Mathematics

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Talking to the moon

This semester I’ve helped teach a biology-meets-math sort of a course. There’s been some debate among my labmates about what such a course ought to include and about the merits of math in biology more generally. It’s true, I’ve never explicitly used control theory to calculate how much liquid I should pipette into my experiment. And I very seldom think about the principle of detailed balance when I peer into a microscope. So why would we take a room full of experimentalists and teach them about dynamical systems on abstract graphs—how could this possible improve their biology?

My students were wondering the same thing when they showed up to the first section this semester. That week we were discussing a paper by van Oudenaarden on yeast’s ability to regulate and maintain its internal osmotic pressure despite living in a varying external environment. The technique they used was indirect but powerful. Assault the cell with pulses of carefully measured salt concentrations and then watch as a judiciously chosen protein read-out accumulate in the nucleus in response. A mess of genes and proteins and other factors have been associated with osmoregulation and the trick was to pick out which of these players are most important and when.

By matching their salt pulse inputs with fluorescently labeled protein outputs, the researchers were able to come up with a simplified model of the cell’s vastly complicated internal logic. And while their model is a cartoon of reality, it was extremely good at predicting previously unmeasured behavior when different cells and different cell types were subjected to new concentrations of salt. Combing the literature helped the researchers identify the parts that popped up in their model. They pared down a hair ball of chemical interactions and were left with a relatively simple mechanism.

This experiment is wholly unlike the classic experiments I usually think to do as a biologist. For example, how would a biologist determine whether the moon affects the tides? Now, hold on. I know that the moon and tides don’t normally reside within the realm of biological experimentation. This shouldn’t worry you. Biology, just like any other discipline, has its own methodologies. And these methodologies make some knowledge easy to dig up and verify and others hard. So, what does biology have to say about the tides?

First off, you’d do a knock-out study. Blow up the moon, but keep everything else the same, and see how the tides changed—if at all. Perfect. They stopped. What next? Well, there’s an obvious follow-up study: over-express the moon. Put two of them up there; maybe three, just in case. Now that the tides are back, we have some strong evidence that the moon in some way seems to influence the tides.

Part of the beauty of the van Oudenaarden input-output approach is that it didn’t require us to muck about with the genetics of the yeast. No knock-outs, no knock-ins. They kept the cells normal and genetically intact. Instead, they did what my friends in the cognitive neuroscience labs do with children subjects. They asked the cells a question: “What will you do with this salt concentration?” And then they listened for the response. From the answers the cells gave them, the researchers were able to infer something about the decision-making process. In this case, they drew it up as second-order linear, time invariant system. Here is an example where math allowed a biologist to do something very surprising. Math was used to talk to cells, ask them questions in their own language (so to speak), and learn something about them from their answers. Doesn’t that sound nice?

Everyday each of us engages in several delicate dances with the other members of society. I secretly long for the days of learned formalities, proper ettiquette, and wide-spread manners. If someone were just to tell me what to do, things would run more smoothly. Take, for instance, the simple act of holding the door for another person.

So far I’ve only noticed one person play the situation correctly. On several occasions, I held the door for my friend Lane, who, by chance, was always a good ten yards away when I first spotted him. Lane usually acknowledged my act of kindness. He might say, “Thanks, Joshie,” but he would never speed up as I stood. The moments seemed to lag as he slowly approached the entrance to the dining hall. Once he arrived, I thanked him with full sincerity. Most people, I explained, sprint once they realize that someone else is holding the door for them. However, that ruins the favor. What sort of charity requires you to break a sweat? Lane had enjoyed my gift as it was intended, and I believe we both appreciated the exchange all the more for it.

