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pastwatcher) wrote2012-04-18 01:30 am
pastwatcher) wrote2012-04-18 01:30 am
7 things #1: The Maths, For Lay people
Landofnowhere and gleameil, you now have your challenges. :)
Tiamat360 asked me to write about "the maths, for lay people". I have decided to take massive poetic license on this, so I hope the following makes *some* sense, and is entertaining. But don't take it too seriously.
If you're truly a "lay person", that is you're not a scientist or a geek, then you are still doing math a lot of the time...
Problem-solving intuitions:
Math is when you wash a floor, but first you work out how to wash it so that you can go around different fixtures of the room but never have to step on a spot that you've already cleaned.
Math is when you are walking from somewhere in a city to somewhere else, and crossing a lot of traffic lights, and you combine zigzaging tactics with walking along major streets' sidewalks, so that you will be able to wait as little as possible for "walk" signs. Or when you are driving and work out a combination of major streets and back roads and avoiding traffic.
Math is when you grab a large bowl with two hands, or a cup with one. How did you know one hand would not be enough to go halfway 'round the bowl? How did you know two hands would be too big for a cup? You don't need to have seen that cup before, only estimate its size.
What I'm getting at is that the purest math for lay people is the problem-solving we do quickly when we aren't really thinking about it.
Abstraction:
Math is also magic tricks, of course; geeky things like making a chart of which members of your social group dated each other; the estimation of how much you can believe a claim in science; seeing a gaping hole in a logical argument instantaneously; being able to extrapolate patterns, take averages of numbers and so on, in split-seconds. Drawing art using known ratios, or in perspective with one viewpoint or two. Various of these skills have improved for me in leaps and bounds as a side effect of studying math.
But to study mathematics is to use a language to help you focus, to take all of these things your brain can just *do*, and bring them up to the surface. One has to talk about logic, and it's really hard and stuttering at first, and people struggle a lot with knowing how to say enough but not too much. I've seen people demonstrate the results of logic countless times, but their reasoning is often so fleeting that they can't remember how they got to their conclusion. Or they give a convoluted explanation that doesn't go in order. The challenge is to keep track of what your brain is doing, to actually *slow down* the process of doing math.
Imagine before you learned algebra. You have a silly question like "if I was twice your age last year, can I ever be twice your age again?" and maybe you just know it's not possible. Or a less silly question. But how can you communicate your thoughts to your friend, or convince yourself, if there was no mathematical model in your head for variables? What did you do to convince yourself? I know that my mind would've gone in circles, or I'd have written down the entire set of possible pairs of my age, and the other person's age. Tedious, for something I knew all along.
Once you can talk abstractly about the cores themselves, then the next time thinking is easier. Your intuitive brain isn't any slower (at least mine isn't), but at every moment that your thinking stops for breath, you know where you are, and you don't go in circles. This is what happens when math is going well. When I don't have a good mathematical model for a real-life issue, I sometimes get lost; when the math is too hard and doesn't all fit in my head yet, I get lost; when the model is too inaccurate, that is most dangerous of all, for I may not realize I've come to the wrong conclusion.
And of course, it's the pure joy of puzzle-solving.
Modeling:
Math is also when you look at a beautiful pattern of tiles on a wall, patches in a quilt, and see different shapes within it, fitting together perfectly. When you don't think about the physical pieces, but the perfect shapes they approximate.
Math is the perfect patterns that approximate the imperfect world. When you see a web page full of comments, or a lifetime of conversations, and your brain tracks the central ideas of "what most people are saying". When you see or hear the "same argument being had over and over again", and you can distinguish the pattern as a sort of game of player A and player B. That's a mathematical model you have in your head, or so I think of it.
Math is the rules of syntax you use to generate your sentences, based on words and connections between them: the sentence is a core of points and connections, supporting a fleshy mass of ideas. Linguists tell me this process is not even *close* to conscious, one can't make it conscious, one can only try to rebuild the same process through conscious mathematical models. Yet without the underlying rules and the fact that our brains group distinct things together to give meanings to words, I don't know how we could communicate at all.
