't have a central position of "power over".
    • Though, of course, a centralized, hierarchical form is one kind of network structure - it's just not the only kind.
    • Classifying structure intro 3 categories: centralized, decentralized and distributed. These are relative terms - how you describe a particular system depends on what you're comparing it with. One network or organization is more decentralized or more distributed than another.
  • So we often hear "network" used to talk about something that is decentralized or distributed, like a coalition or other leaderless organization.
  • Or used to refer to something that uses the Internet, which is the most conspicuous network in our lives presently.
  • Social networks have always existed, but now online social networks are an important part of our daily lives, and there's a lot of debate about whether they're changing the landscape of possibilities for social and political change.
  • There's also the horizontal structure of many social movements, i.e. leaderless. Horizontal vs. vertical. Again, these are relative terms, and there are degrees of horizontal and vertical. (The idea of the tyranny of structurelessness relates to this.)
  • It seems like horizontal structures are becoming more popular, even expected. In social movements there is sometimes a lot of tension between horizontalists and verticalists - for example:

    At the time I was only vaguely aware of the background: that a month before, the Canadian magazine Adbusters had put out the call to “Occupy Wall Street”, but had really just floated the idea on the internet, along with some very compelling graphics, to see if it would take hold; that a local anti-budget cut coalition top-heavy with NGOs, unions, and socialist groups had tried to take possession of the process and called for a “General Assembly” at Bowling Green. The title proved extremely misleading. When I arrived, I found the event had been effectively taken over by a veteran protest group called the Worker’s World Party, most famous for having patched together ANSWER one of the two great anti-war coalitions, back in 2003. They had already set up their banners, megaphones, and were making speeches—after which, someone explained, they were planning on leading the 80-odd assembled people in a march past the Stock Exchange itself.

    The usual reaction to this sort of thing is a kind of cynical, bitter resignation. “I wish they at least wouldn’t advertise a ‘General Assembly’ if they’re not actually going to hold one.” Actually, I think I actually said that, or something slightly less polite, to one of the organizers, a disturbingly large man, who immediately remarked, “well, fine. Why don’t you leave?”

    But as I paced about the Green, I noticed something. To adopt activist parlance: this wasn’t really a crowds of verticals—that is, the sort of people whose idea of political action is to march around with signs under the control of one or another top-down protest movement. They were mostly pretty obviously horizontals: people more sympathetic with anarchist principles of organization, non-hierarchical forms of direct democracy, and direct action. I quickly spotted at least one Wobbly, a young Korean activist I remembered from some Food Not Bomb event, some college students wearing Zapatista paraphernalia, a Spanish couple who’d been involved with the indignados in Madrid… I found my Greek friends, an American I knew from street battles in Quebec during the Summit of the Americas in 2001, now turned labor organizer in Manhattan, a Japanese activist intellectual I’d known for years… My Greek friend looked at me and I looked at her and we both instantly realized the other was thinking the same thing: “Why are we so complacent? Why is it that every time we see something like this happening, we just mutter things and go home?” – though I think the way we put it was more like, “You know something? Fuck this shit. They advertised a general assembly. Let’s hold one.”

    So we gathered up a few obvious horizontals and formed a circle, and tried to get everyone else to join us. Almost immediately people appeared from the main rally to disrupt it, calling us back with promises that a real democratic forum would soon break out on the podium. We complied. It didn’t happen. My Greek friend made an impassioned speech and was effectively shooed off the stage. There were insults and vituperations. After about an hour of drama, we formed the circle again, and this time, almost everyone abandoned the rally and come over to our side.

  • One fundamental result of living experiments like the Occupy movement is that it does seem possible for leaderless organizational structures to work.
  • 1968, groupuscules, wallerstein again... foucault, deleuze + guattari