But that sort of action doesn’t readily transfer to any other person. On Friday a stranger held a door for me, and like most people, I sped up as soon as I noticed that I was inconveniencing another person. The man beckoned me to slow down, but how could I? Then I’d come across as arrogant and entitled. I’ll make no one a doorman for me. Well, at least no stranger. And there lies the fundamental difference. Lane and I are friends. This man and I were not. For some reason it’s easier for me to take advantage of my friends than strangers. I guess that’s a good thing for society at large, though a little strenuous on my immediate circle of friends. I believe sociobiologists would have a thing or two to say about multi-level selection processes at this point, but I don’t.

Instead I have a few questions. Normally we think of selfish behavior as something that individuals inflict on members outside of their group. (The ones who are selfish to those inside their group is called “cheaters” or a “defectors”.) The defectors take advantage of and therefore benefit from the cooperators on the individual level. Locally, the defectors do better. But when it comes to asking for help, it’s easier for me to ask someone I know I can trust. I’m more willing to ask my friends to do me favors than strangers. In the iterated prisoner’s dilemma, it’s in your best interest to cooperate with your partner because you’re going to see them again. If you screw them over, they’ll remember it and be less likely to help you in the future. But in many cases I impose on other precisely because I know I’ll see them again. My willingness to ask others to do things for me increases with my level of comfort with them, and I see it in others, too.

Take it up a notch and look at groups as your fundamental unit rather than individuals. People have noticed before that groups that have more (internal) cooperators do better on average than groups with fewer. That seems to make sense. If more people in your group are willing to help out others in your group, the group should run more smoothly than groups that don’t work well together. Nothing exciting there. But what happens when there’s lots of internal chaos but external altruism—can groups composed of individuals that take advantage of each other but cooperate with members outside of the group coexist given proper inter-group interaction?

Because the comfortable defectors have to act charitably when someone else calls on them, it’s a little unfair to characterize them as defectors. Maybe it’s best to call them comfortable defectors-generous forgivers. This is starting to sound like the win-stay, lose-shift strategy. Maybe the folks studying evolutionary dynamics can clear things up for me. Help me out if you can, especially if I’m already comfortable with you.

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Summer Informal Seminar

No, I haven’t been off on vacation in the Caribbean—but I have seen the third pirates movie. It was great. But I wanted to let anyone out their who is interested that I’m going to lead an informal seminar on general relativity at UMass/Boston this summer starting June 4th. Here are the details.

 What: An informal seminar on differential geometry and general relativity When: MTh 5–6:30pm June 4th — early-August Where: Taffee Tanimoto Conference Room Science Center Third Floor, Room 180 Website: www.gsd.harvard.edu/~jreyes/GR

Games: a Ludic Structure for Problem-Solving

Today I’ve decided to post a journal together with a longer paper about games. You hear all the time that we need to inject more play into education, that we need to return to childhood, etc. But why? You don’t as frequently hear why play is useful in education. People claim things like “If learning is fun, children will learn better.” I’m not sure of the connection. I suppose that if kids are engaged in learning, then they have a better chance of actually picking something new up than if they’re not trying to learn at all. That’s like saying if you look for something you have a better chance of finding it then if you don’t look at all. Sure, I buy that. But why play? By the same argument, we could just as easily pay kids to go to school and do their homework.

Of course some people do give reasons why play is useful. In these two papers, I’m building on some insights found in a 1933 paper by Lev Vygotsky entitled Play and its role in the Mental Development of the Child. (Vygotsky, you may well know, is one of my current heroes.) I remind the reader that in play, you can find all sorts of higher-order thinking skills taking place. Imaginary play is a very natural, distilled, abstractly difficult thing to do. Yet kids seem to do it on their own anyway, and before they even step foot in a classroom. If taught effectively, I think play is a useful vehicle for transfer of skills and tons of that ever-so-hot interdisciplinary work that goes on nowadays. (Wait until I get my genetic algorithmic music up and running.)