When practicing math, any good mathematician knows that their mathematics is not an accurate description of reality; there's always a trade-off between the simplifications of math and the fact that reality is diverse and changeable. We're amazingly good at perceiving patterns, and we use that to simplify memories into knowledge about the world. (Warning: blunt discussion of potentially hurtful issues in next paragraph only.)
It is a measure of this tendency to simplify that we form categories of people, and usually classify them instantaneously and without volition: by gender, race and other things. That is so strange to me, because such processes are mostly unconscious, but when the evidence is unclear, our conscious brain becomes engaged in the problem: "classification error: please update by hand". Of course there are moral implications--our ongoing models associate stereotypes with these categories, and if the pattern is useful enough, we keep seeing the pattern even as reality departs from it. Even the classification of "good person, friend" may make one ignore a person's negative actions! That's confirmation bias for you, it is so much easier not to change a model that is working well enough. But sometimes the evidence challenges the model and makes it conscious again--"error: subject and stereotype mismatch. Please write exception, change subject status, or change stereotype." I criticize people, but especially scientists, when they lose sight of the trade-off between useful statistics and encouraging prejudice. Or more generally, when anyone pushes a mathematical model too much onto reality that may differ, if that causes them to hurt people. I'm not one to say "there is no real truth", but I will say "the truth comes with error bars that repeated assertions may make you forget", and "the patterns one learns early may nevertheless be wrong", and "sometimes it's bad when people enforce their perceived truths on others".
Or maybe I'm wrong about the above psychology. But it's a pattern I've seen, and others have described, pretty clearly, so it's a good model until somebody brings me back down to Earth on this point. :)
What mathematicians do
When a branch of pure mathematics starts out, it starts with a mathematical model for something in the real world. If you can find a mathematical core to a problem and solve that, it is much easier to fit the rest of the problem to it. But once that happens, mathematicians take such a model, and start doing art. We start building more things, that don't have reality around them!
There's a tension in purposes when doing mathematics, between simplifying, consolidating what we know, and exploring. I like mathematics best when it simplifies things, but it is endlessly appealing to make things more complicated. Mathematicians, whatever they may say, tend to keep working on problems because it's aesthetic or fun or they're in the flow of things, not because it's got applications; for me, I think about a problem 100 times as well if I *want* to, if it fascinates me.
But most of us can't wander too far afield for aesthetic reasons, otherwise we will have nobody to talk to, and to me, math is nothing without communication of it. And yet, famous mathematicians of the past have written elaborate ideas independently that have been rediscovered later...it's very strange, what to do? That urge to make things more complicated leads to a *lot* of new math, most of which is useless, until a useful application pops up where you least expect it. It would be maddening if it weren't so much fun and relatively cheap.
So many mathematicians want to take a known concept and "generalize" it to new situations. But in my field often feel like I would kill for some concrete examples of certain very well-developed abstract theories, and for someone to do a computation whose answer is not 0; yet, in the rare cases where computations exist due to heroic analysts, I often find them too long and painful to enjoy following.
Sometimes we want to reformulate a known concept in new terms, shifting the focus. Sometimes 10 types of problems are being talked about and somebody proves they're all the same, and that one particular technique is the simplest way to talk about them. That's what happened with category theory, as far as I know. Now, a lot of definitions in math are made from very abstract categories, which are like teleport stations in a forest: comforting places you can jump between, start exploring in the wilderness from there, and maybe keep a thread anchored to the teleport so you don't get too lost. But there are people like ultrawaffle who love the abstract category theory: in my analogy, they're building spaceships or something.
I just developed a good solid example myself that has so much helped me understand my current problem, it's not even funny. Yet there's no reason anybody should care about my one example, except that it's easier to understand than the more abstract way it comes up; rather, they should care if I get much more solid techniques out of this, techniques that could eventually matter in the physics problems my math is connected to. At my current stage in my problem, I've got a technique and am trying to find out if anyone's used it before, and I may give up and ask Mathoverflow. I've already tried to ask Dusa McDuff though she hasn't answered yet. I'm unable to think directly about this very well, so I've begun a complicated game of dancing around the issue, thinking about things that interest me that might suddenly lead to a connection. My advisor has completely condoned this, though of course his way of thinking isn't quite congruent with mine.