• all that said, let's look at results from network science as well.
  • Graph theory.
    • Is built on top of set theory. A network is a graph, which is a set of "vertices" together with a set of "edges" connecting them, either directed (arrows) or undirected (arcs).
    • Example: social network websites. On Facebook friend relationships are undirected (if I'm your friend, it also means you're my friend). On Google+, Twitter, Diaspora* and some others, they use "follower" relationships which are directed - I can follow you without you following me.
  • A network can be connected or not. It may contain cycles or it may be a tree.
    • A large network may have a big connected component: http://www.newscientist.com/article/mg21228354.500-revealed--the-capitalist-network-that-runs-the-world.html
    • That's an example of network analysis.
  • Degree of a vertex is the number of connections that vertex has.
    • Degree distribution of a network is often important.
  • The problem of mapping out a network
    • they are often opaque, and would like to stay that way.
    • Talk about Mark Lombardi's art works
    • http://www.flickr.com/photos/hragvartanian/5484910150/in/set-72157626038832395/lightbox/, http://media2.moma.org/collection/browse_results.php?criteria=O%3AAD%3AE%3A22980&page_number=3&template_id=1&sort_order=1
    • Populist researchers work to map out the connections among the "1%", while government agencies and corporations - Google, Facebook, the AT&T/US government partnership - record and map the connections among regular people - because knowledge of how people are organized confers power to intervene.
  • networks and control, swarms and strategy
    • Networks and Netwars. Brave New Wars. Distributed vs. centralized. Anonymous, Wikileaks. Anarchy and its discontents. Democracy, Arab Spring, etc.

• From here, I think I will use part of this Barabasi talk (from about 1:45 to ), and pause it in places to add explanations
  • What is $p$ in creating an Erdos-Renyi random graph
  • What is a URL
  • I talked about distributions a bit, let's spell out what a degree distribution looks like. This is statistics. Make a bar chart of how many of the vertices have degree 0, how many have degree 1, and on up.
  • What is a logarithm? What is a log-log plot? Why does a straight line mean there are important hubs in the network that have a lot of spokes?

• Or: do it myself.
  • If you just wire up an E-R random graph, what you get depends on the density of links - there is a threshold, and above it the graph is connected (has a giant cluster)
  • Difference between those graphs and scalefree graphs

Three major ideas from network research

• The small world phenomenon.
  • The Milgram experiment (the small world one, not the fascism one), "six degrees of separation", Kevin Bacon - as in the Barabasi video
  • a small-world network has clustering (friends of friends know each other) and short paths between most points.
  • Two main ways to get one: a localized graph with bridges added, or a scale-free graph.
  • The "strength of weak ties".
    • The less-conspicuous relationships, the ones that aren't active most of the time, and are easily forgotten, can be the most important ones in getting a job, or otherwise making things happen. May be related to the bridge links that make a localized, provincial network into a small-world network. These links are a kind of social capital.
    • If they are related to bridge links in small-world networks, then creating and maintaining them isn't only self-serving, it's also for the common good, because they make it easier for something good to spread across the network and increase potentially fruitful or transformative encounters between disjoint communities.
• Power-law graphs.
  • Airline network vs. highway network.
  • When there are hubs, you need to deal with the network differently.
  • Hubs are influential: use them to get something to happen; they are weak points if one wants to interrupt something that is happening in a network (disrupt an opponent's organization; stop the spread of a disease).
• Structural holes
  • If you are in the special position of connecting two disjoint communities, you are in a privileged position and can use it to your advantage. You have insider information from one that the other doesn't yet have (in both directions), and can broker that information or use it yourself. This is a form of structural power.

• example: Lieberman/Nowak and my work on centralization and consensus
  • illustrates models of something happening on a network; a model that is for multiple things at once; how these models are used to support fundamentally political arguments

other

• statistics for networks, wisdom of crowds?
  • maybe just a few things, like the mean.
  • degree distribution.

• Networks, set theory, and graph theory
  • sets, vertices, edges
    • union and intersection
  • Centralized, decentralized, distributed
  • Vertical and horizontal
  • Definitions and ways of counting
  • Statistics?
  • Hierarchy and control
  • Tyranny, society of control
  • Weak ties, structural holes etc.

reading?

• no: http://www.opendemocracy.net/marianne-maeckelbergh/experiments-in-democracy-and-diversity-within-occupy-movements ?
• friendlier language, but still no: http://stirtoaction.com/?p=1069
• yes: http://www.occupy.com/article/how-occupy-birthed-rhizome
• And some pre-reading on incentives and dilemmas
  • This looks useful: http://socialdilemma.com/content/introduction-social-dilemmas

Week 13: Nov. 21

Self-interest, conflict, and strategy lecture

• The theory of supply and demand, the two curves, the homeostasis that the buyers and sellers do using the curves.
  • This is Adam Smith's "invisible hand".