Peter Elbow introduced concepts of methodological doubt and belief in his book Embracing Contraries: Explorations in Learning and Teaching. They’re central to his believing game and doubting game. Traditionally, doubt has been used as the primary tool in critical thinking. This unbalanced attention really makes a lot of analysis blind to new insights that can be gleaned from a moment of pure, suspended disbelief. (My ego won’t let me pass up an opportunity to say that both games show up automatically in my coffee mug model of classroom education.)

In my first paper I remark that all games require its participants to engage in the believing game—they have to believe that the rules imposed by the game are real and that the game itself is real. There are no consequences in any game if you don’t except them. You can always pick up the ball with your hands in soccer, unless you firmly believe that you can’t. For this reason, we might frame any situation as a game.

In the second paper, I extend my ideas to show that framing a situation as a game can greatly improve your power to predict behavior and arrive at winning strategies by simply considering the acceptable moves in your game. To illustrate my point, I work through a problem of the type sometimes given in consulting or computer science job interviews. The example shows, additionally, how mathematical reasoning (which I believe is no different than plain, old, vanilla reasoning) can be used to solve a problem without once using “math.”

As always, please comment freely. I’d love to get some feedback on this stuff.

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Words and Meanings

Freshman year of college my friend Rebecca tried to explain to me the literary school of deconstruction. After some time I tried to sum up what I had heard in a phrase that (be it my own or not, and whether it be accurate or not) I have kept with me six years later.

Words have meanings, but meanings don’t have words.

Now I’m still not sure what that means, but I do know it has to be true. My friend’s grandmother, sage that she is, disagrees entirely. Meanings are the words they mean—sometimes people misuse words—but that doesn’t detract from their instrinsic definitions. But if that were true, we wouldn’t have any need for dictionaries. If words were their meanings, then words couldn’t be defined in terms of other words. That’d be silly. The other words have their own (other) meanings, after all. Imagine what a dictionary entry might look like in this alternate semantic universe:

apple, n., apple. What don’t you understand? Apple means apple.

Of course, maybe I’m taking too naive an approach. DJ’s grandmother might be onto something. How can you sufficiently define terms like ‘this’, or ‘I’, or ‘you’? This is what it is. It’s nothing else. It’s this. I am who I am. Or am I? Words, like people, take on a meaning that emerges from their use. How words are used, though, follows from larger, guiding principles. Culture helps define who we are. So, too, culture—which is really no more than a vast set of complex and subtle rules—defines what are words mean. So, words do have meaning. But only in relationship to other things (that have meaning). It’s sort of like music.

In music syncopated rhythms accent the beats which normally go unaccented. But without some concept of normal, syncopation doesn’t exist. But it does because in our music there is a structured sense of normal. And if we let loose the structure, we loose some of the meaning. Syncopation just disappears. Ironically, the tighter a straight-jacket we put on rhythm the freer we can be within its constraints: we get things like syncopation back.

In mathematics, too, Kahler manifolds are surfaces that exhibit a rich geometry. It’s thought that the physics of our universe is actually encoded on one of a special class of these surfaces known as Calabi-Yau manifolds. The thing about Kalher manifolds, though, is that their geometry is so highly structured that the surfaces are almost flat. Flat surfaces are the simplest to investigate. It turns out that these guys, by comparison, are notoriously difficult to analyze. There may be something to that—that the most useful, interesting cases often lie just on the cusp between simple and intractable—but I’m not sure what it is.

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To Infinity, and Beyond! On a Llama.

I have a few other posts saved as drafts, and I want to get to them, especially one on technology, but I can’t pretend to have finished talking about that alternative decimal representation of the number one (0.999···) that carries with it an infinite chain of nines. To say that I’m comfortable with that representation of the number is almost as brash as claiming to have solved Zeno’s Paradox. Infinity is a funny thing. In fact, it would be more honest to say, infinities are funny things. After all there are lots of them. And there’s no reason they should all be the same—in fact, they are not.