Math is strange. Thanks for asking. :)
Oh, and for the curious, I'm going to post a link to a beautiful classic article about K-12 math education, even though I disagree with it on many counts.
Tiamat360 asked me to write about "the maths, for lay people". I have decided to take massive poetic license on this, so I hope the following makes *some* sense, and is entertaining. But don't take it too seriously.
If you're truly a "lay person", that is you're not a scientist or a geek, then you are still doing math a lot of the time...
Problem-solving intuitions:
Math is when you wash a floor, but first you work out how to wash it so that you can go around different fixtures of the room but never have to step on a spot that you've already cleaned.
Math is when you are walking from somewhere in a city to somewhere else, and crossing a lot of traffic lights, and you combine zigzaging tactics with walking along major streets' sidewalks, so that you will be able to wait as little as possible for "walk" signs. Or when you are driving and work out a combination of major streets and back roads and avoiding traffic.
Math is when you grab a large bowl with two hands, or a cup with one. How did you know one hand would not be enough to go halfway 'round the bowl? How did you know two hands would be too big for a cup? You don't need to have seen that cup before, only estimate its size.
What I'm getting at is that the purest math for lay people is the problem-solving we do quickly when we aren't really thinking about it.
Abstraction:
Math is also magic tricks, of course; geeky things like making a chart of which members of your social group dated each other; the estimation of how much you can believe a claim in science; seeing a gaping hole in a logical argument instantaneously; being able to extrapolate patterns, take averages of numbers and so on, in split-seconds. Drawing art using known ratios, or in perspective with one viewpoint or two. Various of these skills have improved for me in leaps and bounds as a side effect of studying math.
But to study mathematics is to use a language to help you focus, to take all of these things your brain can just *do*, and bring them up to the surface. One has to talk about logic, and it's really hard and stuttering at first, and people struggle a lot with knowing how to say enough but not too much. I've seen people demonstrate the results of logic countless times, but their reasoning is often so fleeting that they can't remember how they got to their conclusion. Or they give a convoluted explanation that doesn't go in order. The challenge is to keep track of what your brain is doing, to actually *slow down* the process of doing math.
Imagine before you learned algebra. You have a silly question like "if I was twice your age last year, can I ever be twice your age again?" and maybe you just know it's not possible. Or a less silly question. But how can you communicate your thoughts to your friend, or convince yourself, if there was no mathematical model in your head for variables? What did you do to convince yourself? I know that my mind would've gone in circles, or I'd have written down the entire set of possible pairs of my age, and the other person's age. Tedious, for something I knew all along.
Once you can talk abstractly about the cores themselves, then the next time thinking is easier. Your intuitive brain isn't any slower (at least mine isn't), but at every moment that your thinking stops for breath, you know where you are, and you don't go in circles. This is what happens when math is going well. When I don't have a good mathematical model for a real-life issue, I sometimes get lost; when the math is too hard and doesn't all fit in my head yet, I get lost; when the model is too inaccurate, that is most dangerous of all, for I may not realize I've come to the wrong conclusion.
And of course, it's the pure joy of puzzle-solving.
Modeling:
Math is also when you look at a beautiful pattern of tiles on a wall, patches in a quilt, and see different shapes within it, fitting together perfectly. When you don't think about the physical pieces, but the perfect shapes they approximate.
Math is the perfect patterns that approximate the imperfect world. When you see a web page full of comments, or a lifetime of conversations, and your brain tracks the central ideas of "what most people are saying". When you see or hear the "same argument being had over and over again", and you can distinguish the pattern as a sort of game of player A and player B. That's a mathematical model you have in your head, or so I think of it.
Math is the rules of syntax you use to generate your sentences, based on words and connections between them: the sentence is a core of points and connections, supporting a fleshy mass of ideas. Linguists tell me this process is not even *close* to conscious, one can't make it conscious, one can only try to rebuild the same process through conscious mathematical models. Yet without the underlying rules and the fact that our brains group distinct things together to give meanings to words, I don't know how we could communicate at all.
When practicing math, any good mathematician knows that their mathematics is not an accurate description of reality; there's always a trade-off between the simplifications of math and the fact that reality is diverse and changeable. We're amazingly good at perceiving patterns, and we use that to simplify memories into knowledge about the world. (Warning: blunt discussion of potentially hurtful issues in next paragraph only.)