• 1968: enter the tragedy of the commons
  • the scenario
  • the history - Hardin
    • policy prescriptions: privatize, public management
    • Ostrom, Berkes and others: actually many traditional communities do quite well at managing resources held in common
  • the older history - traditional common-resource management, enclosure movement, resistance, dispossession, wage labor, contemporary enclosure struggles.
    • Boston Commons; “They hang the man and flog the woman, That steal the goose from off the common, But let the greater villain loose, That steals the common from the goose”
    • "commoning" and a third form of property
  • There does seem to be something to it sometimes - consider dishes in the sink in a group house.

• Compare to the public goods problem
  • taxes, infrastructure, public radio, etc. Intimately connected to the theory of government.
  • Very similar - people want the results of cooperation but they are tempted to be selfish. In the commons scenario, selfishness is actively degrading the common resources, in the public goods problem it's passively declining to contribute to the shared good.

• The prisoner's dilemma - a theoretical distillation of these issues
  • The standard story - police, suspects.
  • Cooperate, defect.
  • Bottom row dominates.
    • Temptation to defect.
  • Play classroom game 1. http://people.virginia.edu/~cah2k/pdtr.pdf
    • Give each person a red card and a black card. Black = "push" $2 to your partner, take nothing for yourself. Red = "pull" $1 to yourself, give nothing to partner.
    • Lay down one card, then I assign pairs at random. (six-sided die?)
    • Play a few times.
    • Payoff is in fake $100 bills.
  • Play classroom game 2. Same, but know in advance your neighbor is your partner.
    • Play a few times.
  • Play game 1 again. Just one time. See how people play now.
• Return to the matrix, discuss Nash equilibrium
• N-person social dilemma.

• • Tragedy of commons
    • non-excludable: everybody can partake, regardless of whether they contribute
    • subtractible: when people use it it becomes less available for others.
  • Public goods
    • non-excludable but non-rival (the opposite of subtractible): my use of it doesn't diminish your use of it.
    • Public radio, for instance
    • Free riders
  • Collective action problem
    • contribution to public radio, participation in politics (voting, protesting, paying taxes, ...)
• Play social dilemma game: black = push $100 to every other player, red = pull $200 to yourself

• History of prisoner's dilemma studies.
  • repeated play, tournaments, computer tournaments, Tit-for-tat solution (reciprocal altruism). Evolutionary computer tournament.
  • Other solutions: communication, tags, spatial proximity, relatedness, application of heuristics developed in other settings, group selection, reputation, a norm of conformity, punishment, incentives (see http://leeworden.net/pubs/pd.pdf for list with citations).
• mention my escape from prisoner's dilemma and related results, Turner and Chao for example. Save sequential selection for the Darwin stuff, when we will talk about Gaia.
  • Existence of alternative games - chicken (snowdrift; hawk-dove), assurance , byproduct cooperation .
  • Play byproduct cooperation game with your neighbor, for cookies. Use the cards.
• Ways to think critically about claims about cooperation and temptation and altruism. (Numbered items are from Peter Taylor's paper and his book.)
  • Question the structure of the game. Some famous "prisoner's dilemma" scenarios are actually snowdrift, chicken, or even byproduct cooperation games, if they're two-person games at all, and the same caveats and more go for N-person games and "tragedies".
  • Question the assumption of "rational self interest". Sometimes people have other-regarding preferences.
  • Question the limitation to two choices. There are usually others. Positive social change often is often created by discovering or inventing new options when the ones that are given aren't sufficient.
  • 1. Interpret systemness as problematic. What's been declared extraneous to the system?
    • possibility of changing the system's dynamics. For example, changing the game to a different one.
    • connections not acknowledged, for instance social relationships between "game players"
    • boundaries of the system are seen as permeable
    • Inequality among individuals within the system colors their options, including response to developments "outside" the system
  • 2. Interpret the rhetorical effects of models
    • "simpling": "Like sampling, 'simpling' is a technique for reducing the complexity of reality to manageable size. Unlike sampling, 'simpling' does not keep in view the relation between its own scope and the scope of the reality with which it deals ...It then secures a sense of progress by progressively readmitting what it has first denied. 'Simpling' ... is unfortunately easily confused with genuine simplification by valid generalization." (Hymes 1974, See Taylor for citation)
    • reinforcing foundational assumptions, for instance self-interested behavior, liberal idealization of "mankind" all equal
    • privileging certain interests. For instance use of the T.O.C. model strengthens the political position of players with disproportionate power, since it makes the effects of inequality invisible. Who benefits?