But before we dive off the deep end, we should pause to think about what it means for finite numbers to be the same. Have you ever given much thought to statements like “5=5″? So, now, before you read on, turn to the nearest 8 year old you can find and ask her, “Does five equal five?” [That's the easy part.] Now ask her, “How do you know?” and listen for a response. [That's the medium-hard part.] If your 8 year old doesn’t sufficiently convince you that five is, indeed, equal to five, explain to her why it’s true in plain terms that no one could dispute. [That's the hard part.]

When I was taking an abstract algebra course, my professor asked the class what the number three is. We all knew it was a trick, so we waited in nervous silence, each hoping that he wouldn’t start dead-calling members of the audience. After all, college math concentrators should know what the number three is. We were studying math, and numbers, I’m told, are an integral part of mathematics. So what was the Fields medalist‘s definition? He stole his answer from a six year old: the number three is three fire trucks without the fire trucks. Hold up, what? That’s a surprisinglyl useful way to think about number. I think that’s how Frege and Hume viewed number, and by some standards, they’re famous. So maybe this little kid is on to something.

Let’s pretend for a little while. You didn’t know this, but I have a pack of llamas. Every evening I feed them each one carrot for dessert at dinner time. The problem is, I can’t count. I have a bunch of carrots and a pack of llamas. How can I know if I brought out the same number of carrots as the llamas—without counting?

Well, I could feed the llamas each one carrot. If I had more carrots than llamas, then I’ll have carrots left over at the end. If there were more llamas, then I’ll run out of carrots before I finish feeding—the llamas hate that. But if the number of carrots and llamas equal, then after feeding time, each llama will have had exactly one carrot and no carrots would remain. That is, I would be able to put all the carrots in a one-to-one correspondence with the llamas. If I had more carrots than llamas, then some llamas would get the left-overs. That relationship is not one-to-one because some llamas get more than one carrot.

But we’ve got a problem, one-to-oneness is certainly necessary for there to be the same number of carrots and llamas, but it is not sufficient. If I had fewer carrots than llamas, every llama who gets a carrot gets only one. But because I run out of carrots before I run out of llamas, some llamas are left out. In order to know whether there are the same number of carrots and llamas, every llama needs to get exactly one carrot and there can’t be any carrots left over. This is tricky business.

If every llama gets at least one carrot, then we say that the pairing of carrots to llamas is onto. Ontoness is also a necessary condition, but like one-to-oneness it is not sufficient. Having more carrots than llamas leads to a pairing that is onto. Every llama gets one carrot and some get more. What we’re looking for is a matching of carrots and llamas that is both one-to-one and onto. Then for every carrot there is exactly one llama. Likewise, for every llama there is exactly one carrot. The size of the pack of llamas and the size of bunch of carrots is the same. Mathematicians like to have a standard way to talk to one another, so they call the number of elements in a group the group’s cardinality.

So, we’ve done it! We’ve found a way to determine whether two collections of things are the same size, if they have the same cardinality, without resorting to counting. In fact, we’ve secretly discovered a very powerful way of thinking. One thing we can do with our new-found friends, the one-to-one and onto functions, is define what it means to count. I don’t have the room to do it here. In fact, I don’t think we’re even going to get to infinity in this post. Instead, I’ll cop-out and refer you to some of my set theory notes. (That’s what we were doing: set theory.)

Set Theory Lesson Plans

In my notes, I’ve asked just a number of questions. I wrote these questions for a practicum for a class it took this fall. And I used them on real, high school juniors in at Codman Academy. We had a little bit more time to go over each of the questions carefully and resources that allowed dynamic, colorful diagrams—which the students largely produced for me. I was only asking questions, it was up to them to answer them. But please, don’t wait until 11th grade to feed the llamas.

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Inductive Proofs, Constructive Understanding

I remember when I first learned that advertisers will often use glue instead of milk in breakfast cereal commercials. The whole thing blew my mind. Initially, I felt confused. Why would they do something like that? Of course, because glue looks more like what people expect milk to look like than milk does [on camera]. Even after my personal revelation, I still felt confused. Except now my confusion came from within: why had I assumed that things on TV were what they looked like? Even now, I still feel a little uncomfortable thinking about it.