It is a measure of this tendency to simplify that we form categories of people, and usually classify them instantaneously and without volition: by gender, race and other things. That is so strange to me, because such processes are mostly unconscious, but when the evidence is unclear, our conscious brain becomes engaged in the problem: "classification error: please update by hand". Of course there are moral implications--our ongoing models associate stereotypes with these categories, and if the pattern is useful enough, we keep seeing the pattern even as reality departs from it. Even the classification of "good person, friend" may make one ignore a person's negative actions! That's confirmation bias for you, it is so much easier not to change a model that is working well enough. But sometimes the evidence challenges the model and makes it conscious again--"error: subject and stereotype mismatch. Please write exception, change subject status, or change stereotype." I criticize people, but especially scientists, when they lose sight of the trade-off between useful statistics and encouraging prejudice. Or more generally, when anyone pushes a mathematical model too much onto reality that may differ, if that causes them to hurt people. I'm not one to say "there is no real truth", but I will say "the truth comes with error bars that repeated assertions may make you forget", and "the patterns one learns early may nevertheless be wrong", and "sometimes it's bad when people enforce their perceived truths on others".
Or maybe I'm wrong about the above psychology. But it's a pattern I've seen, and others have described, pretty clearly, so it's a good model until somebody brings me back down to Earth on this point. :)
What mathematicians do
When a branch of pure mathematics starts out, it starts with a mathematical model for something in the real world. If you can find a mathematical core to a problem and solve that, it is much easier to fit the rest of the problem to it. But once that happens, mathematicians take such a model, and start doing art. We start building more things, that don't have reality around them!
There's a tension in purposes when doing mathematics, between simplifying, consolidating what we know, and exploring. I like mathematics best when it simplifies things, but it is endlessly appealing to make things more complicated. Mathematicians, whatever they may say, tend to keep working on problems because it's aesthetic or fun or they're in the flow of things, not because it's got applications; for me, I think about a problem 100 times as well if I *want* to, if it fascinates me.
But most of us can't wander too far afield for aesthetic reasons, otherwise we will have nobody to talk to, and to me, math is nothing without communication of it. And yet, famous mathematicians of the past have written elaborate ideas independently that have been rediscovered later...it's very strange, what to do? That urge to make things more complicated leads to a *lot* of new math, most of which is useless, until a useful application pops up where you least expect it. It would be maddening if it weren't so much fun and relatively cheap.
So many mathematicians want to take a known concept and "generalize" it to new situations. But in my field often feel like I would kill for some concrete examples of certain very well-developed abstract theories, and for someone to do a computation whose answer is not 0; yet, in the rare cases where computations exist due to heroic analysts, I often find them too long and painful to enjoy following.
Sometimes we want to reformulate a known concept in new terms, shifting the focus. Sometimes 10 types of problems are being talked about and somebody proves they're all the same, and that one particular technique is the simplest way to talk about them. That's what happened with category theory, as far as I know. Now, a lot of definitions in math are made from very abstract categories, which are like teleport stations in a forest: comforting places you can jump between, start exploring in the wilderness from there, and maybe keep a thread anchored to the teleport so you don't get too lost. But there are people like ultrawaffle who love the abstract category theory: in my analogy, they're building spaceships or something.
I just developed a good solid example myself that has so much helped me understand my current problem, it's not even funny. Yet there's no reason anybody should care about my one example, except that it's easier to understand than the more abstract way it comes up; rather, they should care if I get much more solid techniques out of this, techniques that could eventually matter in the physics problems my math is connected to. At my current stage in my problem, I've got a technique and am trying to find out if anyone's used it before, and I may give up and ask Mathoverflow. I've already tried to ask Dusa McDuff though she hasn't answered yet. I'm unable to think directly about this very well, so I've begun a complicated game of dancing around the issue, thinking about things that interest me that might suddenly lead to a connection. My advisor has completely condoned this, though of course his way of thinking isn't quite congruent with mine.
Math is strange. Thanks for asking. :)
Oh, and for the curious, I'm going to post a link to a beautiful classic article about K-12 math education, even though I disagree with it on many counts.