• Evolution and economics
  • "Self interest", cooperation
  • ecological tragedies and market failures.

• Darwin's history, the basic ideas
  • Natural selection, descent with modification
  • How it works
    1. Variation: some individuals in a population are different from others.
    2. Heredity: offspring resemble their parents more often than they resemble unrelated inviduals.
    3. Natural selection: different variants leave different numbers of offspring
    • examples: peppered moth; finches
    • compare artificial selection
  • models?

• Evolution in independent environment optimizes something.
  • sketch of adaptive dynamics model framework.
• When environment is not fixed all kinds of things can happen
  • general dynamical system
  • We might expect the traits we observe to be adaptations, i.e. the result of optimizing; but they could be an exaptation as Gould argued, and could also be the result of an evolution process that does not optimize -- evolution can even pessimize, even in a one-species system, when the population affects the environment: you can have a tragedy of the commons outcome.
  • Mention the Gaia controversy, but we'll discuss it later

• evolution is a kind of adaptation process
  • evolution of things other than organisms
    • culture, institutions, technologies. universes?
  • other adaptation processes including learning

• The blurring between biology and society in metaphors of interest, politics, economics
  • Frontier of utility http://en.wikipedia.org/wiki/Production%E2%80%93possibility_frontier
  • Optimization on boundary of fitness set [1] for example
    • these are essentially the same model
  • "Diversifying your portfolio" to manage risk, whether you are a pension fund or a milkweed
    • cast seeds far and wide so if there's a drought or landslide, some seeds will be elsewhere
    • put some in the "seed bank", i.e. have them lie dormant for some years so if there are bad years they will be around afterward.
  • "Biological markets", "Bionomics"
  • Is there Capitalism, Marxism in biology?
  • We tend to project features of our current society into nature - biological markets, for instance - but if we were less resistant to considering alternative social possibilities we could also consider things we see in nature as things we could do socially... byproduct mutualism for instance

• Cooperation and "self-interest" in biology
  • the altruism question
  • group selection
  • gene selection
    • the selfish gene
      • example
    • kin selection and altruism
• fitness and self-interest. Payoffs. Evolutionary game theory.
  • conflicts between organisms. Predators and prey. The Red Queen hypothesis; evolutionary arms races.
  • general mashing up of political, economic ideas into evolutionary theory. Capitalist biology.

• Gaia theory
  • has come up several times in the readings by now.
  • Cybernetics proposition: the biosphere is a coherent system that implements negative feedbacks maintaining global conditions beneficial to life
  • Doolittle and Dawkins: why would the participants in the biosphere do that, rather than having a tragedy of the commons?
  • Lovelock and Watson's response: Daisyworld models
  • Daisyworld is vulnerable to evolutionary challenge - why would the world be like that and how easy would it be for a "free rider" to invade
  • Sequential selection provides an answer, why biospheres with negative feedback regulation and without free riders are more likely.

seems like enough.

Other stuff

skip, probably:

• Mendel, genes, the modern synthesis
  • flowers, crosses, recessive/dominant genes?
  • phenotype and genotype
    • the pathway from genes via RNA transcription, folding to proteins, cellular functions, cell types, tissues, bodies, behaviors, survival and reproduction
    • the central dogma; Darwinian vs. Larmarckian ideas
• Genes are selected for/against, tend to fixation/extinction
  • Mutation + selection $\to$ adaptation by natural selection
  • The network of genes' effects; genomics (epigenetics?)
  • Genes "for" traits

• gradual vs. punctuated
  • political resonance of Eldredge/Gould paper

• does evolution make progress?
  • when we say it does we are reifying the politics of Eurocentric domination, faith in science and technology, etc.
  • hence Gould's argument to the contrary. Humans are not "higher" than bacteria, just newer.
  • does complexity increase? It's not clear. Meanwhile, it's not like the bacteria are going away.
  • but Kauffman's adjacent possible is a nice frame.