Again in the the seventh grade, another discovery left me feeling the same way: the infinitely repeating decimal 0.999··· is the same as the whole number one (1). I know that this must be true; my math teacher Mr. Heleen proved it to us. First, let’s hide the infinite string of 9s under a clean variable name, say x. Then we can distract ourselves long enough to arrive at a meaningful conclusion. Here’s what Mr. Heleen did:

 10x = —9.999··· —x = —0.999···

Subtract the two lines (something that is hard to do in HTML) and you’ll get

 9x = 9,

Or, as I claimed earlier, that x = 1. Even now, I find that fact a little bit mysterious. And this is one of my central problems with algebraic methods in general. They’ll tell you that a statement is true, but they seldom lend themselves to obvious readings of just why a statement is true.

In fact, this reminds me of a frequent difference between inductive proofs and constructive proofs: inductive proofs often accompany theorems which speak only about existence—what you’re looking is out there, the proof guarantees it, but you have no idea where; constructive proofs, on the other hand, actually give you what you looking for. Constructive proofs are usually more useful than inductive proofs because they automatically satisfy existence by virtue of demonstration. (Imagine what economists would do with a constructive proof of the Brouwer fixed point theorem; and I’d understand Sard’s lemma a lot better if the proof I learned didn’t rely so heavily on induction.)

So, is it any wonder that I gravitated toward geometry over algebra in college? Geometers use inductive arguments, too, to be sure. However, problems are usually cast in ways that are about as tangible as mathematical problem can be. Some of them even have straight-forward physical interpretations. It’s not (too) hard to imagine that soap bubbles could represent minimal surfaces, for example. However, what does a Dedekind domain look like? (If you can help me visualize a Dedekind domain, I’d be very grateful. Had you helped me three years ago when I was taking algebra, I’d've been even more grateful.) Like I said, algebra is hard.

But let’s get back to our infinitely repeating decimal. Why should it be the same thing as one? Well, I suppose we should ask, what is the number one? There are lots of answers—many of them correct. In this case, one is particularly useful: the number one is a label for a point on the number line.

The decimal 0.999… is also a label. But then again it’s so much more. Both 1 and 0.999··· are directions to the points that they label. How convenient! Here’s how you read the roadmap embedded in every decimal. First you need to arbitrarily pick a point called zero. That’s up to you. Next you need to pick another point that’s a unit length away from zero. This choice is also arbitrary. In the metric system you might use centimeters. In the English system, the unit you pick might be feet. If we were measuring something large, maybe you’d choose lightyears. What you choose is really a matter of convenience.

Now the fun part comes in. The first digit d after the decimal tells you to chop up the unit length into 10 smaller pieces of equal length. This smaller distance (1/10) will become the unit you use in the next step. Then you go to walk to the d-th piece. In this case, we chop up the length 1 into 10 equal pieces and walk to the ninth piece.

In the second stage, you chop up our new unit (1/10) into 10 smaller pieces of equal length. That smaller length becomes our new unit (1/100) for the next iteration, so remember it for later. Now walk over to the piece that the second digit in the decimal tells us to go to. In the third stage, we repeat the process, always taking tinier and tinier steps. For an infinitely repeating decimal we have to take an infinite number of steps to get the point the directions describe. Eventually, the steps we take will be so small that for all practical purposes we stop. This is the idea behind a limit point.