• • • "Putting an optimization program into practice requires a general theory of optimality, which evolutionists have taken directly from the economics of capitalism. It is assumed that organisms are struggling for resources that are in short supply, a postulate introduced by Darwin after he read Malhus's Essay on the Principle of Population. The organism must invest time and energy to acquire these resources, and it reinvests the return from this investment partly in acquiring fresh supplies of resource and partly in reproducing. ... The optimizing theory of allocation assumes that time allocation will be close to optimal for maximizing total investment in reproduction, or growth of the firm. In such theories the criterion of optimality is efficiency, whether of time or invested energy, yet the moralistic and ideological overtones of "efficiency," "waste," "maximum return on investment," and "best use of time" seem never to have come to the consciousness of evolutionists, who adhere to these social norms unquestioningly." The Dialectical Biologist, pp. 25-26

• Evolution vs. religion
  • Monkey trials. Current fights about textbooks and curriculum.
  • Dawkins, PZ Meyer, etc.
  • Orthodox priesthood of science vs. centralized institutions of religion - less hierarchical spiritual traditions are made invisible
  • Some people, even Christians, disdain scientific authority including evolution theory less because of adherence to dogma than because of rejection of technoscientific modernity, in full view of its disastrous consequences and the sound reasons for mistrusting scientistic authority - the Tuskegee experiments, the atomic bomb, X-rays, thalidomide, carcinogenic food, internal combustion and carbon emissions, etc.
  • Rich, powerful white men in the US and Europe heaping scorn on Christians and Muslims alike is politically tone-deaf at best

• Evolutionary theory and structural inequality
  • Darwin was inspired by Malthus's scarcity scenario, which is a flimsy rationale for class oppression and deeply interwoven with racist ideologies of the 19th century
  • Darwinism adopted by racist, elitist Social Darwinists; Kropotkin's book is major response
  • Eugenics; Nazi history, but also racist science in US, Europe
  • Modern-day eugenics; contestations over Human Genome Project; Larry Summers at Harvard; "Gay Gene"; prospect of human genetic enhancement, emergence of elite subspecies; selective abortion of girls, disabled, lgbtq children

• Reductionistic and holistic biology
  • Genes are the atoms of the 21st century, the big funders are looking for the new Manhattan project -- the greatest "triumph" of reductionist science
  • you can't actually do much with genes without understanding them in context - hence systems biology - genomics, proteomics, etc. - which is difficult and tends to confound people's expectations of clear understanding and control.
  • Transgenic experiments, evolutionary escape, ecological impacts - failure to understand what will become of these organisms in context.

Notes

• critiques of Dawkins. critiques of gene ideology. Levins and Lewontin.
• Evolution and economics
• ecology, ecophilosophy. "balance of nature" vs flux.
• game theory and rational man. Nash's claims. von Neumann and Wiener.
• science and democracy. consensus vs. autonomy vs. central authority vs. privatization.

• Evolution vs. religion
  • Monkey trials. Current fights about textbooks and curriculum.
  • Dawkins, PZ Meyer, etc.
  • Orthodox priesthood of science vs. centralized institutions of religion - less hierarchical spiritual traditions are made invisible
  • Some people, even Christians, disdain scientific authority including evolution theory less because of adherence to dogma than because of rejection of technoscientific modernity, in full view of its disastrous consequences and the sound reasons for mistrusting scientistic authority - the Tuskegee experiments, the atomic bomb, X-rays, thalidomide, carcinogenic food, internal combustion and carbon emissions, etc.
  • Rich, powerful white men in the US and Europe heaping scorn on Christians and Muslims alike is politically tone-deaf at best
• Evolutionary theory and structural inequality
  • Darwin was inspired by Malthus's scarcity scenario, which is a flimsy rationale for class oppression and deeply interwoven with racist ideologies of the 19th century
  • Darwinism adopted by racist, elitist Social Darwinists; Kropotkin's book is major response
  • Eugenics; Nazi history, but also racist science in US, Europe
  • Modern-day eugenics; contestations over Human Genome Project; Larry Summers at Harvard; "Gay Gene"; prospect of human genetic enhancement, emergence of elite subspecies; selective abortion of girls, disabled, lgbtq children
• Reductionistic and holistic biology
  • Genes are the atoms of the 21st century, the big funders are looking for the new Manhattan project -- the greatest "triumph" of reductionist science
  • you can't actually do much with genes without understanding them in context - hence systems biology - genomics, proteomics, etc. - which is difficult and tends to confound people's expectations of clear understanding and control.
  • Transgenic experiments, evolutionary escape, ecological impacts - failure to understand what will become of these organisms in context.