Of course I haven’t been terribly rigorous. That’s where the algebra comes in. We already proved that 1 = 0.999··· above, but the geometry is where the understanding is, at least for me. Ideally, I would’ve had some pictures in this post—but modern technology is years behind pencil and paper. But kindergarteners can draw; more importantly they can walk. Maybe limit points aren’t especially useful in most kindergarten curricula, but I think that this shows that they probably have a fair shot at understanding the concepts. And maybe now I can put this demon 0.999··· to rest.

p.s.—Wikipedia also has an entry on 0.999··· with more pictures and deeper, more confusing jargon.

p.p.s.—Now I really need to write up something about infinity. After all, 0.999··· has an infinite number of 9s in it. What does that even mean?

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Critical Thinking Journal/Weak-sene, Strong-sense, and Probabilities

That’s right. It’s time for another installment of “What has Josh been writing for class?” This week I responded mostly to an old article by Richard Paul—who, I think, bears a striking resemblance to Walker Texas Ranger: hold on to that.

He differentiates mainly between two types of styles of problem evaluation: weak-sense and strong-sense critical thinking. To paraphrase, perhaps unfairly, weak-sense is marred by an overly narrow subproblem formulation. It’s atomistic. First you take a big problem, chop it up into smaller problems, and then solve each of the bite-sized pieces one at a time. Paul rightly notes that oftentimes this method misses the larger problem that arrise from the interplay of the otherwise well-behaved subproblems. The mathematician in me has to note that the local-behavior-does-not-imply-global-behavior phenomenon has been a central theme in differential geometry from about its beginning. The same problem creeps up just about everywhere else you look for it. I’ve tried to talk about this before in vague terms relating to urban planning and chaos theory. Maybe I should try again sometime. But for now:

Journal 3: Weak-sense, Strong-sense, and Probabilities

I agree with Paul. Strong-sense thinking is more appropriate for lots modern problems. International conflict, curricular design, and global warming all require strong-sense critical thinking, for example. (Ordering dinner at a restaurant typically does not.) While I like Paul’s network approach to problem solving, I think the primary weakness of weak-sense thinking lies in its absolutist view of truth, not necessarily its divide-and-conquer methodology. Truth, when viewed as a certainty, is rigid and fragile. Today’s demanding social and business landscape calls for something more adaptive, fluid, and functional. (Yes, you were supposed to read that last line with an announcer’s voice.) So how do I amend his framework? With probabilities of course. Really dedicated readers will see that I’ve mostly recycled my blog entry about assumptions. But to keep things fresh, I had to add something. And you knew it would happen eventually. I couldn’t resist.

I center my discussion around a theorem from linear algebra. Gleason’s Theorem tells you exactly what the probabilistic measures on the closed subspaces of a Hilbert space are (basically they’re projection operators). And according to some, it’s central to future research in information retrieval. I use it to show the usefulness of multiple points-of-view with some scientific flare. Of course, my treatment is clumsy—but technically I’m only allowed one page per entry. How thorough could I have been? Maybe later I’ll clean this up and expand it a little. For now, it’s probably okay.

References

Paul, Richard. “Teaching critical thinking in the ‘strong’ sense: A focus on self-deception, world views, and a dialectical mode of analysis.” Informal Logic Newsletter, 1982.

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Grassy Field

Since all I do these days is post my school projects to my blog, here’s another one for you. This week we had to create a collage. The requirements were pretty bare: at least five instances of the picture, one rotation, one rescaling, and at least one color modification. Try to spot each of the requirements in the final product below. (Maybe you’ve seen the original image before.) I had planned on using longer strips than the squares I ended up implementing, but I got lazy. The checkered effect is a little busy for my tastes; hopefully it’ll make the grade.

I tried for freakin’ ever to get the sky to soft clip to the hill top. I was able to adapt the intermediate image technique described in this article to create a tacky sun (not shown for art’s sake), but not for much more. Instead, I used the built-in, jagged setClip() method native to the java.awt.Graphics2D class. In case you were wondering, the clip was made with about six straight lines. I hate spline fitting, and try never to use curves—especially if line segments will do just fine. File that little tidbit away, it could be useful someday.

But convolutions rock. I’ve always thought so. Ever since I started using them to do signal processing in astronomy class. Our professor made us do a lot of convolutions using a visual calculus that really changed the way I thought about calculation in general. Drawing it out refined my sense of geometric interaction and avoided a lot of messy integrals. Here’s to qualitative methods: hurrah!

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Assumptions

A few weeks ago my friend Michelle called me a little after one in the afternoon. The ring of the telephone woke me up and I stumbled across the room to answer her call. I looked down at the little display, saw that it was Michelle, and then put on my best telephone face to accept her “Hello, Joshy.”

Michelle called to tell me that she had accepted a new job—she was in need of a new one, believe you me. This was fantastic news. Usually Michelle can spot my false wakefulness, even over the phone, like one of those empaths from Star Trek (or something). But this time, her excitement—in conjunction with my keen theatrical abilities—distracted her from the reality of my slumber, which is exactly what I was aiming for.

I wasn’t, but the question still remains: could I have justifiably been upset at Michelle for waking me up? Well, yes but mostly no. It was one in the afternoon. Most people are awake when the sun’s up. So, Michelle was right to operate on the assumption “people are awake when the sun’s up.” The world is a crazy and complicated place. We have to live our lives despite only having access to a very small amount of knowledge about our environment. Therefore most of our judgments aren’t certain. Instead, they’re best guesses that approximate what we should do if we actually knew everything there was to know. Thankfully, we’re not totally in the dark.

People are very good at working with probabilities because lots of the events in the world have a high probability of certainty. That tree in the park you saw this morning on your way to work will probably still be there during your commute home later tonight. The position and function of the knobs and buttons on your stove are not going to switch themselves around when you’re not looking—with high probability. So it’s not surprising that people believe that there are certainties in life. And maybe if you were able to know everything about everything at every time, then the world would work according to a small set of fixed laws. Unfortunately, no one—as far as I know—has that sort of depth of knowledge and understanding. So, for practical purposes, we’re left interacting with probabilities.

Now we get into trouble when we confuse probabilities for certainties. Then we become locked into a stereotype. That’s right, I think stereotypes are simply misapplied probabilities. Several years ago some fledging stand-up comedian trying to break it big played the Conan O’Brien Show. He included two “postive stereotypes” that stuck with me. “All Jews can fly and Mexicans are made out of candy,” he claimed. Being (sort of) both Jewish and Mexican, I can say from experience that very few Jews whom I know can fly and even fewer Mexicans are made out of candy. So what makes his stereotypes wrong? Well, probabilistically his claims aren’t well supported.

Here’s another perhaps less inflammatory claim: men are taller than women. I bet a lot of you agree with that. But let’s hold up just a second and see just what the sentence is saying. There are a lot of words missing that really ought to be there. My claim doesn’t mean all men are taller than all women. If you cite your friend from college on the women’s basketball team who towers over everyone else in a crowd, you haven’t disproved anything. What I really mean to say is that on average men are taller than women; i.e., if you pull a random man and a random woman off the street and compare their heights, record the answer, and then repeat the experiment several times over, then in general, you will find that the man is taller than the woman.

So what are assumptions: they are the most probable results from a distribution of possible results that we adopt as fact based on our experience. Experience varies, so assumptions vary. The key is to remember that sometimes outlying events can happen, and we must be open to the possibility that they do. Most Mexicans aren’t made out of candy, but don’t let me fool you into believing that none of us are.

All that said, Michelle should’ve known, given her previous experience, that there was a high likelihood that I would be sleeping at one in the afternoon. Don’t forget that not all assumptions apply in all contexts. These things are conditional, after all. So, she’s only partially excused.

And while I’m on my soapbox, it’s worth pointing out that because people almost exclusively interact with probability distributions, probability and statistics really need to be given more attention in school curricula. Over emphasis on deterministic systems tricks students into believing that the world really operates on certain events. I can’t think of anything further from the truth.

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