Other material

notes

• "game theory" on numb3rs https://www.youtube.com/watch?v=dSOrAQMXTcc

Week 14: Nov. 28

Project presentations.

Notes for a project presentation of my own, on the student debt situation

• Tuitions are rising (440% over past 25 years).
• But college degree is a necessity. Enrollment is up.
• People have to borrow. Stafford loans and Pell grants aren't enough, so people need private loans. Predatory, deceptive practices, high and adjustable interest rates.
• Effectively indentured for life
  • Means you have to take whatever work you can get and can't do work that's meaningful, important, constructive, take risks.
• Student debt exempted from bankruptcy
• The universities are using students' debt to keep afloat and profit from investments secured by them.
  • (See the bunker video: "The banks lend to the students. The students pay us huge tuition... We spend money that is not ours. Then the students pay the banks.").

• Is there a student debt bubble?
  • How bubbles work: Dutch tulip craze example. People believe they are a good investment, therefore they are a good investment. You can buy high and sell even higher to another person just like you. A sort of leaderless pyramid scheme. Recognizable as a positive feedback process.
  • When the mystique fails, people get stuck with bad investments. Lose their fortunes, homes.
• Debt bubble: a certain kind of debt is seen as a good investment by investors. Student debt for instance, because people keep borrowing and they can be forced to pay back no matter how much it hurts them.
  • They sell packages of students' debts to each other and use them as collateral on other investments.
  • But more and more students are defaulting, because the job market sucks so hard and they have medical debt, foreclosure issues, etc.
  • If all those packages of debt lose their value, investors will be in trouble and the trouble will ripple out through the economy, as it did during the housing meltdown.
  • What will happen to students? I don't know. In the case of mortgages, people lost their homes. Student debtors don't have such tangible collateral. But they could end up in debtor's prison or something because of the bankruptcy exception, or just indentured for life as many already are.
  • Loans will become less available, and more people will have to do without college education. It'll become more a privilege of the rich.
  • Tuition keeps universities afloat, whether public or private - they use it directly and to secure other investments. So they will become poorer and continue to cut all "inessentials" such as arts and humanities, becoming more like business schools, to function as a career investment. More of the current trend toward undermining tenure and academic freedom, cutting benefits and commitments by phasing out tenured positions for contingent adjunct and student teaching.

• SFAI tuition was less than $15,440/semester 4 years ago - what is it now? $17,023 or up. $32,424/year last year, $36,618 this year.
  • With housing costs, a student could owe $46000 at end of first year. 99% of applicants are accepted, 46% of those who come graduate. 82% of students receive financial aid. So many will probably end up with serious debt and no degree.

• Graeber's point about the ethics of debt - why should it be more important to pay a monetary debt than to make sure someone has health care or housing, or keep babies from dying?
• Debt is central to Occupy and the idea of the 99%
  • Foreclosure, the financial bailout of the 1%
• Occupy Student Debt's refusal pledge of fall 2011, didn't take off
• There have been debt forgiveness movements, petitions.
• Major student movement in Quebec against massive tuition increase
  • Over 100 days of continual casseroles
  • Completely won 5 months later. New premier signed a tuition freeze.
• student-to-student mutual aid funds in the South
• "Fix UC" proposal to give UC incentive to ensure degrees are a good investment
• Maybe another debt refusal campaign on the way in the US?
  • Debt strike is "like a prisoner's dilemma". Many social movements' hopes are collective action problems.
• the Rolling Jubilee

Week 15: Dec. 5

• swarms, flocks, crowds
  • headless coordination, emergent motion and decision making

• http://www.thomasjacksonphotography.com/Page%2082.htm

• how much math do you need for Condorcet's theorem?
  • that would also let us talk about theorem and proof.
• Collective intelligence and democracy
  • wisdom of crowds...

Final exam

Misc Notes

• some algebra I resources to consider borrowing from: http://www.mathsisfun.com/links/curriculum-high-school-algebra.html http://algebra.mrmeyer.com/ (but don't use that Hot vs. Crazy thing - eww!)